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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 19 /acute put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 53 /five put dup 55 /seven put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put dup 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX6 %!PS-AdobeFont-1.1: CMBX6 1.0 %%CreationDate: 1991 Aug 20 16:35:30 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 122 /z put readonly def /FontBBox{-49 -250 1367 753}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put readonly def /FontBBox{0 -504 2615 1004}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 112 /p put readonly def /FontBBox{0 -250 1171 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 10 /circlemultiply put dup 14 /openbullet put dup 15 /bullet put dup 18 /reflexsubset put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 67 /C put dup 77 /M put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /prime put readonly def /FontBBox{-15 -951 1252 782}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 39 /phi1 put dup 120 /x put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put dup 74 /triangleleftsld put readonly def /FontBBox{8 -463 1331 1003}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 106 /bar put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /Lambda put dup 10 /Omega put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 91 /bracketleft put dup 93 /bracketright put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 67 /C put dup 77 /M put dup 92 /intersection put dup 106 /bar put dup 107 /bardbl put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 15 /epsilon1 put dup 21 /lambda put dup 28 /tau put dup 33 /omega put dup 59 /comma put dup 63 /star put dup 77 /M put dup 82 /R put dup 84 /T put dup 88 /X put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 110 /n put dup 114 /r put dup 116 /t put dup 118 /v put dup 120 /x put dup 121 /y put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 70 /F put dup 73 /I put dup 76 /L put dup 77 /M put dup 80 /P put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-36 -251 1103 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 40 /braceleftBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /Lambda put dup 4 /Xi put dup 10 /Omega put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 61 /equal put dup 77 /M put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 116 /t put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 36 /pi1 put dup 37 /rho1 put dup 39 /phi1 put dup 59 /comma put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 67 /C put dup 75 /K put dup 77 /M put dup 82 /R put dup 84 /T put dup 85 /U put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 93 /sharp put dup 96 /lscript put dup 97 /a put dup 99 /c put dup 100 /d put dup 102 /f put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 115 /s put dup 116 /t put dup 118 /v put dup 120 /x put dup 121 /y put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 4391C9DF440285B8FC159D0E98D4258FC57892DDF753642CD526A96ACEDA4120 788F22B1D09F149794E66DD1AC2C2B3BC6FEC59D626F427CD5AE9C54C7F78F62 C36F49B3C2E5E62AFB56DCEE87445A12A942C14AE618D1FE1B11A9CF9FAA1F32 617B598CE5058715EF3051E228F72F651040AD99A741F247C68007E68C84E9D1 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 36 /pi1 put dup 37 /rho1 put dup 39 /phi1 put dup 42 /arrowrighttophalf put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 84 /T put dup 85 /U put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 2 /Theta put dup 4 /Xi put dup 7 /Upsilon put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 18 /grave put dup 22 /macron put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 79 /O put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put dup 122 /z put readonly def /FontBBox{14 -250 1077 750}readonly def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 18 /grave put dup 19 /acute put dup 38 /ampersand put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def currentdict end currentfile eexec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5434 y Fv(dW)2736 5449 y Fs(t)2782 5434 y Fv(;)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 168 251 a Fq(4)1008 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y Fw(where)25 b FA(v)h Fw(is)f(a)g(b)s(ounded)g(progressiv)m(ely)i (measurable)f(con)m(trol,)g(lying)f(in)g(a)f(closed)i(con)m(v)m(ex)g (set)f Fv(U)39 b Fu(\022)168 566 y Ft(R)240 530 y Fs(m)306 566 y Fw(,)33 b(whic)m(h,)h(without)f(loss)h(of)e(generalit)m(y)-8 b(,)33 b(will)h(b)s(e)e(assumed)j(to)d(con)m(tain)h(the)g(origin)f(0,) 1416 737 y Fv(f)38 b Fw(:)28 b Ft(R)1629 696 y Fs(n)1698 737 y Fu(\002)23 b Ft(R)1870 696 y Fs(n)1939 737 y Fu(\002)f Fv(U)33 b Fu(\002)23 b Fw(\012)28 b Fu(!)f Ft(R)2534 696 y Fs(n)168 908 y Fw(is)33 b(the)g(drift)f(co)s(e\016cien)m(t,)1108 1078 y Fv(\033)g Fw(:)c Ft(R)1322 1037 y Fs(n)1391 1078 y Fu(\002)22 b Ft(R)1562 1037 y Fs(n)1631 1078 y Fu(\002)h Ft(R)1803 1037 y Fr(+)1884 1078 y Fu(\002)g Fv(U)33 b Fu(\002)22 b Fw(\012)28 b Fu(!)g Fw(Mat\()p Fv(n)22 b Fu(\002)h Fv(p)p Fw(\))168 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Fv(x;)17 b(X)r(;)g(v)t(;)g(!)t Fw(\))27 b(:)h Ft(R)1873 1962 y Fs(n)1942 2003 y Fu(\002)23 b Ft(R)2114 1962 y Fs(n)2183 2003 y Fu(\002)f Fv(U)33 b Fu(\002)23 b Fw(\012)28 b Fu(!)f Ft(R)p Fv(;)168 2174 y Fw(where)46 b Fv(U)61 b Fu(\022)49 b Ft(R)787 2138 y Fs(m)899 2174 y Fw(is)d(the)g(con)m(trol)g(space,)k(whic)m(h)d(is)e (tak)m(en)i(to)e(b)s(e)g(closed)i(and)f(con)m(v)m(ex.)84 b(W)-8 b(e)168 2290 y(supp)s(ose)35 b(also)g(that)f Fv(L)g Fw(is)h(con)m(tin)m(uous)h(in)e(\()p Fv(x;)17 b(X)r(;)g(v)t Fw(\))34 b(and)h(b)s(ounded)g(from)f(b)s(elo)m(w.)49 b(Without)34 b(loss)168 2406 y(of)46 b(generalit)m(y)-8 b(,)51 b(w)m(e)d(will)f(assume)h(that)e(inf)24 b Fv(L)52 b Fw(=)f(0.)85 b(F)-8 b(urthermore,)51 b(if)46 b Fv(U)57 b Fw(is)47 b(un)m(b)s(ounded)h(w)m(e)168 2523 y(supp)s(ose)34 b(that)168 2747 y(\(4\))1048 b(lim)1284 2809 y Fs(R)p Fl(!)p Fr(+)p Fl(1)1607 2747 y Fw(inf)1560 2801 y Fk(\()p Fm(x;X)q(;!)r Fk(\))1584 2859 y Fj(j)p Fm(v)r Fj(j\025)p Fm(R)1809 2679 y Fv(L)p Fw(\()p Fv(x;)17 b(X)r(;)g(v)t(;)g(!)t Fw(\))p 1809 2724 528 4 v 2019 2815 a Fu(j)p Fv(v)t Fu(j)2390 2747 y Fw(=)44 b(+)p Fu(1)17 b Fv(:)168 3014 y Fw(F)-8 b(or)41 b(simplicit)m(y)-8 b(,)47 b(w)m(e)d(assume)f(that)f(the)h (terminal)g(cost)g Fv( )j Fw(is)d(a)f(b)s(ounded)h Fv(C)3131 2978 y Fl(1)3248 3014 y Fw(function)f(with)168 3130 y(b)s(ounded)33 b(deriv)-5 b(ativ)m(es)35 b(\(less)e(regular)g(data)f(ma)m(y)i(b)s(e)e (treated)h(as)g(w)m(ell\).)168 3246 y(W)-8 b(e)32 b(supp)s(ose)i(that,) e(for)f Fv(T)42 b(>)27 b Fw(0,)32 b(equation)h(\(1\))f(p)s(ossesses,)j (for)c(an)m(y)i(\014xed)g Fv(!)e Fu(2)d Fw(\012,)33 b(one)f(and)h(only) 168 3362 y(one)39 b(viscosit)m(y)j(solution)e Fv(u)1186 3326 y Fs(\017)1257 3362 y Fw(de\014ned)h(for)e(all)g Fv(t)g Fu(2)h Fw([0)p Fv(;)17 b(T)d Fw(],)40 b(whic)m(h)h(is)f(the)f(v) -5 b(alue)40 b(function)g(of)f(the)168 3479 y(sto)s(c)m(hastic)34 b(con)m(trol)e(problem,)i(that)e(is,)h(it)g(can)g(b)s(e)g(represen)m (ted)i(in)e(the)g(form)842 3635 y Fv(u)898 3594 y Fs(\017)930 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b(let)g(\012)c(=)f Ft(T)1382 3924 y Fs(n)1430 3960 y Fw(,)35 b(and)f(consider)i(in)e(\012)h(the)g(\(normalized\))g (Leb)s(esgue)h(measure.)168 4089 y(Then,)e(if)550 4064 y(^)539 4089 y Fv(L)p Fw(\()p Fv(x;)17 b(X)r(;)g(v)t Fw(\),)1039 4063 y(^)1018 4089 y Fv(f)10 b Fw(\()p Fv(x;)17 b(X)r(;)g(v)t Fw(\))33 b(and)38 b(^)-55 b Fv(\033)5 b Fw(\()p Fv(x;)17 b(X)r(;)g(v)t Fw(\))32 b(are)g(p)s(erio)s(dic)h(in)g Fv(X)8 b Fw(,)33 b(w)m(e)g(can)g(write)1351 4259 y Fv(L)p Fw(\()p Fv(x;)17 b(X)r(;)g(v)t(;)g(!)t Fw(\))27 b(=)2021 4234 y(^)2010 4259 y Fv(L)p Fw(\()p Fv(x;)17 b(X)30 b Fw(+)22 b Fv(!)t(;)17 b(v)t Fw(\))p Fv(;)168 4418 y Fw(with)33 b(similar)g(de\014nitions)h(for)e Fv(f)44 b Fw(and)32 b Fv(\033)t Fw(,)h(and)g(then)g(w)m(e)g(c)m(ho)s(ose)h Fv(\034)2638 4433 y Fs(Y)2699 4418 y Fv(!)d Fw(=)c Fv(!)f Fw(+)c Fv(Y)f Fw(.)168 4534 y(Next,)38 b(w)m(e)g(state)e(the)h(concept) h(of)e(tigh)m(t)g(sequence)k(of)35 b(measures,)40 b(b)m(y)d (generalizing)h(a)e(standard)168 4651 y(de\014nition)d(in)g(probabilit) m(y)h(theory)f(\(see,)h(e.g.,)f(page)f(336)g(of)g([Bil95]\):)168 4826 y FA(De\014nition)38 b(1.)j Fo(A)35 b(net)g(of)f(pr)-5 b(ob)g(ability)35 b(me)-5 b(asur)g(es)34 b Fv(\026)2153 4841 y Fs(\013)2237 4826 y Fo(on)g(a)h(given)e(sp)-5 b(ac)g(e)34 b Fv(Z)29 b Fu(\002)22 b Fw(\012)p Fo(,)35 b(with)g Fv(Z)g Fu(\032)28 b Ft(R)3789 4790 y Fs(d)168 4942 y Fo(is)34 b(said)h(to)g(b)-5 b(e)34 b(tight)h(if)g(for)g(any)f Fv(\017)29 b(>)e Fw(0)35 b Fo(ther)-5 b(e)35 b(exists)f(a)h(c)-5 b(omp)g(act)34 b(set)h Fv(K)2834 4957 y Fs(\017)2901 4942 y Fo(in)g(such)g(a)f(way)h(that)1327 5127 y Fw(lim)17 b(sup)1453 5207 y Fs(\013)1642 5127 y Fv(\026)1701 5142 y Fs(\013)1750 5017 y Fp(\020)1810 5127 y Fw(\()p Fv(Z)29 b Fu(n)22 b Fv(K)2099 5142 y Fs(\017)2131 5127 y Fw(\))h Fu(\002)f Fw(\012)2361 5017 y Fp(\021)2465 5127 y Fu(\024)45 b Fv(\017)17 b(:)168 5353 y Fw(The)34 b(follo)m(wing)g(assumption)g(is) g(simply)h(a)e(compactness)i(prop)s(ert)m(y)f(on)g(measures)h(on)e (\012)g(that)h(is)168 5469 y(necessary)h(to)d(extract)h(w)m(eak)h (limits.)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)612 b(7)0 450 y FA(Assumption)45 b(8.)f Fw(W)-8 b(e)38 b(assume)i(that)e (for)f(eac)m(h)i(tigh)m(t)f(net)h(of)e(probabilit)m(y)i(measures)h Fv(\021)3378 465 y Fs(\013)3428 450 y Fw(\()p Fv(z)t(;)17 b(!)t Fw(\))0 566 y(on)36 b Fv(Z)31 b Fu(\002)25 b Fw(\012,)38 b(with)e Fv(Z)41 b Fu(\032)34 b Ft(R)990 530 y Fs(d)1030 566 y Fw(,)j(there)g(exists)h(a)e(subnet)h(whic)m(h)h(con)m(v)m(erges)g (to)e(a)f(probabilit)m(y)i(mea-)0 683 y(sure)25 b Fv(\021)t Fw(\()p Fv(z)t(;)17 b(!)t Fw(\),)24 b(in)g(the)g(sense)i(that:)39 b(if)23 b Fv( )t Fw(\()p Fv(X)r(;)17 b(z)t(;)g(!)t Fw(\))28 b(:)f Ft(R)1948 646 y Fs(d)1993 683 y Fu(\002)t Fv(Z)11 b Fu(\002)t Fw(\012)29 b Fu(!)e Ft(R)c Fw(is)i(con)m(tin)m(uous)g(in)f (\()p Fv(X)r(;)17 b(z)t Fw(\))24 b(and)0 799 y(stationary)g(\(that)f (is,)g Fv( )t Fw(\()p 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FA(Assumption)k(9.)k Fw(W)-8 b(e)33 b(will)g(assume)h(that)f(the)g(viscosit)m(y)i(co)s(e\016cien)m(t)f Fv(\033)i Fw(is)e(either)173 1829 y(A.)42 b Fv(\033)31 b Fw(=)d Fv(\017\033)599 1844 y Fr(0)177 1945 y Fw(B.)42 b Fv(\033)31 b Fw(=)d Fv(\017)544 1909 y Fr(1)p Fs(=)p Fr(2)654 1945 y Fv(\033)709 1960 y Fr(0)175 2061 y Fw(C.)43 b Fv(\033)31 b Fw(=)d Fv(\033)560 2076 y Fr(0)600 2061 y Fw(,)0 2202 y(with)33 b Fv(\033)277 2217 y Fr(0)345 2202 y Fw(=)27 b Fv(\033)503 2217 y Fr(0)543 2202 y Fw(\()p Fv(x)636 2165 y Fl(0)660 2202 y Fv(;)17 b(X)r(;)g(v)t(;)g(!)t Fw(\))26 b(:)i Ft(R)1182 2165 y Fs(n)1251 2202 y Fu(\002)22 b Ft(R)1422 2165 y Fs(n)1492 2202 y Fu(\002)g Fv(U)33 b Fu(\002)23 b Fw(\012)28 b Fu(!)f Fw(Mat\()p Fv(n)22 b Fu(\002)h Fv(m)p Fw(\).)1025 b Fi(J)0 2380 y Fw(W)-8 b(e)27 b(should)h(remark)g(that)f(these)h(are)f(the)h(relev)-5 b(an)m(t)28 b(scalings,)h(as)e(man)m(y)h(others)g(could)g(b)s(e)f (studied)0 2497 y(with)33 b(similar)g(metho)s(ds.)0 2613 y(From)41 b(section)h(9)f(on,)i(w)m(e)f(will)g(assume)h(that)e(our)g (viscosit)m(y)j(solutions)e(are)f(smo)s(oth,)i(up)f(to)f(a)0 2729 y(small)33 b(uniformly)h(elliptic)f(correction)h(of)e(the)h (Hamiltonian.)44 b(That)32 b(is,)i(w)m(e)f(supp)s(ose:)0 2908 y FA(Assumption)k(10.)j Fw(F)-8 b(or)30 b(an)m(y)i Fv(`)c Fu(2)g Ft(N)p Fw(,)j(there)h(exist)h Fv(u)1986 2872 y Fs(\017)1986 2934 y(`)2046 2908 y Fu(2)28 b Fv(C)2217 2872 y Fr(2)2256 2908 y Fw(\()p Ft(R)2366 2872 y Fs(n)2432 2908 y Fu(\002)20 b Fw(\(0)p Fv(;)d(T)d Fw(\)\),)30 b Fv( )2927 2923 y Fs(`)2988 2908 y Fu(2)e Fv(C)3159 2872 y Fr(2)3199 2908 y Fw(\()p Ft(R)3309 2872 y Fs(n)3355 2908 y Fw(\))j(and)h(a)0 3024 y(Hamiltonian)h Fv(H)642 3039 y Fs(`)707 3024 y Fw(suc)m(h)h(that)f Fv(u)1195 2988 y Fs(\017)1195 3050 y(`)1260 3024 y Fw(is)g(a)f(solution)h(of)f (the)h(follo)m(wing)g(Hamilton-Jacobi)g(equation:)899 3107 y Fp(8)899 3197 y(<)899 3376 y(:)1054 3234 y Fu(\000)23 b Fv(@)1205 3249 y Fs(t)1235 3234 y Fv(u)1291 3193 y Fs(\017)1291 3258 y(`)1345 3234 y Fw(+)f Fv(H)1524 3249 y Fs(`)1574 3123 y Fp(\020)1633 3234 y Fv(D)1717 3193 y Fr(2)1714 3258 y Fs(x)1758 3234 y Fv(u)1814 3193 y Fs(\017)1814 3258 y(`)1847 3234 y Fv(;)17 b(D)1972 3249 y Fs(x)2015 3234 y Fv(u)2071 3193 y Fs(\017)2071 3258 y(`)2104 3234 y Fv(;)g(x;)2256 3166 y(x)p 2256 3211 56 4 v 2264 3302 a(\017)2322 3234 y(;)g(\017;)g(!)2514 3123 y Fp(\021)2600 3234 y Fw(=)28 b(0)1032 3414 y Fv(u)1088 3373 y Fs(\017)1088 3439 y(`)1120 3414 y Fw(\()p Fv(x;)17 b(T)8 b(;)17 b(!)t Fw(\))44 b(=)g Fv( )1696 3429 y Fs(`)1729 3414 y Fw(\()p Fv(x)p Fw(\))17 b Fv(;)0 3592 y Fw(and)33 b(in)f(suc)m(h)j(a)d(w)m(a)m(y)i(that)1081 3756 y(lim)1036 3819 y Fs(`)p Fl(!)p Fr(+)p Fl(1)1277 3756 y Fv(u)1333 3715 y Fs(\017)1333 3781 y(`)1366 3756 y Fw(\()p Fv(x;)17 b(t)p Fw(\))28 b(=)f Fv(u)1763 3715 y Fs(\017)1796 3756 y Fw(\()p Fv(x;)17 b(t)p Fw(\))1081 3948 y(lim)1036 4011 y Fs(`)p Fl(!)p Fr(+)p Fl(1)1277 3948 y Fv( )1340 3963 y Fs(`)1374 3948 y Fw(\()p Fv(x)p Fw(\))28 b(=)f Fv( )t Fw(\()p Fv(x)p Fw(\))17 b Fv(;)653 4180 y Fw(and)271 b(lim)1036 4242 y Fs(`)p Fl(!)p Fr(+)p Fl(1)1277 4180 y Fw(sup)1331 4259 y Fs(x)1441 4065 y Fp(\014)1441 4125 y(\014)1441 4185 y(\014)1474 4180 y Fv(H)1555 4195 y Fs(`)1588 4069 y Fp(\020)1647 4180 y Fv(D)1731 4139 y Fr(2)1728 4204 y Fs(x)1772 4180 y Fw(\011\()p Fv(x)p Fw(\))p Fv(;)17 b(D)2104 4195 y Fs(x)2148 4180 y Fw(\011\()p Fv(x)p Fw(\))p Fv(;)g(x;)2508 4112 y(x)p 2508 4157 V 2516 4248 a(\017)2573 4180 y(;)g(\017;)g(!)2765 4069 y Fp(\021)1426 4421 y Fu(\000)p Fv(H)1592 4310 y Fp(\020)1652 4421 y Fv(D)1736 4380 y Fr(2)1733 4445 y Fs(x)1776 4421 y Fw(\011\()p Fv(x)p Fw(\))p Fv(;)g(D)2108 4436 y Fs(x)2152 4421 y Fw(\011\()p Fv(x)p Fw(\))p Fv(;)g(x;)2512 4353 y(x)p 2512 4398 V 2520 4489 a(\017)2578 4421 y(;)g(\017;)g(!)2770 4310 y Fp(\021)2828 4306 y(\014)2828 4366 y(\014)2828 4426 y(\014)2889 4421 y Fw(=)28 b(0)0 4634 y(for)k(an)m(y)h Fv(x)28 b Fu(2)g Ft(R)582 4598 y Fs(n)629 4634 y Fw(,)33 b Fv(t)28 b Fu(2)g Fw([0)p Fv(;)17 b(T)d Fw(])32 b(and)h(\011)27 b Fu(2)h Fv(C)1560 4598 y Fr(2)1600 4634 y Fw(\()p Ft(R)1710 4598 y Fs(n)1756 4634 y Fw(\))33 b(with)g(b)s(ounded)g Fv(C)2524 4598 y Fr(2)2564 4634 y Fw(-norm.)43 b(Also,)p 3136 4554 89 4 v 33 w Fv(H)3225 4649 y Fs(`)3290 4634 y Fw(satis\014es)0 4800 y(\(14\))1138 b Fu(\000)23 b Fv(D)1493 4815 y Fs(M)1561 4825 y Fm(ij)1620 4800 y Fv(H)1701 4815 y Fs(`)1761 4800 y Fv(\030)1804 4815 y Fs(i)1849 4800 y Fv(\030)1892 4815 y Fs(j)1972 4800 y Fu(\025)45 b Fv(c)2136 4815 y Fs(`)2185 4800 y Fu(j)p Fv(\030)5 b Fu(j)2289 4759 y Fr(2)2344 4800 y Fv(;)0 4964 y Fw(for)32 b(some)h Fv(c)435 4979 y Fs(`)496 4964 y Fv(>)28 b Fw(0.)0 5081 y(W)-8 b(e)33 b(also)g(assume)h(that)0 5245 y(\(15\))1278 b(sup)1312 5331 y Fs(x)p Fl(2)p Fn(R)1447 5312 y Fm(n)1489 5331 y Fs(;)12 b(t)p Fl(2)p Fr([0)p Fs(;T)e Fr(])1755 5245 y Fu(j)p Fv(D)s(u)1923 5204 y Fs(\017)1923 5270 y(`)1955 5245 y Fu(j)44 b Fv(<)g Fw(+)p Fu(1)17 b Fv(:)3584 5469 y Fi(J)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 168 251 a Fq(8)1008 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y Fw(A)48 b(natural)g(p)s(ossibilit)m(y)i(for)e(suc)m(h)i(Hamiltonian)e Fv(H)2184 465 y Fs(`)2265 450 y Fw(is)h(of)f(course)h Fv(H)40 b Fu(\000)34 b Fw(tr)16 b Fv(M)5 b(=`)p Fw(,)52 b(but)d(other)168 566 y(regularizations)30 b(are)h(p)s(ossible)g(\(for) e(general)i(regularization)f(results,)i(see,)g(e.g.,)e([Ev)-5 b(a82,)30 b(Ev)-5 b(a97,)168 683 y(W)d(an90)o(,)33 b(W)-8 b(an92a)o(,)33 b(W)-8 b(an92b,)32 b(W)-8 b(an92c,)33 b(CC95,)g(LS05]\).)168 799 y(W)-8 b(e)27 b(remark)h(that)e(\(15\))h(is) g(not)g(assumed,)j(a-priori,)d(to)g(b)s(e)g(uniform)g(in)h Fv(\017)f Fw(and)g Fv(`)g Fw(\(in)g(fact,)h(uniform)168 915 y(estimates)41 b(will)e(b)s(e)h(pro)m(v)m(ed)g(in)g(Prop)s(osition) f(20)g(b)s(elo)m(w\).)64 b(In)39 b(man)m(y)h(applications,)i(b)s(ounds) e(as)168 1031 y(in)j(\(15\))f(are)h(quite)g(natural)g(and)g(ma)m(y)g(b) s(e)g(obtained)h(b)m(y)f(comparing)g(tra)5 b(jectories)44 b(\(see,)j(e.g.,)168 1147 y(page)32 b(555)g(of)g([Ev)-5 b(a98])33 b(and)f(Lemma)i(8.1)e(of)g([FS06)o(]\).)168 1264 y(The)26 b(next)g(three)g(assumptions)h(will)g(b)s(e)e(used)i(in)e (section)i(9)e(to)g(pro)m(v)m(e)h(v)-5 b(arious)26 b(uniform)f (estimates)168 1380 y(in)33 b Fv(\017)p Fw(.)168 1552 y FA(Assumption)38 b(11.)k Fw(W)-8 b(e)33 b(assume)h(that)e Fv(H)40 b Fw(enjo)m(ys)34 b(one)f(of)f(the)h(follo)m(wing)g(prop)s (erties:)288 1688 y(P1\))41 b Fv(H)8 b Fw(\()p Fv(M)d(;)17 b(p;)g(x;)g(X)r(;)g(\017;)g(!)t Fw(\))24 b(do)s(es)i(not)f(dep)s(end)h (on)f Fv(M)36 b Fw(and)25 b(satis\014es)i(the)f(co)s(ercivit)m(y)h(h)m (yp)s(othesis)168 1845 y(\(16\))1106 b(lim)1408 1912 y Fl(j)p Fs(p)p Fl(j!1)1641 1845 y Fv(H)8 b Fw(\()p Fv(p;)17 b(x;)g(X)r(;)g(\017;)g(!)t Fw(\))26 b(=)i(+)p Fu(1)p Fv(;)482 2064 y Fw(uniformly)34 b(in)e(\()p Fv(x;)17 b(X)r(;)g(\017;)g(!)t Fw(\))28 b Fu(2)g Ft(R)1684 2027 y Fs(n)1753 2064 y Fu(\002)22 b Ft(R)1924 2027 y Fs(n)1993 2064 y Fu(\002)h Ft(R)2165 2027 y Fr(+)2246 2064 y Fu(\002)g Fw(\012.)288 2180 y(P2\))41 b(There)h(exists)h(a)e(function)g Fv(F)14 b Fw(\()p Fv(p;)j(\020)8 b Fw(\))40 b(suc)m(h)i(that)f(for)f (all)h(\()p Fv(M)5 b(;)17 b(p;)g(x;)g(X)r(;)g(\017;)g(!)t Fw(\))41 b Fu(2)i Fw(Mat\()p Fv(n)28 b Fu(\002)482 2296 y Fv(n)p Fw(\))23 b Fu(\002)f Ft(R)772 2260 y Fs(n)841 2296 y Fu(\002)h Ft(R)1013 2260 y Fs(n)1082 2296 y Fu(\002)g Ft(R)1254 2260 y Fs(n)1323 2296 y Fu(\002)f Ft(R)1494 2260 y Fr(+)1575 2296 y Fu(\002)h Fw(\012)33 b(and)g Fv(\020)h Fu(2)29 b Ft(R)j Fw(suc)m(h)i(that)1436 2453 y Fv(\020)c Fw(+)22 b Fv(H)8 b Fw(\()p Fv(M)d(;)17 b(p;)g(x;)g(X)r(;)g (\017;)g(!)t Fw(\))26 b(=)i(0)482 2610 y(implies)661 2810 y Fu(\000)748 2742 y Fw(1)p 748 2787 49 4 v 753 2878 a Fv(\017)807 2810 y(D)888 2825 y Fs(M)956 2837 y Fm(j)s(k)1027 2810 y Fv(H)1116 2699 y Fp(\020)1185 2742 y Fw(1)p 1185 2787 V 1190 2878 a Fv(\017)1243 2810 y(M)5 b(;)17 b(p;)g(x;)g(X)r(;)g(\017;)g(!)1853 2699 y Fp(\021)1912 2810 y Fv(M)2006 2825 y Fs(ij)2067 2810 y Fv(M)2161 2825 y Fs(ik)2250 2810 y Fw(+)22 b Fv(\017p)2436 2825 y Fs(i)2465 2810 y Fv(D)2546 2825 y Fs(x)2586 2835 y Fm(i)2616 2810 y Fv(H)2705 2699 y Fp(\020)2774 2742 y Fw(1)p 2774 2787 V 2779 2878 a Fv(\017)2833 2810 y(M)5 b(;)17 b(p;)g(x;)g(X)r(;)g(\017;)g(!)3443 2699 y Fp(\021)661 3047 y Fw(+)p Fv(p)786 3062 y Fs(i)814 3047 y Fv(D)895 3062 y Fs(X)953 3072 y Fm(i)984 3047 y Fv(H)1073 2936 y Fp(\020)1142 2980 y Fw(1)p 1142 3024 V 1147 3115 a Fv(\017)1200 3047 y(M)5 b(;)17 b(p;)g(x;)g(X)r(;)g(\017;)g(!)1810 2936 y Fp(\021)1914 3047 y Fu(\025)44 b Fv(F)14 b Fw(\()p Fv(p;)j(\020)8 b Fw(\))p Fv(;)482 3234 y Fw(and,)33 b(furthermore,)g (if)g Fv(K)39 b Fw(is)34 b(a)e(\014xed)i(compact)f(set)g(of)f Ft(R)2618 3197 y Fs(n)2665 3234 y Fw(,)168 3391 y(\(17\))1271 b(inf)1599 3453 y Fs(\020)5 b Fl(2)p Fs(K)1762 3391 y Fv(F)14 b Fw(\()p Fv(p;)j(\020)8 b Fw(\))26 b Fu(!)h Fw(+)p Fu(1)p Fv(;)482 3600 y Fw(as)33 b Fu(j)p Fv(p)p Fu(j)27 b(!)g Fw(+)p Fu(1)p Fw(.)3752 3736 y Fi(J)168 3908 y Fw(Examples)e(of)d(Hamiltonians)h(satisfying)h(P2)f(are)g (easily)h(obtained,)h(for)e(instance,)j(b)m(y)e(considering)168 4024 y(the)33 b(case)g Fv(\033)f Fw(:=)c(Id)849 4039 y Fs(n)896 4024 y Fw(,)33 b Fv(f)38 b Fw(:=)28 b Fv(v)t Fw(,)k Fv(L)c Fw(:=)g Fu(j)p Fv(v)t Fu(j)1615 3988 y Fr(2)1653 4024 y Fv(=)p Fw(2)22 b Fu(\000)g Fv(U)10 b Fw(\()p Fv(x;)17 b(X)r(;)g(!)t Fw(\).)44 b(In)33 b(suc)m(h)h(a)e(case,) i(\(6\))e(yields)i(that)168 4230 y(\(18\))897 b Fv(H)35 b Fw(=)28 b Fu(\000)1546 4163 y Fw(1)p 1546 4207 V 1546 4298 a(2)1621 4230 y(tr)17 b Fv(M)33 b Fw(+)1949 4163 y(1)p 1949 4207 V 1949 4298 a(2)2007 4230 y Fu(j)p Fv(p)p Fu(j)2112 4189 y Fr(2)2173 4230 y Fw(+)22 b Fv(U)10 b Fw(\()p Fv(x;)17 b(X)r(;)g(!)t Fw(\))g Fv(;)168 4416 y Fw(whic)m(h)34 b(is)f(plainly)g(seen)h(to)e(satisfy)i(P2.)168 4533 y(The)f(next)h(assumption)g(will)f(b)s(e)g(used)g(in)g(section)h (9)e(to)g(pro)m(v)m(e)i(that)f(the)g(limit)g(is)g(non-random.)168 4705 y FA(Assumption)46 b(12.)f Fw(W)-8 b(e)39 b(supp)s(ose)i(that)e (there)g(exists)i(a)e(constan)m(t)h Fv(C)46 b Fw(indep)s(enden)m(t)c (of)c Fv(\017)i Fw(suc)m(h)168 4821 y(that)32 b(for)g(an)m(y)h(\()p Fv(x;)17 b(z)t Fw(\))29 b Fu(2)f Ft(R)1131 4785 y Fs(n)1200 4821 y Fu(\002)23 b Ft(R)1372 4785 y Fs(n)1451 4821 y Fw(w)m(e)34 b(ha)m(v)m(e)1055 4895 y Fp(\014)1055 4954 y(\014)1055 5014 y(\014)1088 5009 y Fv(D)1169 5024 y Fs(x)1213 5009 y Fv(H)1318 4899 y Fp(\020)1378 5009 y Fv(D)1462 4968 y Fr(2)1501 5009 y Fv(u)1557 4968 y Fs(\017)1589 5009 y Fw(\()p Fv(z)t(;)17 b(t)p Fw(\))p Fv(;)g(D)s(u)1977 4968 y Fs(\017)2009 5009 y Fw(\()p Fv(z)t(;)g(t)p Fw(\))p Fv(;)g(z)t(;)2361 4942 y(x)p 2361 4986 56 4 v 2369 5078 a(\017)2426 5009 y(;)g(\017;)g(!)2618 4899 y Fp(\021)2677 4895 y(\014)2677 4954 y(\014)2677 5014 y(\014)2738 5009 y Fu(\024)28 b Fv(C)r(:)3752 5181 y Fi(J)168 5353 y Fw(Note)34 b(that)f(\(18\))g(pro)m(vides)j(an)d(example)j(of)d(an)g(Hamiltonian)i (satisfying)f(Assumption)i(12,)d(due)168 5469 y(to)f(Prop)s(osition)h (20)f(b)s(elo)m(w.)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)612 b(9)0 450 y FA(Assumption)33 b(13.)38 b Fw(W)-8 b(e)28 b(assume)h(the)f(follo)m(wing)g(uniform)g(ergo)s(dicit)m(y)h(h)m(yp)s (othesis:)43 b(for)28 b(an)m(y)g Fv(\016)k(>)0 566 y Fw(0)48 b(there)i(exists)g Fv(M)66 b(>)55 b Fw(0,)e(suc)m(h)d(that)f (for)f(an)m(y)h Fv(!)1958 581 y Fr(0)1997 566 y Fw(,)k Fv(!)2138 581 y Fr(1)2233 566 y Fu(2)i Fw(\012,)f Fv(x)h Fu(2)h Ft(R)2809 530 y Fs(n)2856 566 y Fw(,)c Fv(t)k Fu(2)f Fw([0)p Fv(;)17 b(T)d Fw(],)53 b(there)0 683 y(exists)34 b Fv(y)d Fu(2)d Ft(R)514 646 y Fs(n)561 683 y Fw(,)33 b Fu(j)p Fv(y)t Fu(j)26 b(\024)i Fv(M)10 b Fw(,)34 b(suc)m(h)g(that)761 771 y Fp(\014)761 831 y(\014)761 891 y(\014)794 886 y Fv(H)883 775 y Fp(\020)942 886 y Fv(D)1026 845 y Fr(2)1066 886 y Fv(u)1122 845 y Fs(\017)1154 886 y Fw(\()p Fv(x;)17 b(t;)g(!)1431 901 y Fr(1)1470 886 y Fw(\))p Fv(;)g(D)s(u)1692 845 y 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2271 y Fw(\026)2131 2296 y Fv(C)2201 2311 y Fr(1)2240 2296 y Fv(;)212 b Fu(j)p Fv(\033)t Fw(\()p Fv(x)2659 2255 y Fl(0)2682 2296 y Fv(;)17 b(X)r(;)g Fw(0)p Fv(;)g(!)t Fw(\))p Fu(j)26 b(\024)3230 2271 y Fw(\026)3208 2296 y Fv(C)3278 2311 y Fr(2)3317 2296 y Fv(:)0 2468 y Fw(for)32 b(an)m(y)h(\()p Fv(x)426 2432 y Fl(0)450 2468 y Fv(;)17 b(X)r(;)g(!)t Fw(\))27 b Fu(2)h Ft(R)917 2432 y Fs(n)986 2468 y Fu(\002)22 b Ft(R)1157 2432 y Fs(n)1227 2468 y Fu(\002)g Fw(\012.)2161 b Fi(J)0 2651 y FA(Assumption)38 b(15.)k Fw(Let)33 b Fv(p)27 b(>)h Fw(1)k(b)s(e)h(as)g(in)g(Assumption)h(4.)43 b(W)-8 b(e)33 b(supp)s(ose)h(that)436 2823 y Fu(j)p Fv(f)11 b Fw(\()p Fv(x)616 2782 y Fl(0)639 2823 y Fv(;)17 b(X)r(;)g(v)t(;)g(!)t Fw(\))p Fu(j)1036 2782 y Fr(2)1101 2823 y Fu(\024)28 b Fv(C)7 b Fw(\(1)22 b(+)g Fv(L)p Fw(\()p Fv(x)1649 2782 y Fl(0)1673 2823 y Fv(;)17 b(X)r(;)g(v)t(;)g(!)t Fw(\)\))g Fv(;)210 b Fu(j)p Fv(\033)t Fw(\()p Fv(x)2514 2782 y Fl(0)2537 2823 y Fv(;)17 b(X)r(;)g(\017;)g(v)t(;)g(!)t Fw(\))p 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Fw(\()p Fv(x;)17 b(X)8 b Fw(\))27 b(=)16 b Fv(f)33 b Fu(\001)22 b Fv(D)1290 5161 y Fs(x)1334 5146 y Fv(\036)g Fw(+)1522 5079 y Fv(\033)t(\033)1640 5043 y Fs(T)p 1522 5124 174 4 v 1584 5215 a Fw(2)1733 5146 y(:)27 b Fv(D)1871 5105 y Fr(2)1868 5171 y Fs(x)1912 5146 y Fv(\036)1101 5401 y Fw(+)1209 5334 y(1)p 1209 5378 49 4 v 1214 5469 a Fv(\017)1267 5401 y(f)33 b Fu(\001)22 b Fv(D)1479 5416 y Fs(X)1546 5401 y Fv(\036)g Fw(+)1734 5334 y(1)p 1734 5378 V 1739 5469 a Fv(\017)1793 5401 y Fw(\()p Fv(\033)t(\033)1949 5360 y Fs(T)2004 5401 y Fw(\))28 b(:)f Fv(D)2208 5360 y Fr(2)2205 5426 y Fs(xX)2312 5401 y Fv(\036)22 b Fw(+)2515 5334 y(1)p 2500 5378 79 4 v 2500 5469 a Fv(\017)2539 5441 y Fr(2)2599 5334 y Fv(\033)t(\033)2717 5298 y Fs(T)p 2599 5378 174 4 v 2661 5469 a Fw(2)2810 5401 y(:)28 b Fv(D)2949 5360 y Fr(2)2946 5426 y Fs(X)3013 5401 y Fv(\036:)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 168 251 a Fq(10)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y Fw(This)33 b(extended)i(generator)e(satis\014es)h(the)f(follo)m(wing)g (iden)m(tit)m(y)1356 663 y(\()1420 638 y(~)1394 663 y Fv(A\036)p Fw(\))1580 552 y Fp(\020)1639 663 y Fv(x;)1748 595 y(x)p 1748 640 56 4 v 1756 731 a(\017)1813 552 y Fp(\021)1900 663 y Fw(=)28 b Fv(A)2094 552 y Fp(\020)2153 663 y Fv(\036)2228 552 y Fp(\020)2287 663 y Fv(x;)2396 595 y(x)p 2396 640 V 2404 731 a(\017)2462 552 y Fp(\021\021)2614 663 y Fv(:)168 879 y Fw(Observ)m(e)34 b(further)f(that)168 1072 y(\(20\))1565 1047 y(~)1540 1072 y Fv(A)1629 962 y Fp(\020)1689 1072 y Fv(')1770 962 y Fp(\020)1829 1072 y Fv(X)d Fu(\000)2049 1005 y Fv(x)p 2049 1050 V 2057 1141 a(\017)2115 962 y Fp(\021\021)2261 1072 y Fw(=)e(0)17 b Fv(;)168 1299 y Fw(for)32 b(all)g Fv(C)529 1262 y Fr(2)601 1299 y Fw(functions)i Fv(')p Fw(\()p Fv(x)p Fw(\))28 b(:)f Ft(R)1371 1262 y Fs(n)1446 1299 y Fu(!)g Ft(R)p Fw(.)168 1415 y(Before)46 b(pro)s(ceeding,)k(w)m(e)d(need)g(some)g (further)f(de\014nitions.)86 b(If)45 b Fv(U)57 b Fw(is)47 b(b)s(ounded,)j(set)c Fv(\015)56 b Fw(=)50 b(1,)168 1531 y(otherwise,)34 b(let)f Fv(\015)g Fw(:)27 b Fv(U)39 b Fu(!)27 b Ft(R)1212 1495 y Fr(+)1303 1531 y Fw(b)s(e)33 b(a)f(function)i(whic)m(h)g(satis\014es)1412 1771 y(lim)1372 1838 y Fl(j)p Fs(v)r Fl(j!1)1607 1771 y Fw(inf)1605 1833 y Fs(x;X)1754 1704 y Fv(L)p Fw(\()p Fv(x;)17 b(X)r(;)g(v)t(;)g(!)t Fw(\))p 1754 1748 528 4 v 1927 1840 a Fv(\015)5 b Fw(\()p Fv(v)t Fw(\))2319 1771 y(=)28 b(+)p Fu(1)p Fv(;)168 2013 y Fw(and)1591 2209 y(lim)1550 2276 y Fl(j)p Fs(v)r Fl(j!1)1820 2117 y Fw(~)1794 2142 y Fv(A)1867 2106 y Fs(v)1908 2142 y Fv(\036)p Fw(\()p Fv(x;)17 b(X)8 b Fw(\))p 1794 2186 436 4 v 1920 2278 a Fv(\015)d Fw(\()p Fv(v)t Fw(\))2267 2209 y(=)28 b(0)p Fv(;)168 2440 y Fw(for)c(all)h Fv(\036)i Fu(2)i Fv(C)694 2404 y Fr(2)733 2440 y Fw(\()p Ft(R)843 2404 y Fs(n)896 2440 y Fu(\002)7 b Ft(R)1052 2404 y Fs(n)1100 2440 y Fw(\).)40 b(Let)26 b Fv(p)f Fw(b)s(e)g(the)g(exp)s(onen)m(t)i (for)d(whic)m(h)j(Assumption)f(4)f(holds.)42 b(Consider)168 2556 y(the)e(spaces)i Fu(P)48 b Fw(and)40 b Fu(Q)g Fw(of)f(signed)1468 2518 y Fr(1)1545 2556 y Fw(measures,)k(resp.)e Fv(\026)f Fw(on)f Ft(R)2553 2520 y Fs(n)2628 2556 y Fu(\002)27 b Ft(R)2804 2520 y Fs(n)2878 2556 y Fu(\002)h Fv(U)38 b Fu(\002)27 b Fw([)p Fv(t;)17 b(T)d Fw(])40 b(and)g Fv(\027)46 b Fw(on)168 2672 y Ft(R)240 2636 y Fs(n)309 2672 y Fu(\002)22 b Ft(R)480 2636 y Fs(n)527 2672 y Fw(,)33 b(whic)m(h)h(satisfy)711 2771 y Fp(Z)827 2906 y Fv(\015)5 b Fw(\()p Fv(v)t Fw(\))22 b(+)g Fu(j)p Fv(x)p Fu(j)1241 2865 y Fs(p)1302 2906 y Fw(+)g Fv(\017)p Fu(j)p Fv(X)8 b Fu(j)1584 2865 y Fs(p)1623 2906 y Fv(d)p Fu(j)p Fv(\026)p Fu(j)27 b Fv(<)g Fu(1)p Fv(;)2258 2771 y Fp(Z)2374 2906 y Fu(j)p Fv(x)p Fu(j)2485 2865 y Fs(p)2547 2906 y Fw(+)22 b Fv(\017)p Fu(j)p Fv(X)8 b Fu(j)2829 2865 y Fs(p)2868 2906 y Fv(d)p Fu(j)p Fv(\027)e Fu(j)27 b Fv(<)h Fu(1)p Fv(:)168 3146 y Fw(W)-8 b(e)26 b(remark)h(that)f Fu(P)34 b Fw(is)27 b(the)f(dual)h(of)e(the)i(set)g Fv(C)1910 3161 y Fl(\003)1949 3146 y Fw(\()p Ft(R)2059 3109 y Fs(n)2115 3146 y Fu(\002)9 b Ft(R)2273 3109 y Fs(n)2329 3146 y Fu(\002)g Fv(U)20 b Fu(\002)9 b Fw([)p Fv(t;)17 b(T)d Fw(]\))26 b(of)f(con)m(tin)m(uous)j(functions)168 3262 y Fv(\037)g Fw(:)f Ft(R)383 3226 y Fs(n)452 3262 y Fu(\002)c Ft(R)624 3226 y Fs(n)693 3262 y Fu(\002)g Fv(U)32 b Fu(\002)23 b Fw([)p Fv(t;)17 b(T)d Fw(])28 b Fu(!)f Ft(R)p Fw(,)32 b(whic)m(h)j(satisfy)1329 3508 y(lim)1142 3574 y Fl(j)p Fs(x)p Fl(j)p Fr(+)p Fl(j)p Fs(X)5 b Fl(j)p Fr(+)p Fl(j)p Fs(v)r Fl(j!1)1667 3508 y Fw(sup)1717 3590 y Fs(t)1742 3571 y Fj(0)1981 3440 y Fv(\037)p Fw(\()p Fv(x;)17 b(X)r(;)g(v)t(;)g(t) 2436 3404 y Fl(0)2459 3440 y Fw(\))p 1841 3485 797 4 v 1841 3576 a Fv(\015)5 b Fw(\()p Fv(v)t Fw(\))22 b(+)g Fu(j)p Fv(x)p Fu(j)2255 3547 y Fs(p)2316 3576 y Fw(+)g Fv(\017)p Fu(j)p Fv(X)8 b Fu(j)2598 3547 y Fs(p)2675 3508 y Fw(=)28 b(0)p Fv(;)168 3754 y Fw(and)j Fu(Q)h Fw(is)f(the)h(dual)f(of)g(the)h(set)g Fv(C)1445 3769 y Fs(])1476 3754 y Fw(\()p Ft(R)1586 3718 y Fs(n)1652 3754 y Fu(\002)20 b Ft(R)1821 3718 y Fs(n)1868 3754 y Fw(\))31 b(of)g(con)m(tin)m(uous)i(functions)f(on)f Fv(\021)g Fw(:)d Ft(R)3293 3718 y Fs(n)3359 3754 y Fu(\002)20 b Ft(R)3528 3718 y Fs(n)3603 3754 y Fu(!)27 b Ft(R)p Fw(,)168 3871 y(whic)m(h)34 b(satisfy)1562 4070 y(lim)1441 4137 y Fl(j)p Fs(x)p Fl(j)p Fr(+)p Fl(j)p Fs(X)5 b Fl(j!1)1934 4003 y Fv(\021)t Fw(\()p Fv(x;)17 b(X)8 b Fw(\))p 1845 4047 494 4 v 1845 4138 a Fu(j)p Fv(x)p Fu(j)1956 4110 y Fs(p)2018 4138 y Fw(+)22 b Fv(\017)p Fu(j)p Fv(X)8 b Fu(j)2300 4110 y Fs(p)2376 4070 y Fw(=)28 b(0)p Fv(:)168 4297 y Fw(W)-8 b(e)46 b(further)f(consider)i(the)f(space)h Fv(C)1607 4312 y Fs(?)1646 4297 y Fw(\()p Ft(R)1756 4260 y Fs(n)1834 4297 y Fu(\002)31 b Ft(R)2014 4260 y Fs(n)2092 4297 y Fu(\002)g Fw([)p Fv(t;)17 b(T)d Fw(]\))46 b(of)f(con)m(tin)m (uous)i(functions)f(on)g Fv(\036)j Fw(:)168 4413 y Ft(R)240 4377 y Fs(n)309 4413 y Fu(\002)22 b Ft(R)480 4377 y Fs(n)555 4413 y Fu(!)27 b Ft(R)p Fw(,)33 b(whic)m(h)h(satisfy)1480 4653 y(lim)1359 4719 y Fl(j)p Fs(x)p Fl(j)p Fr(+)p Fl(j)p Fs(X)5 b Fl(j!1)1753 4653 y Fw(sup)1803 4736 y Fs(t)1828 4717 y Fj(0)1964 4586 y Fv(\036)p Fw(\()p Fv(x;)17 b(X)r(;)g(t)2321 4549 y Fl(0)2345 4586 y Fw(\))p 1927 4630 V 1927 4721 a Fu(j)p Fv(x)p Fu(j)2038 4693 y Fs(p)2099 4721 y Fw(+)22 b Fv(\017)p Fu(j)p Fv(X)8 b Fu(j)2381 4693 y Fs(p)2458 4653 y Fw(=)28 b(0)p Fv(:)168 4894 y Fw(In)45 b(this)g(section,)j(w)m (e)e(relax)f(the)g(original)f(sto)s(c)m(hastic)i(con)m(trol)f(problem)g (b)m(y)h(lo)s(oking)e(at)g(gen-)168 5010 y(eralized)50 b(tra)5 b(jectories)50 b(represen)m(ted)h(b)m(y)f(certain)f(measures,) 55 b(whic)m(h)50 b(will)g(b)s(e)f(elemen)m(ts)i(of)e Fu(P)168 5127 y Fw(and)41 b Fu(Q)p Fw(.)68 b(T)-8 b(o)41 b(do)g(this,)j(w)m(e)e(m)m(ust)g(consider)g(measures)h(that)d(satisfy)i (a)f(generalized)h(holonom)m(y)p 168 5271 499 4 v 267 5338 a Fr(1)303 5370 y Fz(All)32 b(the)f(measures)f(will)i(b)r(e)g (tacitly)f(assumed)g(to)g(b)r(e)h(Radon.)48 b(If)32 b(not)f(di\013eren) n(tly)h(sp)r(eci\014ed,)g(they)g(are)e(also)168 5469 y(assumed)d(to)g(b)r(e)h(p)r(ositiv)n(e.)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(11)0 450 y Fw(constrain)m(t.)64 b(More)40 b(precisely)-8 b(,)43 b(let)d Fv(\027)1403 414 y Fs(\017)1397 475 y(t)1475 450 y Fw(b)s(e)g(a)f(giv)m(en)h(probabilit)m(y)g(measure.)65 b(W)-8 b(e)40 b(lo)s(ok)f(for)f(mea-)0 566 y(sures)c(\()p Fv(\026)342 530 y Fs(\017)374 566 y Fw(\()p Fv(x)467 530 y Fl(0)491 566 y Fv(;)17 b(X)r(;)g(v)t(;)g(s)p Fw(\))p Fv(;)g(\027)939 530 y Fs(\017)933 592 y(T)987 566 y Fw(\()p Fv(x)1080 530 y Fl(0)1104 566 y Fv(;)g(X)8 b Fw(\)\))27 b Fu(2)h(P)i(\002)23 b(Q)33 b Fw(whic)m(h)h(satisfy)638 669 y Fp(Z)693 894 y Fn(R)741 875 y Fm(n)784 894 y Fl(\002)p Fn(R)887 875 y Fm(n)929 894 y Fl(\002)p Fs(U)7 b Fl(\002)p Fr([)p Fs(t;T)j Fr(])1250 804 y Fv(@)1301 819 y Fs(t)1331 804 y Fv(\036)p Fw(\()p Fv(x)1482 763 y Fl(0)1505 804 y Fv(;)17 b(X)r(;)g(s)p Fw(\))22 b(+)1906 779 y(~)1880 804 y Fv(A\036)p Fw(\()p Fv(x)2104 763 y Fl(0)2128 804 y Fv(;)17 b(X)r(;)g(s)p Fw(\))g Fv(d\026)2510 763 y Fs(\017)2541 804 y Fw(\()p Fv(x)2634 763 y Fl(0)2658 804 y Fv(;)g(X)r(;)g(v)t(;)g(s) p Fw(;)g Fv(!)t Fw(\))-3117 b(\(21\))479 1084 y(=)638 948 y Fp(Z)693 1174 y Fn(R)741 1155 y Fm(n)784 1174 y Fl(\002)p Fn(R)887 1155 y Fm(n)950 1084 y Fv(\036)17 b Fw(\()o Fv(x)1117 1043 y Fl(0)1141 1084 y Fv(;)g(X)r(;)g(T)d Fw(\))33 b Fv(d\027)1559 1043 y Fs(\017)1553 1109 y(T)1608 1084 y Fw(\()p Fv(x)1701 1043 y Fl(0)1724 1084 y Fv(;)17 b(X)8 b Fw(\))22 b Fu(\000)2016 948 y Fp(Z)2072 1174 y Fn(R)2120 1155 y Fm(n)2162 1174 y Fl(\002)p Fn(R)2265 1155 y Fm(n)2328 1084 y Fv(\036)17 b Fw(\()p Fv(x)2496 1043 y Fl(0)2519 1084 y Fv(;)g(X)r(;)g(t)p Fw(\))34 b Fv(d\027)2902 1043 y Fs(\017)2896 1109 y(t)2934 1084 y Fw(\()p Fv(x)3027 1043 y Fl(0)3051 1084 y Fv(;)17 b(X)8 b Fw(\))17 b Fv(;)0 1349 y Fw(for)29 b(an)m(y)i Fv(\036)c Fu(2)h Fv(C)584 1313 y Fr(2)624 1349 y Fw(\()p Ft(R)734 1313 y Fs(n)803 1349 y Fu(\002)22 b Ft(R)974 1313 y Fs(n)1038 1349 y Fu(\002)16 b Fw([)p Fv(t;)h(T)d Fw(]\))30 b(suc)m(h)h(that)f Fv(@)1880 1364 y Fs(t)1910 1349 y Fv(\036)p Fw(\()p Fv(x)2061 1313 y Fl(0)2085 1349 y Fv(;)17 b(X)r(;)g(s)p Fw(\))f(+)2474 1324 y(~)2448 1349 y Fv(A\036)p Fw(\()p Fv(x)2672 1313 y Fl(0)2696 1349 y Fv(;)h(X)r(;)g(s)p Fw(\))27 b Fu(2)h Fv(C)3142 1364 y Fl(\003)3181 1349 y Fw(\()p Ft(R)3291 1313 y Fs(n)3355 1349 y Fu(\002)16 b Ft(R)3520 1313 y Fs(n)3584 1349 y Fu(\002)0 1466 y Fv(U)33 b Fu(\002)22 b Fw([)p Fv(t;)17 b(T)d Fw(]\))33 b(and)f Fv(\036)17 b Fw(\()p Fv(x)830 1430 y Fl(0)853 1466 y Fv(;)g(X)r(;)g(t)p Fw(\))g Fv(;)g(\036)g Fw(\()o Fv(x)1325 1430 y Fl(0)1349 1466 y Fv(;)g(X)r(;)g(T)d Fw(\))27 b Fu(2)h Fv(C)1820 1481 y Fs(])1851 1466 y Fw(\()p Ft(R)1961 1430 y Fs(n)2030 1466 y Fu(\002)23 b Ft(R)2202 1430 y Fs(n)2249 1466 y Fw(\).)0 1582 y(The)37 b(measure)h Fv(\027)642 1546 y Fs(\017)636 1607 y(t)711 1582 y Fw(enco)s(des)g(the)e(giv)m(en)i (initial)e(condition)h FA(x)p Fw(\()p Fv(t)p Fw(\))d(=)g Fv(x)p Fw(,)k(allo)m(wing)f(for)e(probabilit)m(y)0 1698 y(distributions)e(of)d(initial)i(conditions,)h(whereas)f(the)g(measure) g Fv(\027)2418 1662 y Fs(\017)2412 1723 y(T)2468 1698 y Fw(,)f(whic)m(h)i(is)f(arbitrary)-8 b(,)31 b(enco)s(des)0 1814 y(the)i(p)s(ossible)h(probabilit)m(y)f(distribution)h(of)e (terminal)h(conditions.)45 b(W)-8 b(e)33 b(will)g(alw)m(a)m(ys)h(c)m (ho)s(ose)0 2000 y(\(22\))1293 b Fv(\027)1521 1959 y Fs(\017)1515 2024 y(t)1582 2000 y Fw(=)28 b Fv(\016)1729 2015 y Fs(x)1773 2000 y Fw(\()p Fv(x)1866 1959 y Fl(0)1889 2000 y Fw(\))p Fv(\016)1980 1988 y Fm(x)p 1981 2000 35 3 v 1985 2041 a(\017)2029 2000 y Fw(\()p Fv(X)8 b Fw(\))0 2187 y(and)33 b(w)m(e)g(will)h(omit)e(the)h(dep)s(endence)i(of)d Fv(\027)1601 2151 y Fs(\017)1595 2212 y(t)1667 2187 y Fw(and)h Fv(\027)1911 2151 y Fs(\017)1905 2212 y(T)1992 2187 y Fw(up)s(on)g(the)g(prescrib)s(ed)h(initial)f(p)s(oin)m(t)g Fv(x)p Fw(.)0 2303 y(The)38 b(motiv)-5 b(ation)37 b(for)g(this)h (holonom)m(y)g(prop)s(ert)m(y)g(is)g(the)g(follo)m(wing:)53 b(let)38 b(us)g(consider)g(a)f(tra)5 b(jec-)0 2420 y(tory)41 b(\()p FA(x)p Fw(\()p Fu(\001)p Fw(\))p Fv(;)17 b FA(v)r Fw(\()p Fu(\001)p Fw(\)\))40 b(of)g(the)h(con)m(trolled)h(dynamics)h (\(3\);)i(to)c(this)g(tra)5 b(jectory)-8 b(,)44 b(one)d(can)h(asso)s (ciate)0 2536 y(measures)34 b Fv(\026)f Fw(and)f Fv(\027)747 2551 y Fs(T)835 2536 y Fw(in)h(the)g(follo)m(wing)g(w)m(a)m(y)1190 2633 y Fp(Z)1306 2768 y Fv(\036)p Fw(\()p Fv(x)1457 2727 y Fl(0)1480 2768 y Fv(;)17 b(X)r(;)g(v)t(;)g(s)p Fw(\))g Fv(d\026)1957 2727 y Fs(\017)1988 2768 y Fw(\()p Fv(x)2081 2727 y Fl(0)2105 2768 y Fv(;)g(X)r(;)g(v)t(;)g(s)p Fw(;)g Fv(!)t Fw(\))1070 3046 y(:=)g Fv(E)1284 2911 y Fp(Z)1384 2937 y Fs(T)1340 3136 y(t)1455 3046 y Fv(\036)1530 2906 y Fp(\022)1603 3046 y FA(x)p Fw(\()p Fv(s)p Fw(\))p Fv(;)1838 2979 y FA(x)p Fw(\()p Fv(s)p Fw(\))p 1838 3023 181 4 v 1909 3114 a Fv(\017)2029 3046 y(;)g FA(v)r Fw(\()p Fv(s)p Fw(\))p Fv(;)g(s)2346 2906 y Fp(\023)2451 3046 y Fv(ds)g(;)0 2911 y Fw(\(23\))0 3287 y(and)0 3509 y(\(24\))892 3374 y Fp(Z)1008 3509 y Fv(\036)p Fw(\()p Fv(x)1159 3468 y Fl(0)1182 3509 y Fv(;)g(X)8 b Fw(\))17 b Fv(d\027)1475 3468 y Fs(\017)1469 3534 y(T)1523 3509 y Fw(\()p Fv(x)1616 3468 y Fl(0)1640 3509 y Fv(;)g(X)8 b Fw(\))27 b(:=)h Fv(E)6 b(\036)2122 3369 y Fp(\022)2194 3509 y FA(x)p Fw(\()p Fv(T)14 b Fw(\))p Fv(;)2454 3442 y FA(x)p Fw(\()p Fv(T)g Fw(\))p 2454 3486 206 4 v 2537 3577 a Fv(\017)2669 3369 y Fp(\023)2759 3509 y Fv(:)0 3756 y Fw(By)37 b(Dynkin's)i(F)-8 b(orm)m(ula)36 b(\(see,)j(e.g.,)g(form)m(ula)e(\(3\))f(on)h(page)g(105) f(of)g([Ev)-5 b(a06b]\),)38 b(the)f(constrain)m(t)0 3872 y(in)c(\(21\))f(is)h(satis\014ed)h(b)m(y)f(these)h(measures.)0 3988 y(Of)43 b(course,)k(b)m(y)e(considering)f(measures)i(instead)e(of) f(tra)5 b(jectories,)47 b(w)m(e)e(are,)h(in)e(principle,)j(en-)0 4105 y(larging)35 b(the)h(minimizing)g(set.)53 b(Ho)m(w)m(ev)m(er,)38 b(as)d(w)m(e)i(will)f(see)g(later,)g(the)g(v)-5 b(alue)36 b(of)e(the)i(in\014m)m(um)h(is)0 4221 y(unc)m(hanged)d(\(see)g(Prop)s (ositions)f(4)f(and)h(9)f(b)s(elo)m(w)i(for)e(further)h(details)g(on)g (this\).)0 4337 y(In)42 b(the)h(next)g(prop)s(osition,)h(w)m(e)f(sho)m (w)h(that)d(a)h(measure)h Fv(\026)f Fw(satisfying)j(\(21\))c(has)h(alw) m(a)m(ys)i(total)0 4453 y(mass)34 b Fv(T)h Fu(\000)23 b Fv(t)p Fw(.)0 4644 y FA(Prop)s(osition)k(2.)35 b Fo(Supp)-5 b(ose)25 b(that)i Fv(\026)g Fo(is)f(a)g(me)-5 b(asur)g(e)26 b(on)g Ft(R)2100 4608 y Fs(n)2151 4644 y Fu(\002)t Ft(R)2304 4608 y Fs(n)2354 4644 y Fu(\002)t Fv(U)14 b Fu(\002)t Fw([)p Fv(t;)j(T)d Fw(])27 b Fo(which)e(satis\014es)34 b Fw(\(21\))p Fo(,)0 4760 y(for)h(a)f(suitable)h(pr)-5 b(ob)g(ability)35 b(me)-5 b(asur)g(e)34 b Fv(\027)1495 4724 y Fs(\017)1489 4785 y(T)1579 4760 y Fo(on)g Ft(R)1790 4724 y Fs(n)1860 4760 y Fu(\002)22 b Ft(R)2031 4724 y Fs(n)2078 4760 y Fo(.)45 b(Then)0 5017 y Fw(\(25\))1349 4882 y Fp(Z)1465 5017 y Fv(\035)t Fw(\()p Fv(s)p Fw(\))p Fv(d\026)26 b Fw(=)1883 4882 y Fp(Z)1983 4908 y Fs(T)1939 5107 y(t)2055 5017 y Fv(\035)t Fw(\()p Fv(s)p Fw(\))p Fv(ds)0 5259 y Fo(for)35 b(any)f Fv(\035)d Fu(2)d Fv(C)7 b Fw(\([)p Fv(t;)17 b(T)d Fw(]\))p Fo(.)45 b(A)n(lso,)0 5445 y Fw(\(26\))938 b Fv(\026)p Fw(\()p Ft(R)1281 5403 y Fs(n)1350 5445 y Fu(\002)22 b Ft(R)1521 5403 y Fs(n)1590 5445 y Fu(\002)h Fv(U)33 b Fu(\002)22 b Fw([)p Fv(t;)17 b(T)d Fw(]\))44 b(=)h Fv(T)35 b Fu(\000)23 b Fv(t:)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 168 251 a Fq(12)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y Fo(Pr)k(o)g(of.)41 b Fw(Cho)s(ose)33 b(\007\()p Fv(t)p Fw(\))g(suc)m(h)h(that)e Fv(\035)t Fw(\()p Fv(t)p Fw(\))27 b(=)g(\007)1830 414 y Fl(0)1854 450 y Fw(\()p Fv(t)p Fw(\).)43 b(Then,)34 b(from)e(\(21\),)g(w)m(e)i(ha)m(v)m(e)g(that)570 577 y Fp(Z)686 713 y Fv(\035)t Fw(\()p Fv(s)p Fw(\))p Fv(d\026)26 b Fw(=)1105 577 y Fp(Z)1221 713 y Fw(\007\()p Fv(T)14 b Fw(\))p Fv(d\027)1549 671 y Fs(\017)1543 737 y(T)1620 713 y Fu(\000)1719 577 y Fp(Z)1835 713 y Fw(\007\()p Fv(t)p Fw(\))p Fv(d\027)2127 671 y Fs(\017)2121 737 y(t)2188 713 y Fw(=)27 b(\007\()p Fv(T)14 b Fw(\))22 b Fu(\000)h Fw(\007\()p Fv(t)p Fw(\))k(=)2954 577 y Fp(Z)3054 603 y Fs(T)3009 803 y(t)3125 713 y Fv(\035)t Fw(\()p Fv(s)p Fw(\))p Fv(ds;)168 960 y Fw(yielding)35 b(\(25\))o(.)44 b(Then)33 b(\(26\))f(follo)m(ws)i(b)m(y)f(applying)i(\(25\))d(to)g Fv(\035)f Fw(:=)d(1.)1041 b Fi(\003)168 1175 y Fw(W)-8 b(e)33 b(no)m(w)g(pro)m(v)m(e)h(that)e(b)s(oth)g Fv(\027)1297 1139 y Fs(\017)1291 1200 y(T)1379 1175 y Fw(and)h Fv(\026)1628 1139 y Fs(\017)1693 1175 y Fw(are)f(supp)s(orted)i(on)e Fu(f)p Fv(X)k Fw(=)2729 1136 y Fs(x)2769 1112 y Fj(0)p 2729 1152 63 4 v 2746 1209 a Fs(\017)2801 1175 y Fu(g)p Fw(,)d(as)f(long)h(as)f(so)h(is)g Fv(\027)3633 1139 y Fs(\017)3627 1200 y(t)3667 1175 y Fw(.)168 1386 y FA(Prop)s(osition)52 b(3.)c Fo(Supp)-5 b(ose)47 b Fv(\027)1361 1349 y Fs(\017)1355 1410 y(t)1441 1386 y Fo(is)f(supp)-5 b(orte)g(d)47 b(in)f(the)h(set)g Fv(X)57 b Fw(=)2747 1346 y Fs(x)2787 1323 y Fj(0)p 2747 1363 V 2764 1420 a Fs(\017)2819 1386 y Fo(,)50 b(and)c(that)h Fw(\()p Fv(\026)3408 1349 y Fs(\017)3441 1386 y Fv(;)17 b(\027)3539 1349 y Fs(\017)3533 1411 y(T)3587 1386 y Fw(\))47 b Fo(sat-)168 1519 y(isfy)c Fw(\(21\))p Fo(.)h(Then)34 b(b)-5 b(oth)35 b Fv(\027)1121 1482 y Fs(\017)1115 1544 y(T)1205 1519 y Fo(and)g Fv(\026)1454 1482 y Fs(\017)1521 1519 y Fo(ar)-5 b(e)34 b(supp)-5 b(orte)g(d)35 b(in)g(the)g(set)f Fv(X)i Fw(=)2784 1479 y Fs(x)2824 1456 y Fj(0)p 2784 1496 V 2801 1553 a Fs(\017)2856 1519 y Fo(.)168 1720 y(Pr)-5 b(o)g(of.)41 b Fw(Let)33 b Ft({)g Fw(:)28 b Ft(R)867 1684 y Fs(n)943 1720 y Fu(!)g Ft(R)1143 1679 y Fr(+)1143 1745 y(0)1234 1720 y Fw(b)s(e)34 b(a)e Fv(C)1526 1684 y Fr(2)1599 1720 y Fw(function)h(suc)m(h)i(that)d Ft({)t Fw(\(0\))d(=)f(0)33 b(and)g Ft({)t Fw(\()p Fv(x)3176 1684 y Fl(0)3200 1720 y Fw(\))28 b Fv(>)g Fw(0)33 b(if)g Fv(x)3597 1684 y Fl(0)3649 1720 y Fu(6)p Fw(=)28 b(0.)168 1844 y(Assume)i(that)f Ft({)820 1763 y Fp(\000)866 1844 y Fv(X)h Fu(\000)1086 1804 y Fs(x)1126 1781 y Fj(0)p 1086 1821 V 1103 1878 a Fs(\017)1158 1763 y Fp(\001)1232 1844 y Fu(2)e Fv(C)1396 1859 y Fs(])1427 1844 y Fw(\()p Ft(R)1537 1808 y Fs(n)1599 1844 y Fu(\002)15 b Ft(R)1763 1808 y Fs(n)1810 1844 y Fw(\).)42 b(T)-8 b(o)29 b(pro)m(v)m(e)h(the)f (desired)i(claim,)f(w)m(e)g(will)f(sho)m(w)h(that)168 1960 y(if)1345 2011 y Fp(Z)1461 2147 y Ft({)1548 2006 y Fp(\022)1621 2147 y Fv(X)g Fu(\000)1842 2079 y Fv(x)1897 2043 y Fl(0)p 1842 2124 79 4 v 1861 2215 a Fv(\017)1930 2006 y Fp(\023)2020 2147 y Fv(d\027)2125 2105 y Fs(\017)2119 2171 y(t)2158 2147 y Fw(\()p Fv(x)2251 2105 y Fl(0)2275 2147 y Fv(;)17 b(X)8 b Fw(\))27 b(=)g(0)p Fv(;)168 2373 y Fw(then)1336 2424 y Fp(Z)1453 2560 y Ft({)1540 2419 y Fp(\022)1613 2560 y Fv(X)j Fu(\000)1834 2492 y Fv(x)1889 2456 y Fl(0)p 1834 2537 V 1853 2628 a Fv(\017)1922 2419 y Fp(\023)2012 2560 y Fv(d\027)2117 2519 y Fs(\017)2111 2584 y(T)2166 2560 y Fw(\()p Fv(x)2259 2519 y Fl(0)2283 2560 y Fv(;)17 b(X)8 b Fw(\))27 b(=)g(0)p Fv(;)168 2786 y Fw(and)1199 2837 y Fp(Z)1315 2973 y Ft({)1402 2832 y Fp(\022)1476 2973 y Fv(X)j Fu(\000)1696 2905 y Fv(x)1751 2869 y Fl(0)p 1696 2950 V 1716 3041 a Fv(\017)1785 2832 y Fp(\023)1875 2973 y Fv(d\026)1985 2932 y Fs(\017)2017 2973 y Fw(\()p Fv(x)2110 2932 y Fl(0)2134 2973 y Fv(;)17 b(X)r(;)g(v)t(;)g(s)p Fw(;)g Fv(!)t Fw(\))25 b(=)j(0)p Fv(:)168 3220 y Fw(Set)33 b Fv(\036)p Fw(\()p Fv(x)487 3184 y Fl(0)510 3220 y Fv(;)17 b(X)8 b Fw(\))27 b(:=)h Ft({)926 3140 y Fp(\000)972 3220 y Fv(X)h Fu(\000)1192 3181 y Fs(x)1232 3158 y Fj(0)p 1192 3197 63 4 v 1209 3255 a Fs(\017)1264 3140 y Fp(\001)1310 3220 y Fw(.)43 b(Since,)34 b(b)m(y)h(\(20\))o(,)168 3429 y(\(27\))1847 3404 y(~)1821 3429 y Fv(A\036)28 b Fw(=)f(0)17 b Fv(;)168 3620 y Fw(the)33 b(constrain)m(t)g(in)g(\(21\))f(implies)i(that)168 3873 y(\(28\))999 3737 y Fp(Z)1116 3873 y Ft({)1203 3732 y Fp(\022)1276 3873 y Fv(X)c Fu(\000)1496 3805 y Fv(x)1551 3769 y Fl(0)p 1496 3850 79 4 v 1516 3941 a Fv(\017)1585 3732 y Fp(\023)1692 3873 y Fv(d\027)1797 3831 y Fs(\017)1791 3897 y(T)1873 3873 y Fw(=)1977 3737 y Fp(Z)2093 3873 y Ft({)2180 3732 y Fp(\022)2253 3873 y Fv(X)g Fu(\000)2474 3805 y Fv(x)2529 3769 y Fl(0)p 2474 3850 V 2494 3941 a Fv(\017)2563 3732 y Fp(\023)2669 3873 y Fv(d\027)2774 3831 y Fs(\017)2768 3897 y(t)2835 3873 y Fw(=)d(0)p Fv(:)168 4140 y Fw(No)m(w)33 b(c)m(ho)s(ose)g Fv(\036)p Fw(\()p Fv(x)850 4104 y Fl(0)874 4140 y Fv(;)17 b(X)r(;)g(t)p Fw(\))27 b(:=)h Fv(t)p Ft({)1398 4060 y Fp(\000)1444 4140 y Fv(X)i Fu(\000)1664 4101 y Fs(x)1704 4078 y Fj(0)p 1664 4118 63 4 v 1681 4175 a Fs(\017)1736 4060 y Fp(\001)1782 4140 y Fw(.)43 b(Then,)34 b(using)h(\(27\))o(,)e(\(28\))f(and)g(\(21\)) g(once)h(more,)g(w)m(e)168 4257 y(gather)f(that)1093 4327 y Fp(Z)1209 4463 y Ft({)1296 4322 y Fp(\022)1370 4463 y Fv(X)e Fu(\000)1590 4395 y Fv(x)1645 4359 y Fl(0)p 1590 4440 79 4 v 1610 4531 a Fv(\017)1679 4322 y Fp(\023)1786 4463 y Fv(d\026)1896 4422 y Fs(\017)1955 4463 y Fw(=)2059 4327 y Fp(Z)2175 4463 y Fv(@)2226 4478 y Fs(t)2256 4463 y Fv(\036)22 b Fw(+)2460 4438 y(~)2434 4463 y Fv(A\036)17 b(d\026)2692 4422 y Fs(\017)2751 4463 y Fw(=)28 b(0)168 4689 y(as)k(w)m(e)i(wished.)3008 b Fi(\003)168 4883 y FA(Prop)s(osition)37 b(4.)42 b Fo(F)-7 b(or)34 b Fv(t)28 b Fu(2)g Fw(\(0)p Fv(;)17 b(T)d Fw(\))p Fo(,)34 b(let)652 5126 y Fw(~)-55 b Fv(u)702 5085 y Fs(\017)734 5126 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))83 b(:=)g(inf)1456 4991 y Fp(Z)1512 5216 y Fn(R)1560 5197 y Fm(n)1602 5216 y Fl(\002)p Fn(R)1705 5197 y Fm(n)1747 5216 y Fl(\002)p Fs(U)7 b Fl(\002)p Fr([)p Fs(t;T)j Fr(])2068 5126 y Fv(L)p Fw(\()p Fv(x)2227 5085 y Fl(0)2251 5126 y Fv(;)17 b(X)r(;)g(v)t(;)g(!)t Fw(\))g Fv(d\026)2747 5085 y Fs(\017)2777 5126 y Fw(\()p Fv(x)2870 5085 y Fl(0)2894 5126 y Fv(;)g(X)r(;)g(v)t(;)g(s)p Fw(;)g Fv(!)t Fw(\))1322 5406 y(+)1415 5270 y Fp(Z)1469 5496 y Fn(R)1517 5477 y Fm(n)1560 5496 y Fl(\002)p Fn(R)1663 5477 y Fm(n)1726 5406 y Fv( )t Fw(\()p Fv(x)1886 5365 y Fl(0)1910 5406 y Fw(\))g Fv(d\027)2070 5365 y Fs(\017)2064 5430 y(T)2118 5406 y Fw(\()p Fv(x)2211 5365 y Fl(0)2235 5406 y Fv(;)g(X)8 b Fw(;)17 b Fv(!)t Fw(\))g Fv(;)-2391 b Fw(\(29\))p eop end %%Page: 13 13 TeXDict begin 13 12 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(13)0 450 y Fo(wher)-5 b(e)37 b(the)g(in\014mum)g(is)g(taken)g(over)g (al)5 b(l)37 b(me)-5 b(asur)g(es)37 b Fw(\()p Fv(\026)2079 414 y Fs(\017)2111 450 y Fv(;)17 b(\027)2209 414 y Fs(\017)2203 475 y(T)2258 450 y Fw(\))37 b Fo(which,)g(for)g(every)h Fv(!)d Fu(2)e Fw(\012)p Fo(,)38 b(satisfy)0 566 y(the)f(fol)5 b(lowing:)35 b Fv(\026)668 530 y Fs(\017)737 566 y Fo(is)i(a)g(me)-5 b(asur)g(e)36 b(on)h Ft(R)1524 530 y Fs(n)1594 566 y Fu(\002)24 b Ft(R)1767 530 y Fs(n)1838 566 y Fu(\002)g Fv(U)34 b Fu(\002)24 b Fw([)p Fv(t;)17 b(T)d Fw(])37 b Fo(with)g(total)g(mass)f Fv(T)h Fu(\000)24 b Fv(t)p Fo(,)38 b Fv(\027)3419 530 y Fs(\017)3413 592 y(T)3505 566 y Fo(is)f(a)0 683 y(pr)-5 b(ob)g(ability)34 b(me)-5 b(asur)g(e)35 b(on)f Ft(R)1058 646 y Fs(n)1127 683 y Fu(\002)23 b Ft(R)1299 646 y Fs(n)1346 683 y Fo(,)35 b(and)f Fv(\026)1659 646 y Fs(\017)1726 683 y Fo(and)h Fv(\027)1970 646 y Fs(\017)1964 708 y(T)2054 683 y Fo(satisfy)f(the)h (fol)5 b(lowing)34 b(c)-5 b(onstr)g(aint:)523 766 y Fp(Z)579 992 y Fn(R)627 973 y Fm(n)669 992 y Fl(\002)p Fn(R)772 973 y Fm(n)814 992 y Fl(\002)p Fs(U)7 b Fl(\002)p Fr([)p Fs(t;T)j Fr(])1135 902 y Fv(@)1186 917 y Fs(t)1216 902 y Fv(\036)p Fw(\()p Fv(x)1367 860 y Fl(0)1390 902 y Fv(;)17 b(X)r(;)g(s)p Fw(\))22 b(+)1791 876 y(~)1765 902 y Fv(A\036)p Fw(\()p Fv(x)1989 860 y Fl(0)2013 902 y Fv(;)17 b(X)r(;)g(s)p Fw(\))g Fv(d\026)2395 860 y Fs(\017)2426 902 y Fw(\()p Fv(x)2519 860 y Fl(0)2543 902 y Fv(;)g(X)r(;)g(v)t(;)g(s)p Fw(;)g Fv(!)t Fw(\))403 1181 y(=)523 1045 y Fp(Z)579 1271 y Fn(R)627 1252 y Fm(n)669 1271 y Fl(\002)p Fn(R)772 1252 y Fm(n)835 1181 y Fv(\036)g Fw(\()o Fv(x)1002 1140 y Fl(0)1026 1181 y Fv(;)g(X)r(;)g(T)d Fw(\))33 b Fv(d\027)1444 1140 y Fs(\017)1438 1206 y(T)1493 1181 y Fw(\()p Fv(x)1586 1140 y Fl(0)1609 1181 y Fv(;)17 b(X)8 b Fw(;)17 b Fv(!)t Fw(\))k Fu(\000)2010 1045 y Fp(Z)2065 1271 y Fn(R)2113 1252 y Fm(n)2155 1271 y Fl(\002)p Fn(R)2258 1252 y Fm(n)2322 1181 y Fv(\036)c Fw(\()o Fv(x)2489 1140 y Fl(0)2513 1181 y Fv(;)g(X)r(;)g(t)p Fw(\))33 b Fv(d\027)2895 1140 y Fs(\017)2889 1206 y(t)2928 1181 y Fw(\()p Fv(x)3021 1140 y Fl(0)3044 1181 y Fv(;)17 b(X)8 b Fw(\))17 b Fv(;)0 1046 y Fw(\(30\))0 1415 y Fo(for)35 b(any)f Fv(\036)28 b Fu(2)g Fv(C)599 1379 y Fr(2)638 1415 y Fw(\()p Ft(R)748 1379 y Fs(n)817 1415 y Fu(\002)23 b Ft(R)989 1379 y Fs(n)1058 1415 y Fu(\002)g Fw([)p Fv(t;)17 b(T)d Fw(]\))p Fo(.)44 b(Then,)34 b Fv(u)1814 1379 y Fs(\017)1874 1415 y Fu(\025)g Fw(~)-55 b Fv(u)2035 1379 y Fs(\017)2067 1415 y Fo(.)0 1595 y(Pr)-5 b(o)g(of.)41 b Fw(Let)d Fv(u)534 1559 y Fs(\017)605 1595 y Fw(b)s(e)g(as)g(in)h(\(5\))f(and)44 b(~)-55 b Fv(u)1402 1559 y Fs(\017)1472 1595 y Fw(giv)m(en)39 b(b)m(y)i(\(29\))o(.)60 b(Fix)39 b Fv(\016)i(>)c Fw(0,)i(and)f (consider)i(a)e(b)s(ounded)0 1711 y(progressiv)m(ely)i(measurable)f (con)m(trol)f FA(v)1490 1726 y Fs(\016)1528 1711 y Fw(,)g(and)g(the)g (corresp)s(onding)g(tra)5 b(jectory)38 b FA(x)3111 1726 y Fs(\016)3187 1711 y Fw(of)44 b(\(3\))37 b(with)0 1827 y(the)c(initial)g(condition)g FA(x)937 1842 y Fs(\016)975 1827 y Fw(\()p Fv(t)p Fw(\))28 b(=)f Fv(x)p Fw(,)33 b(suc)m(h)h(that) 671 1985 y Fv(u)727 1944 y Fs(\017)759 1985 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))22 b(+)g Fv(\016)699 2216 y Fu(\025)28 b Fv(E)899 2076 y Fp(\024)951 2081 y(Z)1051 2107 y Fs(T)1007 2306 y(t)1123 2216 y Fv(L)1206 2076 y Fp(\022)1279 2216 y FA(x)1338 2231 y Fs(\016)1376 2216 y Fw(\()p Fv(s)p Fw(\))p Fv(;)1551 2149 y FA(x)1610 2164 y Fs(\016)1648 2149 y Fw(\()p Fv(s)p Fw(\))p 1551 2193 219 4 v 1641 2284 a Fv(\017)1780 2216 y(;)17 b FA(v)1883 2231 y Fs(\016)1921 2216 y Fw(\()p Fv(s)p Fw(\))p Fv(;)g(!)2152 2076 y Fp(\023)2257 2216 y Fv(ds)22 b Fw(+)g Fv( )f Fw(\()p FA(x)2655 2231 y Fs(\016)2693 2216 y Fw(\()p Fv(T)14 b Fw(\)\))2877 2076 y Fp(\025)2963 2216 y Fv(:)0 2135 y Fw(\(31\))0 2444 y(W)-8 b(e)25 b(in)m(tro)s(duce)h(the)f(probabilit)m (y)g(measures)i Fv(\026)1706 2459 y Fs(\016)1768 2444 y Fw(and)d Fv(\027)1997 2459 y Fs(\016)2060 2444 y Fw(on)h Ft(R)2260 2408 y Fs(n)2312 2444 y Fu(\002)6 b Ft(R)2467 2408 y Fs(n)2520 2444 y Fu(\002)g Fv(U)16 b Fu(\002)6 b Fw([)p Fv(t;)17 b(T)d Fw(])26 b(and)f(on)f Ft(R)3379 2408 y Fs(n)3432 2444 y Fu(\002)6 b Ft(R)3587 2408 y Fs(n)3634 2444 y Fw(,)0 2560 y(resp)s(ectiv)m(ely)-8 b(,)36 b(de\014ned)e(b)m(y)935 2638 y Fp(Z)991 2863 y Fn(R)1039 2844 y Fm(n)1081 2863 y Fl(\002)p Fn(R)1184 2844 y Fm(n)1227 2863 y Fl(\002)p Fs(U)7 b Fl(\002)p Fr([)p Fs(t;T)j Fr(])1547 2773 y Fv(\036)p Fw(\()p Fv(x)1698 2732 y Fl(0)1722 2773 y Fv(;)17 b(X)r(;)g(v)t(;)g(s)p Fw(\))g Fv(d\026)2199 2788 y Fs(\016)2234 2773 y Fw(\()p Fv(x)2327 2732 y Fl(0)2351 2773 y Fv(;)g(X)r(;)g(v)t(;)g(s)p Fw(\))1142 3072 y(:=)27 b Fv(E)1367 2936 y Fp(Z)1467 2962 y Fs(T)1422 3162 y(t)1538 3072 y Fv(\036)1613 2931 y Fp(\022)1686 3072 y FA(x)1745 3087 y Fs(\016)1783 3072 y Fw(\()p Fv(s)p Fw(\))p Fv(;)1958 3004 y FA(x)2017 3019 y Fs(\016)2055 3004 y Fw(\()p Fv(s)p Fw(\))p 1958 3049 V 2048 3140 a Fv(\017)2187 3072 y(;)17 b FA(v)2290 3087 y Fs(\016)2328 3072 y Fw(\()p Fv(s)p Fw(\))p Fv(;)g(s)2540 2931 y Fp(\023)2646 3072 y Fv(ds)0 3294 y Fw(and)742 3320 y Fp(Z)797 3546 y Fn(R)845 3527 y Fm(n)887 3546 y Fl(\002)p Fn(R)990 3527 y Fm(n)1054 3456 y Fv(')p Fw(\()p Fv(x)1211 3415 y Fl(0)1234 3456 y Fv(;)g(X)8 b Fw(\))17 b Fv(d\027)1521 3471 y Fs(\016)1558 3456 y Fw(\()p Fv(x)1651 3415 y Fl(0)1675 3456 y Fv(;)g(X)8 b Fw(\))27 b(:=)g Fv(E)6 b(')2162 3315 y Fp(\022)2235 3456 y FA(x)2294 3471 y Fs(\016)2332 3456 y Fw(\()p Fv(T)14 b Fw(\))p Fv(;)2533 3388 y FA(x)2592 3403 y Fs(\016)2630 3388 y Fw(\()p Fv(T)g Fw(\))p 2533 3433 244 4 v 2635 3524 a Fv(\017)2786 3315 y Fp(\023)2893 3456 y Fv(;)0 3665 y Fw(for)32 b(an)m(y)h Fv(\036)28 b Fu(2)g Fv(C)7 b Fw(\()p Ft(R)700 3629 y Fs(n)769 3665 y Fu(\002)22 b Ft(R)940 3629 y Fs(n)1009 3665 y Fu(\002)h Fv(U)33 b Fu(\002)22 b Fw([)p Fv(t;)17 b(T)d Fw(]\))33 b(and)f Fv(')c Fu(2)g Fv(C)7 b Fw(\()p Ft(R)2144 3629 y Fs(n)2213 3665 y Fu(\002)23 b Ft(R)2385 3629 y Fs(n)2432 3665 y Fw(\).)0 3781 y(Then,)48 b(if)c Fv(\036)j Fu(2)h Fv(C)693 3745 y Fr(2)732 3781 y Fw(\()p Ft(R)842 3745 y Fs(n)919 3781 y Fu(\002)31 b Ft(R)1099 3745 y Fs(n)1175 3781 y Fu(\002)g Fw([)p Fv(t;)17 b(T)d Fw(]\),)47 b(b)m(y)e(Dynkin's)g(F)-8 b(orm)m(ula)44 b(\(see,)k(e.g.,)g(form)m(ula)c(\(3\))f(on)0 3898 y(page)33 b(105)e(of)i([Ev)-5 b(a06b]\),)535 3981 y Fp(Z)590 4206 y Fn(R)638 4187 y Fm(n)681 4206 y Fl(\002)p Fn(R)784 4187 y Fm(n)826 4206 y Fl(\002)p Fs(U)7 b Fl(\002)p Fr([)p Fs(t;T)j Fr(])1147 4116 y Fv(@)1198 4131 y Fs(t)1228 4116 y Fv(\036)22 b Fw(+)1431 4091 y(~)1406 4116 y Fv(A\036d\026)1647 4131 y Fs(\016)546 4405 y Fw(=)27 b Fv(E)744 4264 y Fp(\024)797 4405 y Fv(\036)872 4264 y Fp(\022)944 4405 y FA(x)1003 4420 y Fs(\016)1041 4405 y Fw(\()p Fv(T)14 b Fw(\))p Fv(;)1242 4337 y FA(x)1301 4352 y Fs(\016)1339 4337 y Fw(\()p Fv(T)g Fw(\))p 1242 4382 V 1344 4473 a Fv(\017)1495 4405 y(;)j(T)1610 4264 y Fp(\023)1705 4405 y Fu(\000)23 b Fv(\036)1880 4264 y Fp(\022)1952 4405 y FA(x)2011 4420 y Fs(\016)2049 4405 y Fw(\()p Fv(t)p Fw(\))p Fv(;)2214 4337 y FA(x)2273 4352 y Fs(\016)2311 4337 y Fw(\()p Fv(t)p Fw(\))p 2214 4382 209 4 v 2299 4473 a Fv(\017)2432 4405 y(;)17 b(t)2511 4264 y Fp(\023)q(\025)546 4672 y Fw(=)649 4537 y Fp(Z)705 4762 y Fn(R)753 4743 y Fm(n)795 4762 y Fl(\002)p Fn(R)898 4743 y Fm(n)961 4672 y Fv(\036)p Fw(\()p Fv(x)1112 4631 y Fl(0)1136 4672 y Fv(;)g(X)r(;)g(T)d Fw(\))p Fv(d\027)1515 4687 y Fs(\016)1552 4672 y Fw(\()p Fv(x)1645 4631 y Fl(0)1669 4672 y Fv(;)j(X)8 b Fw(\))21 b Fu(\000)1961 4537 y Fp(Z)2016 4762 y Fn(R)2064 4743 y Fm(n)2107 4762 y Fl(\002)p Fn(R)2210 4743 y Fm(n)2273 4672 y Fv(\036)p Fw(\()p Fv(x)2424 4631 y Fl(0)2447 4672 y Fv(;)c(X)r(;)g(t)p Fw(\))p Fv(d\027)2796 4631 y Fs(\017)2790 4697 y(t)2829 4672 y Fw(\()p Fv(x)2922 4631 y Fl(0)2946 4672 y Fv(;)g(X)8 b Fw(\))p Fv(;)0 4897 y Fw(where)34 b Fv(\027)336 4861 y Fs(\017)330 4921 y(t)401 4897 y Fw(is)g(tak)m(en)f(as)1010 4980 y Fp(Z)1126 5115 y Fv(\036)p Fw(\()p Fv(x)1277 5074 y Fl(0)1301 5115 y Fv(;)17 b(X)r(;)g(t)p Fw(\))g Fv(d\027)1667 5074 y Fs(\017)1661 5140 y(t)1699 5115 y Fw(\()p Fv(x)1792 5074 y Fl(0)1816 5115 y Fv(;)g(X)8 b Fw(\))27 b(:=)g Fv(\036)2219 5005 y Fp(\020)2278 5115 y Fv(x;)2387 5048 y(x)p 2387 5092 56 4 v 2395 5183 a(\017)2453 5115 y(;)17 b(t)2532 5005 y Fp(\021)2624 5115 y Fv(;)0 5334 y Fw(whic)m(h)36 b(sho)m(ws)h(that)d Fv(\026)836 5349 y Fs(\016)908 5334 y Fw(and)h Fv(\027)1148 5349 y Fs(\016)1221 5334 y Fw(satisfy)h(the)f(constrain)m(t)h(in)f(\(30\))o (.)50 b(Accordingly)-8 b(,)36 b(\(29\))e(and)h(\(31\))0 5450 y(giv)m(e)f(that)e Fv(u)468 5414 y Fs(\017)522 5450 y Fw(+)22 b Fv(\016)32 b Fu(\025)i Fw(~)-55 b Fv(u)856 5414 y Fs(\017)888 5450 y Fw(,)33 b(and)f(so)h(the)g(claim)g(follo)m (ws)h(b)m(y)f(sending)h Fv(\016)i Fw(to)d(zero.)691 b Fi(\003)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 168 251 a Fq(14)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)1472 450 y Fw(5.)48 b Fx(Moment)38 b(estima)-7 b(tes)168 624 y Fw(In)41 b(this)g(section,)j(w)m(e)d(discuss)i(some)e(a-priori)f(momen) m(t)i(estimates)g(for)e(minimizing)i(measures)168 741 y(\()p Fv(\026)265 705 y Fs(\017)297 741 y Fv(;)17 b(\027)395 705 y Fs(\017)389 766 y(T)444 741 y Fw(\),)46 b(as)e(giv)m(en)h(in)f (\(29\),)i(that)e(will)g(b)s(e)g(essen)m(tial)i(to)d(establish)i(the)f (tigh)m(tness)i(necessary)168 857 y(to)41 b(our)h(scop)s(es.)73 b(More)43 b(re\014ned)g(estimates,)j(needed)d(to)f(pro)m(v)m(e)h(that)f (the)g(homogenized)i(limit)168 973 y(is)35 b(a)g(viscosit)m(y)i (solution)f(of)e(a)h(Hamilton-Jacobi)g(equation,)h(will)g(b)s(e)f(dev)m (elop)s(ed)i(in)e(section)h(11.)168 1089 y(From)g(this)g(section)i(on,) f(w)m(e)g(will)g(assume)h(Assumptions)g(4)e(and)g(5.)54 b(By)37 b(comparison)g(with)g(the)168 1206 y(con)m(trol)c FA(v)c Fw(:=)f(0)k(in)h(\(3\))o(,)g(recalling)g(Assumption)h(5,)f(w)m (e)g(see)h(that)168 1479 y(\(32\))1459 1343 y Fp(Z)1575 1479 y Fv(L)p Fw(\()p Fv(x)1734 1438 y Fl(0)1758 1479 y Fv(;)17 b(X)r(;)g(v)t(;)g(!)t Fw(\))p Fv(d\026)2237 1438 y Fs(\017)2312 1479 y Fu(\024)45 b Fv(C)23 b(;)168 1746 y Fw(with)33 b Fv(C)i(>)27 b Fw(0)32 b(indep)s(enden)m(t)j(of)d Fv(\017)p Fw(.)168 1957 y FA(Prop)s(osition)k(5.)41 b Fo(L)-5 b(et)34 b Fw(\()p Fv(\026)1164 1921 y Fs(\017)1196 1957 y Fv(;)17 b(\027)1294 1921 y Fs(\017)1288 1982 y(T)1343 1957 y Fw(\))34 b Fo(b)-5 b(e)34 b(me)-5 b(asur)g(es)33 b(satisfying)42 b Fw(\(30\))o Fo(,)34 b(with)g Fv(\027)2893 1921 y Fs(\017)2887 1982 y(t)2960 1957 y Fo(given)f(by)43 b Fw(\(22\))o Fo(.)i(L)-5 b(et)34 b Fv(L)168 2073 y Fo(satisfy)k Fw(\(32\))o Fo(.)43 b(L)-5 b(et)30 b Fv(p)g Fo(b)-5 b(e)29 b(as)g(in)h(Assumption)f(4.)43 b(Then,)30 b(ther)-5 b(e)29 b(exists)g Fv(C)35 b(>)28 b Fw(0)p Fo(,)i(p)-5 b(ossibly)29 b(dep)-5 b(ending)168 2189 y(on)34 b Fv(t)h Fo(and)g Fv(T)14 b Fo(,)34 b(but)h(indep)-5 b(endent)34 b(of)g Fv(\017)p Fo(,)i(in)e(such)h(a)g(way)f(that)1135 2321 y Fp(Z)1251 2457 y Fu(j)p Fv(x)1334 2416 y Fl(0)1357 2457 y Fu(j)1385 2416 y Fs(p)1441 2457 y Fv(d\026)1551 2416 y Fs(\017)1605 2457 y Fw(+)1703 2321 y Fp(Z)1820 2457 y Fu(j)p Fv(x)1903 2416 y Fl(0)1926 2457 y Fu(j)1954 2416 y Fs(p)2010 2457 y Fv(d\027)2115 2416 y Fs(\017)2109 2482 y(T)2208 2457 y Fu(\024)45 b Fv(C)23 b Fw(\(1)f(+)g Fu(j)p Fv(x)p Fu(j)2741 2416 y Fs(p)2781 2457 y Fw(\))17 b Fv(:)168 2715 y Fo(Pr)-5 b(o)g(of.)41 b Fw(In)e(what)h(follo)m(ws,)h (w)m(e)g(will)e(denote)h(b)m(y)g Fv(C)2074 2730 y Fs(i)2141 2715 y Fw(suitable)g(p)s(ositiv)m(e)h(quan)m(tities,)i(whic)m(h)d(ma)m (y)168 2831 y(dep)s(end)33 b(on)g Fv(t)p Fw(,)g Fv(T)14 b Fw(,)32 b(but)h(that)f(are)h(indep)s(enden)m(t)i(of)d Fv(\017)p Fw(.)168 2947 y(W)-8 b(e)33 b(observ)m(e)h(that)168 3168 y(\(33\))862 b Fu(j)p Fv(x)1287 3127 y Fl(0)1310 3168 y Fu(j)1338 3127 y Fs(p)1405 3168 y Fu(\024)28 b Fw(\(1)22 b(+)g Fu(j)p Fv(x)1800 3127 y Fl(0)1824 3168 y Fu(j)1852 3127 y Fr(2)1891 3168 y Fw(\))1929 3127 y Fs(p=)p Fr(2)2067 3168 y Fu(\024)28 b Fw(2)2221 3127 y Fs(p=)p Fr(2)2331 3168 y Fw(\(1)22 b(+)g Fu(j)p Fv(x)2621 3127 y Fl(0)2644 3168 y Fu(j)2672 3127 y Fs(p)2711 3168 y Fw(\))17 b Fv(:)168 3391 y Fw(W)-8 b(e)37 b(apply)h(the)f(constrain)m (t)h(in)f(\(30\))g(to)f(the)i(function)f Fv(\036)p Fw(\()p Fv(x)2410 3355 y Fl(0)2433 3391 y Fw(\))f(:=)f(\(1)24 b(+)h Fu(j)p Fv(x)2940 3355 y Fl(0)2964 3391 y Fu(j)2992 3355 y Fr(2)3031 3391 y Fw(\))3069 3355 y Fs(p=)p Fr(2)3216 3391 y Fw(and)37 b(w)m(e)h(obtain)168 3508 y(that)1474 3620 y Fp(Z)1591 3756 y Fv(p)p Fw(\(1)21 b(+)h Fu(j)p Fv(x)1929 3715 y Fl(0)1953 3756 y Fu(j)1981 3715 y Fr(2)2020 3756 y Fw(\))2058 3715 y Fr(\()p Fs(p)p Fl(\000)p Fr(2\))p Fs(=)p Fr(2)2313 3756 y Fv(f)33 b Fu(\001)22 b Fv(x)2499 3715 y Fl(0)471 4042 y Fw(+)557 3975 y(1)p 557 4019 49 4 v 557 4110 a(2)694 3917 y Fs(n)643 3947 y Fp(X)632 4157 y Fs(i;j)t Fr(=1)799 4042 y Fw(\()p Fv(\033)t(\033)955 4001 y Fs(T)1010 4042 y Fw(\))1048 4057 y Fs(ij)1108 3931 y Fp(\020)1168 4042 y Fv(p)p Fw(\()p Fv(p)g Fu(\000)g Fw(2\)\(1)g(+)g Fu(j)p Fv(x)1802 4001 y Fl(0)1825 4042 y Fu(j)1853 4001 y Fr(2)1892 4042 y Fw(\))1930 4001 y Fr(\()p Fs(p)p Fl(\000)p Fr(4\))p Fs(=)p Fr(2)2185 4042 y Fv(x)2240 4001 y Fl(0)2240 4067 y Fs(i)2269 4042 y Fv(x)2324 4001 y Fl(0)2324 4067 y Fs(j)2383 4042 y Fw(+)g Fv(p)p Fw(\(1)g(+)g Fu(j)p Fv(x)2820 4001 y Fl(0)2843 4042 y Fu(j)2871 4001 y Fr(2)2911 4042 y Fw(\))2949 4001 y Fr(\()p Fs(p)p Fl(\000)p Fr(2\))p Fs(=)p Fr(2)3204 4042 y Fv(\016)3247 4057 y Fs(ij)3308 3931 y Fp(\021)3384 4042 y Fv(d\026)3494 4001 y Fs(\017)1082 4353 y Fw(=)1185 4218 y Fp(Z)1285 4353 y Fw(\(1)g(+)g Fu(j)p Fv(x)1575 4312 y Fl(0)1598 4353 y Fu(j)1626 4312 y Fr(2)1665 4353 y Fw(\))1703 4312 y Fs(p=)p Fr(2)1830 4353 y Fv(d\027)1935 4312 y Fs(\017)1929 4378 y(T)2006 4353 y Fu(\000)2106 4218 y Fp(Z)2205 4353 y Fw(\(1)g(+)g Fu(j)p Fv(x)2495 4312 y Fl(0)2518 4353 y Fu(j)2546 4312 y Fr(2)2586 4353 y Fw(\))2624 4312 y Fs(p=)p Fr(2)2750 4353 y Fv(d\027)2855 4312 y Fs(\017)2849 4378 y(t)2888 4353 y Fv(;)168 4626 y Fw(and)32 b(so,)h(b)m(y)i(\(33\),)1057 4764 y Fp(Z)1173 4899 y Fu(j)p Fv(x)1256 4858 y Fl(0)1280 4899 y Fu(j)1308 4858 y Fs(p)1347 4899 y Fv(d\027)1452 4858 y Fs(\017)1446 4924 y(T)1556 4899 y Fu(\024)1689 4764 y Fp(Z)1789 4899 y Fw(\(1)22 b(+)g Fu(j)p Fv(x)2079 4858 y Fl(0)2102 4899 y Fu(j)2130 4858 y Fr(2)2169 4899 y Fw(\))2207 4858 y Fs(p=)p Fr(2)2317 4899 y Fv(d\027)2422 4858 y Fs(\017)2416 4924 y(T)897 5158 y Fu(\024)83 b Fv(C)1127 5173 y Fr(2)1166 5048 y Fp(\020)1243 5023 y(Z)1359 5158 y Fu(j)p Fv(f)11 b Fu(j)17 b Fw(\(1)j(+)i Fu(j)p Fv(x)1779 5117 y Fl(0)1803 5158 y Fu(j)1831 5117 y Fr(2)1870 5158 y Fw(\))1908 5117 y Fr(\()p Fs(p)p Fl(\000)p Fr(1\))p Fs(=)p Fr(2)1057 5417 y Fw(+)p Fu(j)p Fv(\033)t(\033)1279 5376 y Fs(T)1334 5417 y Fu(j)17 b Fw(\(1)k(+)h Fu(j)p Fv(x)1668 5376 y Fl(0)1691 5417 y Fu(j)1719 5376 y Fr(2)1758 5417 y Fw(\))1796 5376 y Fr(\()p Fs(p)p Fl(\000)p Fr(2\))p Fs(=)p Fr(2)2052 5417 y Fv(d\026)2162 5376 y Fs(\017)2216 5417 y Fw(+)2314 5281 y Fp(Z)2414 5417 y Fw(\(1)f(+)h Fu(j)p Fv(x)2703 5376 y Fl(0)2727 5417 y Fu(j)2755 5376 y Fr(2)2794 5417 y Fw(\))2832 5376 y Fs(p=)p Fr(2)2942 5417 y Fv(d\027)3047 5376 y Fs(\017)3041 5441 y(t)3080 5306 y Fp(\021)3156 5417 y Fv(:)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(15)0 450 y Fw(Consequen)m(tly)-8 b(,)37 b(Y)-8 b(oung's)34 b(inequalit)m(y)-8 b(,)35 b(Assumption)h(4,)d(estimates)i(\(32\))e(and) g(\(33\),)h(and)f(Prop)s(o-)0 566 y(sition)g(2)f(giv)m(e)i(that)529 638 y Fp(Z)645 773 y Fu(j)p Fv(x)728 732 y Fl(0)751 773 y Fu(j)779 732 y Fs(p)819 773 y Fv(d\027)924 732 y Fs(\017)918 798 y(T)418 1046 y Fu(\024)17 b Fv(C)582 1061 y Fr(3)638 905 y Fp(\022)711 910 y(Z)828 1046 y Fu(j)p Fv(f)11 b Fu(j)943 1004 y Fs(p)1003 1046 y Fw(+)1242 978 y Fu(j)p Fv(\033)t(\033)1388 942 y Fs(T)1442 978 y Fu(j)1470 942 y Fs(p)p 1111 1023 529 4 v 1111 1114 a Fw(\(1)22 b(+)g Fu(j)p Fv(x)1401 1085 y Fl(0)1425 1114 y Fu(j)1453 1085 y Fr(2)1492 1114 y Fw(\))1530 1085 y Fs(p=)p Fr(2)1672 1046 y Fw(+)g(\(1)g(+)g Fu(j)p Fv(x)2060 1004 y Fl(0)2083 1046 y Fu(j)2111 1004 y Fr(2)2150 1046 y Fw(\))2188 1004 y Fs(p=)p Fr(2)2315 1046 y Fv(d\026)2425 1004 y Fs(\017)2479 1046 y Fw(+)2577 910 y Fp(Z)2677 1046 y Fw(\(1)g(+)g Fu(j)p Fv(x)2967 1004 y Fl(0)2990 1046 y Fu(j)3018 1004 y Fr(2)3057 1046 y Fw(\))3095 1004 y Fs(p=)p Fr(2)3206 1046 y Fv(d\027)3311 1004 y Fs(\017)3305 1070 y(t)3343 905 y Fp(\023)418 1322 y Fu(\024)17 b Fv(C)582 1337 y Fr(4)638 1182 y Fp(\022)711 1186 y(Z)828 1322 y Fw(1)k(+)h Fv(L)h Fw(+)f Fu(j)p Fv(x)1266 1281 y Fl(0)1289 1322 y Fu(j)1317 1281 y Fs(p)1373 1322 y Fv(d\026)1483 1281 y Fs(\017)1538 1322 y Fw(+)1636 1186 y Fp(Z)1752 1322 y Fw(1)g(+)g Fu(j)p 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Fl(0)2773 2438 y Fu(j)2801 2397 y Fr(2)2840 2438 y Fw(\))2878 2397 y Fr(\()p Fs(p)p Fl(\000)p Fr(2\))p Fs(=)p Fr(2)3133 2438 y Fv(\016)3176 2453 y Fs(ij)3237 2327 y Fp(\021)q(i)3360 2438 y Fv(d\026)3470 2397 y Fs(\017)242 2749 y Fw(=)83 b(\()p Fv(T)36 b Fu(\000)22 b Fv(t)p Fw(\))721 2614 y Fp(Z)821 2749 y Fw(\(1)g(+)g Fu(j)p Fv(x)1111 2708 y Fl(0)1134 2749 y Fu(j)1162 2708 y Fr(2)1201 2749 y Fw(\))1239 2708 y Fs(p=)p Fr(2)1349 2749 y Fv(d\027)1454 2708 y Fs(\017)1448 2774 y(t)1487 2749 y Fv(:)0 2962 y Fw(Th)m(us,)34 b(using)h(\(33\))d(and)g(a)h(scaled)h(Y)-8 b(oung's)33 b(inequalit)m(y)-8 b(,)34 b(w)m(e)g(obtain)e(that)893 3039 y Fp(Z)992 3174 y Fw(\(1)22 b(+)g Fu(j)p Fv(x)1282 3133 y Fl(0)1305 3174 y Fu(j)1333 3133 y Fr(2)1373 3174 y Fw(\))1411 3133 y Fs(p=)p Fr(2)1537 3174 y Fv(d\026)1647 3133 y Fs(\017)732 3446 y Fu(\024)84 b Fv(C)963 3461 y Fr(6)1018 3306 y Fp(\022)1092 3446 y Fw(1)22 b(+)1261 3311 y Fp(Z)1377 3446 y Fu(j)p Fv(x)1460 3405 y Fl(0)1483 3446 y Fu(j)1511 3405 y 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Fw(,)h(see)h(\(22\))o(,)f(yield)g(the)g(desired)i(result.)688 b Fi(\003)0 5075 y Fw(As)43 b(a)f(consequence)k(of)c(this)h(result,)j (using)e(\(22\),)h(w)m(e)e(ha)m(v)m(e,)k(using)c(Prop)s(osition)g(3,)h (that)f(an)m(y)0 5191 y(minimizing)34 b(sequence)h(\()p Fv(\026)1005 5155 y Fs(\017)1005 5216 y(n)1052 5191 y Fv(;)17 b(\027)1150 5155 y Fs(\017)1144 5216 y(T)6 b(;n)1257 5191 y Fw(\))33 b(will)g(satisfy)h(the)f(b)s(ound)0 5417 y(\(35\))644 5281 y Fp(Z)761 5417 y Fu(j)p Fv(x)844 5376 y Fl(0)867 5417 y Fu(j)895 5376 y Fs(p)956 5417 y Fw(+)22 b Fv(\017)1093 5376 y Fs(p)1133 5417 y Fu(j)p Fv(X)8 b Fu(j)1278 5376 y Fs(p)1317 5417 y Fv(d\026)1427 5376 y Fs(\017)1427 5441 y(n)1501 5417 y Fu(\024)28 b Fv(C)r(;)1917 5281 y Fp(Z)2033 5417 y Fu(j)p Fv(x)2116 5376 y Fl(0)2139 5417 y Fu(j)2167 5376 y Fs(p)2229 5417 y Fw(+)22 b Fv(\017)2366 5376 y Fs(p)2406 5417 y Fu(j)p Fv(X)8 b Fu(j)2551 5376 y Fs(p)2590 5417 y Fv(d\027)2695 5376 y Fs(\017)2689 5441 y(T)e(;n)2830 5417 y Fu(\024)28 b Fv(C)r(:)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 168 251 a Fq(16)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y Fw(The)38 b(ab)s(o)m(v)m(e)f(tigh)m(tness)i(and)e(Assumption)i(8)d (imply)i(that)f(there)h(are)f(minimizing)h(measures)h Fv(\026)3797 414 y Fs(\017)168 566 y Fw(and)32 b Fv(\027)411 530 y Fs(\017)405 592 y(T)493 566 y Fw(attaining)g(the)h(in\014m)m(um)i (in)d(\(29\).)1733 1099 y(6.)49 b Fx(Duality)168 1274 y Fw(In)32 b(addition)g(to)g(the)h(Hamiltonian)f Fv(H)39 b Fw(as)33 b(de\014ned)g(in)f(\(6\),)g(w)m(e)h(need)h(also)e(the)g (extended)i(Hamil-)168 1390 y(tonian)e(op)s(erator)168 1667 y(\(36\))905 1642 y(~)880 1667 y Fv(H)7 b Fw(\()p Fv(D)1090 1626 y Fr(2)1129 1667 y Fv(\036)p Fw(\()p Fv(x;)17 b(X)8 b Fw(\))p Fv(;)17 b(D)s(\036)p Fw(\()p Fv(x;)g(X)8 b Fw(\))p Fv(;)17 b(x;)g(X)r(;)g(\017;)g(!)t Fw(\))42 b(:=)28 b(sup)2534 1748 y Fs(v)r Fl(2)p Fs(U)2693 1667 y Fu(\000)2797 1642 y Fw(~)2770 1667 y Fv(A)q(\036)22 b Fu(\000)h Fv(L;)168 2008 y Fw(whic)m(h)34 b(is)f(de\014ned)h(for)e Fv(\036)27 b Fu(2)h Fv(C)1286 1972 y Fr(2)1326 2008 y Fw(\()p Ft(R)1436 1972 y Fs(n)1505 2008 y Fu(\002)22 b Ft(R)1676 1972 y Fs(n)1723 2008 y Fw(\).)44 b(It)32 b(will)i(b)s(e)e(useful)i(later)f(the)g(observ)-5 b(ation)33 b(that)1227 2265 y(~)1202 2291 y Fv(H)7 b Fw(\()p Fv(D)1412 2249 y Fr(2)1451 2291 y Fv(\036)1509 2306 y Fs(\025)1555 2291 y Fw(\()p Fv(x;)17 b(X)8 b Fw(\))p Fv(;)17 b(D)s(\036)2005 2306 y Fs(\025)2049 2291 y Fw(\()p Fv(x;)g(X)8 b Fw(\))p Fv(;)17 b(x;)g(X)r(;)g(\017;)g(!)t Fw(\))p Fv(;)168 2565 y Fw(with)168 2842 y(\(37\))1064 b Fv(\036)1464 2857 y Fs(\025)1536 2842 y Fw(=)28 b Fv(\036)p Fw(\()p Fv(x;)17 b(X)8 b Fw(\))22 b(+)g Fv(\025')p Fw(\()p Fv(X)29 b Fu(\000)2461 2774 y Fv(x)p 2461 2819 56 4 v 2469 2910 a(\017)2526 2842 y Fw(\))p Fv(;)168 3155 y Fw(is)k(constan)m(t)g(in)g Fv(\025)p Fw(,)g(for)f(an)m(y)h Fv(C)1300 3119 y Fr(2)1372 3155 y Fw(function)g Fv(')p Fw(,)f(thanks)i(to)e(\(20\).)168 3271 y(Recalling)j(the)f(framew)m(ork)i(in)m(tro)s(duced)f(on)g(page)f (10,)h(w)m(e)g(de\014ne)h(the)f(set)g Fv(C)3091 3235 y Fr(2)3084 3297 y(\003)3171 3271 y Fw(to)f(b)s(e)h(the)f(set)i(of)168 3388 y(functions)29 b Fv(\036)e Fu(2)i Fv(C)841 3352 y Fr(2)880 3388 y Fw(\()p Ft(R)990 3352 y Fs(n)1050 3388 y Fu(\002)14 b Ft(R)1213 3352 y Fs(n)1274 3388 y Fu(\002)g Fw([)p Fv(t;)j(T)d Fw(]\))28 b(suc)m(h)i(that)e Fv(\036)p Fw(\()p Fv(x)2209 3352 y Fl(0)2233 3388 y Fv(;)17 b(X)r(;)g(t)p Fw(\))p Fv(;)g(\036)p Fw(\()p Fv(x)2672 3352 y Fl(0)2695 3388 y Fv(;)g(X)r(;)g(T)d Fw(\))27 b Fu(2)h Fv(C)3166 3403 y Fs(])3197 3388 y Fw(\()p Ft(R)3307 3352 y Fs(n)3368 3388 y Fu(\002)14 b Ft(R)3531 3352 y Fs(n)3578 3388 y Fw(\),)29 b(and)168 3515 y Fv(@)219 3530 y Fs(t)249 3515 y Fv(\036)22 b Fw(+)452 3490 y(~)427 3515 y Fv(A)500 3479 y Fs(v)541 3515 y Fv(\036)27 b Fu(2)h Fv(C)790 3530 y Fl(\003)829 3515 y Fw(\()p Ft(R)939 3479 y Fs(n)1008 3515 y Fu(\002)23 b Ft(R)1180 3479 y Fs(n)1249 3515 y Fu(\002)g Fv(U)33 b Fu(\002)22 b Fw([)p Fv(t;)17 b(T)d Fw(]\).)168 3631 y(In)35 b(order)g(to)g(pro)m(v)m(e)i(the)e(rev)m(erse) i(inequalit)m(y)g(of)e(the)h(one)f(in)g(Prop)s(osition)h(4,)f(w)m(e)i (in)m(tro)s(duce)f(the)168 3748 y(dual)c(problem)i(of)39 b(\(29\))o(:)168 3989 y FA(Prop)s(osition)e(6.)42 b Fo(In)34 b(the)h(notation)g(of)f(Pr)-5 b(op)g(osition)34 b(4,)h(we)f(have)h (that)561 4263 y Fw(~)-55 b Fv(u)611 4222 y Fs(\017)643 4263 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))27 b(=)-897 b(\(38\))743 4484 y(sup)671 4570 y Fs(\036)p Fr(\()p Fs(x)780 4551 y Fj(0)803 4570 y Fs(;X)q(;s)p Fr(\))962 4484 y Fw(\()p Fv(T)35 b Fu(\000)23 b Fv(t)p Fw(\))1282 4344 y Fp(\024)1644 4484 y Fw(inf)1334 4551 y Fr(\()p Fs(x)1401 4532 y Fj(0)1424 4551 y Fs(;X)q(;s)p Fr(\))p Fl(2)p Fn(R)1678 4532 y Fm(n)1720 4551 y Fl(\002)p Fn(R)1823 4532 y Fm(n)1865 4551 y Fl(\002)p Fr(\()p Fs(t;T)10 b Fr(\))2088 4484 y Fv(@)2139 4499 y Fs(t)2169 4484 y Fv(\036)22 b Fu(\000)2374 4459 y Fw(~)2348 4484 y Fv(H)2453 4404 y Fp(\000)2499 4484 y Fv(D)2583 4443 y Fr(2)2622 4484 y Fv(\036;)17 b(D)s(\036;)g(x)2965 4443 y Fl(0)2987 4484 y Fv(;)g(X)r(;)g(\017;)g(!)3306 4404 y Fp(\001)3351 4344 y(\025)777 4770 y Fw(+)35 b(inf)875 4833 y Fs(x)915 4814 y Fj(0)937 4833 y Fs(;X)1036 4770 y Fw([)p Fv( )t Fw(\()p Fv(x)1223 4728 y Fl(0)1247 4770 y Fv(;)17 b(X)r(;)g(!)t Fw(\))k Fu(\000)i Fv(\036)p Fw(\()p Fv(x)1793 4728 y Fl(0)1816 4770 y Fv(;)17 b(X)r(;)g(T)d Fw(\)])22 b(+)2243 4634 y Fp(Z)2298 4860 y Fn(R)2346 4841 y Fm(n)2389 4860 y Fl(\002)p Fn(R)2492 4841 y Fm(n)2555 4770 y Fv(\036)p Fw(\()p Fv(x)2706 4728 y Fl(0)2729 4770 y Fv(;)17 b(X)r(;)g(t)p Fw(\))g Fv(d\027)3095 4728 y Fs(\017)3089 4794 y(t)3127 4770 y Fw(\()p Fv(x)3220 4728 y Fl(0)3244 4770 y Fv(;)g(X)8 b Fw(\))p Fv(:)168 5112 y Fo(wher)-5 b(e)34 b(the)h(supr)-5 b(emum)34 b(is)h(taken)f(over)h(any)g Fv(\036)27 b Fu(2)h Fv(C)2099 5076 y Fr(2)2092 5137 y(\003)2145 5112 y Fo(.)168 5353 y Fw(T)-8 b(o)23 b(heuristically)i(motiv)-5 b(ate)23 b(the)h(statemen)m(t)g(in)f(\(38\),)i(w)m(e)f(apply)f(the)h(minimax)g (principle,)j(that)22 b(is,)168 5469 y(w)m(e)29 b(exc)m(hange)i (formally)e(the)g(in\014m)m(um)h(with)g(the)f(suprem)m(um)i(in)e(the)g (v)-5 b(ariational)28 b(problem)i(\(29\))o(,)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(17)0 450 y Fw(b)m(y)33 b(using)h(the)f(constrain)m(t)g(in)g(\(30\))o (,)g(th)m(us)h(obtaining:)696 619 y(~)-55 b Fv(u)746 578 y Fs(\017)778 619 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))531 826 y(=)112 b(inf)690 887 y Fs(\026)732 868 y Fm(\017)763 887 y Fs(;\027)822 864 y Fm(\017)818 910 y(T)950 826 y Fw(sup)882 912 y Fs(\036)p Fr(\()p Fs(x)991 893 y Fj(0)1014 912 y Fs(;X)q(;t)p Fr(\))1182 685 y Fp(\024)1234 690 y(Z)1290 916 y Fn(R)1338 897 y Fm(n)1380 916 y Fl(\002)p Fn(R)1483 897 y Fm(n)1525 916 y Fl(\002)p Fs(U)7 b Fl(\002)p Fr(\012)p Fl(\002)p Fr(\()p Fs(t;T)j Fr(\))1968 826 y Fv(L)p Fw(\()p Fv(x)2127 785 y Fl(0)2151 826 y Fv(;)17 b(X)r(;)g(v)t(;)g(!)t Fw(\))j(+)2664 801 y(~)2638 826 y Fv(A\036)i Fw(+)g Fv(@)2940 841 y Fs(t)2970 826 y Fv(\036)17 b(d\026)3155 785 y Fs(\017)700 1120 y Fw(+)793 984 y Fp(Z)848 1210 y Fn(R)896 1191 y Fm(n)938 1210 y Fl(\002)p Fn(R)1041 1191 y Fm(n)1084 1210 y Fl(\002)p Fr(\012)1194 1120 y Fw(\()p Fv( )26 b Fu(\000)c Fv(\036)p Fw(\))p Fv(d\027)1621 1079 y Fs(\017)1615 1145 y(T)1692 1120 y Fw(+)1790 984 y Fp(Z)1845 1210 y Fn(R)1893 1191 y Fm(n)1936 1210 y Fl(\002)p Fn(R)2039 1191 y Fm(n)2081 1210 y Fl(\002)p Fr(\012)2208 1120 y Fv(\036d\027)2371 1079 y Fs(\017)2365 1145 y(t)2403 980 y Fp(\025)690 1411 y Fw(\(using)33 b(the)g(minimax)h(principle\))531 1738 y(=)155 b(sup)690 1823 y Fs(\036)p Fr(\()p Fs(x)799 1805 y Fj(0)822 1823 y Fs(;X)q(;s)p Fr(\))997 1597 y Fp(\024)1049 1602 y(Z)1105 1828 y Fn(R)1153 1809 y Fm(n)1195 1828 y Fl(\002)p Fn(R)1298 1809 y Fm(n)1340 1828 y Fl(\002)p Fr(\012)1467 1738 y Fv(\036d\027)1630 1697 y Fs(\017)1624 1762 y(t)690 2023 y Fw(+)17 b(inf)805 2083 y Fs(\026)847 2064 y Fm(\017)917 1887 y Fp(Z)972 2113 y Fn(R)1020 2094 y Fm(n)1063 2113 y Fl(\002)p Fn(R)1166 2094 y Fm(n)1208 2113 y Fl(\002)p Fs(U)7 b Fl(\002)p Fr(\012)p Fl(\002)p Fr(\()p Fs(t;T)j Fr(\))1650 2023 y Fv(L)p Fw(\()p Fv(x)1809 1982 y Fl(0)1833 2023 y Fv(;)17 b(X)r(;)g(v)t(;)g(!)t Fw(\))k(+)2347 1998 y(~)2321 2023 y Fv(A\036)h Fw(+)g Fv(@)2623 2038 y Fs(t)2653 2023 y Fv(\036)17 b(d\026)2838 1982 y Fs(\017)700 2311 y Fw(+)g(inf)810 2372 y Fs(\027)849 2349 y Fm(\017)845 2395 y(T)927 2176 y Fp(Z)982 2401 y Fn(R)1030 2382 y Fm(n)1073 2401 y Fl(\002)p Fn(R)1176 2382 y Fm(n)1218 2401 y Fl(\002)p Fr(\012)1328 2311 y Fw(\()p Fv( )26 b Fu(\000)d Fv(\036)p Fw(\))p Fv(d\027)1756 2270 y Fs(\017)1750 2336 y(T)1805 2171 y Fp(\025)1874 2311 y Fv(;)0 2539 y Fw(and)30 b(then,)h(taking)f(in)m(to)g(accoun)m(t)h(that)e(b)s(oth)h Fv(\026)1780 2502 y Fs(\017)1842 2539 y Fw(and)g Fv(\027)2083 2502 y Fs(\017)2077 2564 y(T)2161 2539 y Fw(are)g(measures,)j(with)d Fv(\027)3038 2502 y Fs(\017)3032 2564 y(T)3117 2539 y Fw(a)f(probabilit)m(y)0 2663 y(measure,)34 b(and)f Fv(\026)656 2627 y Fs(\017)721 2663 y Fw(with)g(mass)g Fv(T)j Fu(\000)23 b Fv(t)p Fw(,)33 b(the)g(de\014nition)g(of)2208 2638 y(~)2182 2663 y Fv(A)g Fw(and)2503 2638 y(~)2478 2663 y Fv(H)40 b Fw(w)m(e)33 b(obtain)g(\(38\))o(.)0 2844 y Fo(Pr)-5 b(o)g(of)35 b(\(of)f(Pr)-5 b(op)g(osition)34 b(6\).)41 b Fw(The)26 b(rigorous)f(pro)s(of)f(of)h(Prop)s(osition)g(6)g (will)h(b)s(e)f(quite)h(long)f(\(ending)0 2960 y(on)31 b(page)g(20\))f(and)h(it)g(will)h(mak)m(e)g(use)g(of)e(the)h (Legendre-F)-8 b(enc)m(hel-Ro)s(c)m(k)j(afellar)33 b(Theorem.)44 b(Let)31 b Fu(P)0 3077 y Fw(and)i Fu(Q)g Fw(b)s(e)f(de\014ned)i(as)f (in)g(page)g(10.)43 b(De\014ne)33 b(the)g(follo)m(wing)g(sets:)478 3301 y Fu(M)598 3316 y Fr(1)665 3301 y Fw(:=)796 3161 y Fp(\032)870 3301 y Fw(\()p Fv(\026;)17 b(\027)6 b Fw(\))28 b Fu(2)g(P)i(\002)23 b(Q)28 b Fw(:)1588 3166 y Fp(Z)1704 3301 y Fv(d\026)f Fw(=)h Fv(T)36 b Fu(\000)22 b Fv(t;)2249 3166 y Fp(Z)2365 3301 y Fv(d\027)34 b Fw(=)27 b(1)p Fv(;)49 b(\026;)17 b(\027)34 b Fu(\025)28 b Fw(0)3065 3161 y Fp(\033)3156 3301 y Fv(;)0 3526 y Fw(and)286 3732 y Fu(M)406 3747 y Fr(2)473 3732 y Fw(:=)604 3591 y Fp(\032)678 3732 y Fw(\()p Fv(\026;)17 b(\027)6 b Fw(\))28 b Fu(2)g(P)i(\002)23 b(Q)28 b Fw(:)1396 3596 y Fp(Z)1538 3707 y Fw(~)1512 3732 y Fv(A\036)22 b Fw(+)g Fv(@)1814 3747 y Fs(t)1844 3732 y Fv(\036d\026)27 b Fw(=)2143 3596 y Fp(Z)2259 3732 y Fv(\036d\027)h Fu(\000)2544 3596 y Fp(Z)2660 3732 y Fv(\036d\027)2823 3691 y Fs(\017)2817 3757 y(t)2855 3732 y Fv(;)17 b Fu(8)p Fv(\036)28 b Fu(2)g Fv(C)3211 3691 y Fr(2)3204 3757 y(\003)3257 3591 y Fp(\033)3348 3732 y Fv(:)0 3962 y Fw(Giv)m(en)33 b Fv(\037)28 b Fu(2)g Fv(C)535 3977 y Fl(\003)575 3962 y Fw(\()p Ft(R)685 3926 y Fs(n)754 3962 y Fu(\002)22 b Ft(R)925 3926 y Fs(n)994 3962 y Fu(\002)h Fv(U)33 b Fu(\002)22 b Fw([)p Fv(t;)17 b(T)d Fw(]\))33 b(and)f Fv(\036)c Fu(2)g Fv(C)2006 3977 y Fs(?)2045 3962 y Fw(\()p Ft(R)2155 3926 y Fs(n)2224 3962 y Fu(\002)23 b Ft(R)2396 3926 y Fs(n)2465 3962 y Fu(\002)f Fw([)p Fv(t;)17 b(T)d Fw(]\),)33 b(w)m(e)g(set)544 4131 y Fv(h)p Fw(\()p Fv(\037;)17 b(\036)p Fw(\))83 b(:=)g(\()p Fv(T)36 b Fu(\000)22 b Fv(t)p Fw(\))85 b(sup)1428 4217 y Fr(\()p Fs(x;X)q(;v)r(;s)p Fr(\))1727 4131 y Fw([)p Fu(\000)p Fv(\037)p Fw(\()p Fv(x;)17 b(X)r(;)g(v)t(;)g(s)p Fw(\))22 b Fu(\000)h Fv(L)p Fw(\()p Fv(x;)17 b(X)r(;)g(v)t(;)g(s;)g(!)t Fw(\)])1108 4359 y(+)43 b(sup)1201 4445 y Fr(\()p Fs(x)1268 4426 y Fj(0)1290 4445 y Fs(;X)5 b Fr(\))1400 4359 y Fw([)p Fv( )t Fw(\()p Fv(x)1587 4318 y Fl(0)1611 4359 y Fv(;)17 b(X)r(;)g(!)t Fw(\))k Fu(\000)i Fv(\036)p Fw(\()p Fv(x)2157 4318 y Fl(0)2180 4359 y Fv(;)17 b(X)r(;)g(T)d Fw(\)])j Fv(:)-2531 b Fw(\(39\))0 4602 y(Note)33 b(that,)f(for)g(short,)h(w)m(e) h(ha)m(v)m(e)g(omitted)f(the)g(dep)s(endence)i(of)d Fv(h)h Fw(on)f Fv(!)t Fw(.)43 b(W)-8 b(e)33 b(consider)h(the)f(set)791 4797 y Fu(C)h Fw(=)27 b(cl)1067 4686 y Fp(n)1134 4797 y Fw(\()p Fv(\037;)17 b(\036)p Fw(\))59 b(:)i Fv(\036)27 b Fu(2)h Fv(C)1776 4756 y Fr(2)1769 4821 y(\003)1855 4797 y Fv(;)49 b(\037)28 b Fw(=)2149 4772 y(~)2123 4797 y Fv(A\036)22 b Fw(+)g Fv(@)2425 4812 y Fs(t)2455 4797 y Fv(\036)28 b Fu(2)g Fv(C)2705 4812 y Fl(\003)2744 4686 y Fp(o)2844 4797 y Fv(;)0 5002 y Fw(where)34 b(the)f(closure)h(is)f (tak)m(en)g(in)g(the)g Fv(C)1498 4965 y Fr(2)1491 5027 y(\003)1566 5002 y Fu(\002)23 b Fv(C)1736 5017 y Fl(\003)1807 5002 y Fw(top)s(ology)-8 b(.)0 5118 y(Let)33 b(\()p Fv(\026)272 5133 y Fl(\003)311 5118 y Fv(;)17 b(\027)403 5133 y Fl(\003)442 5118 y Fw(\))28 b Fu(2)g(M)722 5133 y Fr(1)783 5118 y Fu(\\)23 b(M)992 5133 y Fr(2)1063 5118 y Fw(and)33 b(de\014ne)677 5378 y Fv(g)t Fw(\()p Fv(\037;)17 b(\036)p Fw(\))27 b(:=)1124 5208 y Fp(\()1205 5231 y(R)1288 5311 y Fv(\037d\026)1459 5326 y Fl(\003)1520 5311 y Fu(\000)1619 5231 y Fp(R)1702 5311 y Fv(\036)p Fw(\()p Fv(x;)17 b(X)r(;)g(T)d Fw(\))p Fv(d\027)2232 5326 y Fl(\003)2466 5311 y Fw(if)33 b(\()p Fv(\037;)17 b(\036)p Fw(\))27 b Fu(2)h(C)1205 5451 y(\0001)1084 b Fw(otherwise)r Fv(:)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 168 251 a Fq(18)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y Fw(W)h(e)33 b(will)g(no)m(w)g(compute)h(the)f(dual)g(functions)g(of)f Fv(h)h Fw(and)g Fv(g)t Fw(,)e(that)i(is)900 677 y Fv(h)956 636 y Fl(\003)996 677 y Fw(\()p Fv(\026;)17 b(\027)6 b Fw(\))44 b(:=)171 b(sup)1404 763 y Fr(\()p Fs(\037;\036)p Fr(\))p Fl(2)p Fs(C)1661 771 y Fj(\003)1697 763 y Fl(\002)p Fs(C)1802 771 y Fm(?)1855 677 y Fu(\000)1949 541 y Fp(Z)2065 677 y Fv(\037d\026)22 b Fu(\000)2357 541 y Fp(Z)2474 677 y Fv(\036d\027)28 b Fu(\000)22 b Fv(h)p Fw(\()p Fv(\037;)17 b(\036)p Fw(\))g Fv(;)906 962 y(g)957 921 y Fl(\003)996 962 y Fw(\()p Fv(\026;)g(\027)6 b Fw(\))44 b(:=)186 b(inf)1404 1029 y Fr(\()p Fs(\037;\036)p Fr(\))p Fl(2)p Fs(C)1661 1037 y Fj(\003)1697 1029 y Fl(\002)p Fs(C)1802 1037 y Fm(?)1855 962 y Fu(\000)1949 826 y Fp(Z)2065 962 y Fv(\037d\026)22 b Fu(\000)2357 826 y Fp(Z)2474 962 y Fv(\036d\027)28 b Fu(\000)22 b Fv(g)t Fw(\()p Fv(\037;)17 b(\036)p Fw(\))g Fv(:)168 823 y Fw(\(40\))168 1204 y FA(Lemma)39 b(7.)j Fo(We)35 b(have)168 1450 y Fw(\(41\))661 b Fv(h)1059 1409 y Fl(\003)1098 1450 y Fw(\()p Fv(\026;)17 b(\027)6 b Fw(\))28 b(=)1462 1279 y Fp(\()1543 1303 y(R)1626 1383 y Fv(Ld\026)21 b Fw(+)1921 1303 y Fp(R)2004 1383 y Fv( )t(d\027)206 b Fo(if)35 b Fw(\()p Fv(\026;)17 b(\027)6 b Fw(\))27 b Fu(2)h(M)2945 1398 y Fr(1)1543 1522 y Fw(+)p Fu(1)657 b Fo(otherwise)168 1715 y(and)168 1961 y Fw(\(42\))892 b Fv(g)1285 1920 y Fl(\003)1324 1961 y Fw(\()p Fv(\026;)17 b(\027)6 b Fw(\))27 b(=)1688 1791 y Fp(\()1768 1894 y Fw(0)327 b Fo(if)35 b Fw(\()p Fv(\026;)17 b(\027)6 b Fw(\))28 b Fu(2)g(M)2714 1909 y Fr(2)1768 2034 y Fu(\0001)199 b Fo(otherwise)p Fv(:)168 2240 y Fo(Pr)-5 b(o)g(of.)41 b Fw(W)-8 b(e)29 b(\014rst)g(claim)h(that)e(if)h(either)h Fv(\026)e Fw(or)h Fv(\027)35 b Fw(are)29 b(non-p)s(ositiv)m(e)g(then)h Fv(h)2916 2204 y Fl(\003)2955 2240 y Fw(\()p Fv(\026;)17 b(\027)6 b Fw(\))28 b(=)f(+)p Fu(1)p Fw(.)42 b(T)-8 b(o)29 b(see)168 2356 y(this,)k(w)m(e)h(c)m(ho)s(ose)f(a)f(sequence)k(of)c (non-negativ)m(e)h(functions)h(\()p Fv(\037)2529 2371 y Fs(j)2565 2356 y Fv(;)17 b(\036)2667 2371 y Fs(j)2703 2356 y Fw(\))33 b(suc)m(h)h(that)1392 2591 y Fu(\000)1486 2456 y Fp(Z)1603 2591 y Fv(\037)1664 2606 y Fs(j)1700 2591 y Fv(d\026)22 b Fu(\000)1931 2456 y Fp(Z)2048 2591 y Fv(\036)2106 2606 y Fs(j)2142 2591 y Fv(d\027)34 b Fu(!)27 b Fw(+)p Fu(1)p Fv(:)168 2818 y Fw(Since)33 b Fv(L)28 b Fu(\025)h Fw(0)j(w)m(e)i(ha)m(v)m(e)f(that)1631 2957 y(sup)17 b Fu(\000)p Fv(\037)1932 2972 y Fs(j)1991 2957 y Fu(\000)23 b Fv(L)28 b Fu(\024)g Fw(0)p Fv(;)168 3116 y Fw(and,)k(furthermore,)1572 3255 y(sup)17 b Fv( )26 b Fu(\000)d Fv(\036)1982 3270 y Fs(j)2046 3255 y Fu(\024)28 b Fw(sup)17 b Fv( )k(:)168 3414 y Fw(As)33 b(a)f(consequence,)k(since)e Fv( )i Fw(is)d(b)s(ounded,)1274 3594 y Fv(h)p Fw(\()p Fv(\037)1429 3609 y Fs(j)1465 3594 y Fv(;)17 b(\036)1567 3609 y Fs(j)1603 3594 y Fw(\))28 b Fu(\024)g(\000)p Fw(\()p Fv(T)36 b Fu(\000)23 b Fv(t)p Fw(\))17 b(inf)23 b Fv(L)g Fw(+)f(sup)17 b Fv( )168 3774 y Fw(is)33 b(b)s(ounded)g(from)f(ab)s(o)m (v)m(e)i(uniformly)f(in)g Fv(j)6 b Fw(,)33 b(and)f(so)h Fv(h)2200 3737 y Fl(\003)2240 3774 y Fw(\()p Fv(\026;)17 b(\027)6 b Fw(\))27 b(=)h(+)p Fu(1)p Fw(.)168 3890 y(No)m(w)33 b(w)m(e)g(claim)h(that)e(the)h(follo)m(wing)g(inequalit)m(y)h(holds:) 538 4117 y Fv(h)594 4075 y Fl(\003)634 4117 y Fw(\()p Fv(\026;)17 b(\027)6 b Fw(\))27 b Fu(\025)1000 3981 y Fp(Z)1116 4117 y Fv(Ld\026)22 b Fw(+)1412 3981 y Fp(Z)1528 4117 y Fv( )t(d\027)1084 4384 y Fw(+)g(sup)1242 4465 y Fr(~)-43 b Fs(\037)1345 4244 y Fp(\024)1398 4249 y(Z)1526 4384 y Fw(~)-61 b Fv(\037d\026)22 b Fu(\000)h Fw(\()p Fv(T)35 b Fu(\000)23 b Fv(t)p Fw(\))17 b(sup)29 b(~)-61 b Fv(\037)2351 4244 y Fp(\025)2426 4384 y Fw(+)22 b(sup)2586 4465 y Fr(~)2576 4482 y Fs(\036)2687 4244 y Fp(\024)2740 4249 y(Z)2869 4358 y Fw(~)2856 4384 y Fv(\036d\027)28 b Fu(\000)23 b Fw(sup)3317 4358 y(~)3304 4384 y Fv(\036)3362 4244 y Fp(\025)3431 4384 y Fv(:)168 4267 y Fw(\(43\))168 4651 y(T)-8 b(o)29 b(establish)i(\(43\))o(,)g(consider)f(a)f(sequence)j Fv(L)1859 4666 y Fs(j)1924 4651 y Fu(\025)c Fw(inf)c Fv(L)k Fw(=)f(0)i(\(see)i(Assumption)g(1\))e(of)g(compactly)168 4768 y(supp)s(orted)35 b(functions,)i(increasing)f(p)s(oin)m(t)m(wise)g (to)f Fv(L)p Fw(.)50 b(F)-8 b(or)34 b(an)m(y)i Fv(\037)c Fu(2)f Fv(C)2859 4783 y Fl(\003)2933 4768 y Fw(and)k Fv(\036)c Fu(2)h Fv(C)3382 4783 y Fs(?)3422 4768 y Fw(,)j(de\014ne)48 b(~)-61 b Fv(\037)168 4888 y Fw(and)370 4862 y(~)357 4888 y Fv(\036)33 b Fw(b)m(y)g Fv(\037)28 b Fw(=)f Fu(\000)p Fv(L)918 4903 y Fs(j)978 4888 y Fu(\000)34 b Fw(~)-60 b Fv(\037)p Fw(,)32 b(and)h Fv(\036)27 b Fw(=)h Fu(\000)p Fv( )f Fu(\000)1856 4862 y Fw(~)1843 4888 y Fv(\036)p Fw(.)43 b(Note)33 b(that)44 b(~)-61 b Fv(\037)28 b Fu(2)g Fv(C)2671 4903 y Fl(\003)2743 4888 y Fw(and)2945 4862 y(~)2933 4888 y Fv(\036)f Fu(2)h Fv(C)3182 4903 y Fs(?)3221 4888 y Fw(,)33 b(so)g(that)883 5132 y Fv(h)939 5091 y Fl(\003)979 5132 y Fw(\()p Fv(\026;)17 b(\027)6 b Fw(\))27 b Fu(\025)h Fw(sup)1373 5230 y Fr(~)-43 b Fs(\037;)1438 5212 y Fr(~)1429 5230 y Fs(\036)1508 4991 y Fp(\024)1560 4996 y(Z)1677 5132 y Fv(L)1743 5147 y Fs(j)1780 5132 y Fv(d\026)21 b Fw(+)2009 4996 y Fp(Z)2137 5132 y Fw(~)-61 b Fv(\037d\026)22 b Fw(+)2416 4996 y Fp(Z)2532 5132 y Fv( )t(d\027)29 b Fw(+)2825 4996 y Fp(Z)2953 5106 y Fw(~)2941 5132 y Fv(\036)o(d\027)1433 5405 y Fu(\000)p Fw(\()p Fv(T)37 b Fu(\000)22 b Fv(t)p Fw(\))17 b(sup)g([)p Fv(L)2087 5420 y Fs(j)2147 5405 y Fw(+)33 b(~)-60 b Fv(\037)22 b Fu(\000)g Fv(L)p Fw(])h Fu(\000)g Fw(sup)2819 5378 y(~)2806 5405 y Fv(\036)2864 5294 y Fp(i)2927 5405 y Fv(:)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(19)0 450 y Fw(Since)34 b Fv(L)321 465 y Fs(j)380 450 y Fu(\000)22 b Fv(L)28 b Fu(\024)h Fw(0)j(w)m(e)h(gather)g(that)371 669 y Fv(h)427 628 y Fl(\003)466 669 y Fw(\()p Fv(\026;)17 b(\027)6 b Fw(\))28 b Fu(\025)832 534 y Fp(Z)948 669 y Fv(L)1014 684 y Fs(j)1051 669 y Fv(d\026)21 b Fw(+)1280 534 y Fp(Z)1397 669 y Fv( )t(d\027)916 937 y Fw(+)h(sup)1074 1017 y Fr(~)-43 b Fs(\037)1178 796 y Fp(\024)1230 801 y(Z)1358 937 y Fw(~)-60 b Fv(\037d\026)21 b Fu(\000)i Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(t)p Fw(\))17 b(sup)29 b(~)-61 b Fv(\037)2184 796 y Fp(\025)2258 937 y Fw(+)22 b(sup)2418 1017 y Fr(~)2409 1035 y Fs(\036)2520 796 y Fp(\024)2572 801 y(Z)2701 910 y Fw(~)2689 937 y Fv(\036)o(d\027)29 b Fu(\000)22 b Fw(sup)3149 910 y(~)3137 937 y Fv(\036)3194 796 y Fp(\025)3264 937 y Fv(:)0 1188 y Fw(By)33 b(letting)g Fv(j)h Fu(!)27 b Fw(+)p Fu(1)32 b Fw(w)m(e)i(obtain)e(\(43\))o(,)h(due) g(to)g(the)g(Monotone)f(Con)m(v)m(ergence)k(Theorem.)0 1304 y(No)m(w)27 b(w)m(e)f(claim)h(that)f(if)f(the)i(mass)g(of)e Fv(\026)h Fw(is)g(not)g Fv(T)c Fu(\000)9 b Fv(t)26 b Fw(or)g(the)g(mass)h(of)e Fv(\027)33 b Fw(is)26 b(not)g(1,)h(then)g Fv(h)3286 1268 y Fl(\003)3325 1304 y Fw(\()p Fv(\026;)17 b(\027)6 b Fw(\))28 b(=)0 1430 y(+)p Fu(1)p Fw(.)43 b(T)-8 b(o)33 b(see)g(this)g(just)g(add)g(constan)m(ts)h(to)44 b(~)-61 b Fv(\037)33 b Fw(or)f(to)2016 1403 y(~)2003 1430 y Fv(\036)p Fw(.)0 1550 y(Finally)-8 b(,)32 b(if)e(the)i(mass)g (of)e Fv(\026)h Fw(is)h Fv(T)g Fu(\000)20 b Fv(t)31 b Fw(and)g(the)g(mass)h(of)f Fv(\027)37 b Fw(is)32 b(1,)f(b)m(y)h(c)m(ho) s(osing)43 b(~)-61 b Fv(\037)28 b Fw(:=)g(0)i(and)3409 1524 y(~)3397 1550 y Fv(\036)d Fw(:=)h(0)0 1666 y(in)33 b(\(43\))o(,)g(w)m(e)g(obtain)1221 1841 y Fv(h)1277 1800 y Fl(\003)1316 1841 y Fw(\()p Fv(\026;)17 b(\027)6 b Fw(\))28 b Fu(\025)1682 1706 y Fp(Z)1798 1841 y Fv(L)1864 1856 y Fs(j)1901 1841 y Fv(d\026)22 b Fw(+)2131 1706 y Fp(Z)2247 1841 y Fv( )t(d\027)q(:)0 2037 y Fw(T)-8 b(o)30 b(obtain)f(the)h(opp)s(osite)g(inequalit)m(y)-8 b(,)32 b(just)e(use)h(\(39\))e(and)g(\(40\),)h(and)g(observ)m(e)h (that,)f(if)f(the)h(mass)0 2153 y(of)i Fv(\026)g Fw(is)h Fv(T)j Fu(\000)23 b Fv(t)33 b Fw(and)f(the)h(mass)h(of)e Fv(\027)39 b Fw(is)33 b(1,)f(then,)i(for)e(eac)m(h)h(\()p Fv(\037;)17 b(\036)p Fw(\),)32 b(w)m(e)i(ha)m(v)m(e)1040 2234 y Fp(Z)1156 2370 y Fu(\000)p Fv(\037)23 b Fu(\000)f Fv(Ld\026)28 b Fu(\024)g Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(t)p Fw(\))17 b(sup)q(\()p Fu(\000)p Fv(\037)23 b Fu(\000)f Fv(L)p Fw(\))p Fv(;)0 2580 y Fw(as)33 b(w)m(ell)h(as)1274 2595 y Fp(Z)1390 2731 y Fv( )26 b Fu(\000)d Fv(\036d\027)33 b Fu(\024)c Fw(sup\()p Fv( )e Fu(\000)22 b Fv(\036)p Fw(\))17 b Fv(;)0 2921 y Fw(so)33 b(that)604 3112 y Fu(\000)698 2976 y Fp(Z)815 3112 y Fv(\037d\026)21 b Fu(\000)1107 2976 y Fp(Z)1223 3112 y Fv(\036d\027)28 b Fu(\000)23 b Fv(h)p Fw(\()p Fv(\037;)17 b(\036)p Fw(\))444 3371 y Fu(\024)604 3235 y Fp(Z)721 3371 y Fv(Ld\026)22 b Fw(+)1017 3235 y Fp(Z)1133 3371 y Fv( )t(d\027)28 b Fw(+)22 b(\()p Fv(T)36 b Fu(\000)23 b Fv(t)p Fw(\))17 b(sup\()p Fu(\000)p Fv(\037)23 b Fu(\000)f Fv(L)p Fw(\))h(+)f(sup)q(\()p Fv( )k Fu(\000)c Fv(\036)p Fw(\))g Fu(\000)h Fv(h)p Fw(\()p Fv(\037;)17 b(\036)p Fw(\))445 3629 y(=)604 3494 y Fp(Z)721 3629 y Fv(Ld\026)22 b Fw(+)1017 3494 y Fp(Z)1133 3629 y Fv( )t(d\027)q(:)0 3845 y Fw(This)34 b(completes)g(the)f(pro)s(of)f (of)39 b(\(41\))o(.)0 3962 y(T)-8 b(o)33 b(compute)h(the)f(Legendre)h (transform)e(of)h Fv(g)t Fw(,)f(observ)m(e)i(that)f(if)f(\()p Fv(\026;)17 b(\027)6 b Fw(\))28 b Fu(62)g(M)2919 3977 y Fr(2)2991 3962 y Fw(then)34 b(there)f(exist)0 4090 y(functions)h(\()12 b(^)-61 b Fv(\037;)576 4064 y Fw(^)564 4090 y Fv(\036)o Fw(\))28 b Fu(2)g(C)39 b Fw(suc)m(h)34 b(that)0 4306 y(\(44\))1093 4171 y Fp(Z)1221 4306 y Fw(^)-61 b Fv(\037d)p Fw(\()p Fv(\026)1418 4321 y Fl(\003)1479 4306 y Fu(\000)23 b Fv(\026)p Fw(\))e Fu(\000)1797 4171 y Fp(Z)1926 4280 y Fw(^)1913 4306 y Fv(\036d)p Fw(\()p Fv(\027)2108 4321 y Fl(\003)2169 4306 y Fu(\000)i Fv(\027)6 b Fw(\))28 b Fu(6)p Fw(=)g(0)17 b Fv(;)0 4543 y Fw(thanks)30 b(to)f(the)h(fact)f(that)g(\()p Fv(\026)1089 4558 y Fl(\003)1128 4543 y Fv(;)17 b(\027)1220 4558 y Fl(\003)1260 4543 y Fw(\))27 b Fu(2)h(M)1539 4558 y Fr(2)1579 4543 y Fw(.)42 b(P)m(ossibly)31 b(c)m(hanging)f(the)g(signs)g(of)41 b(^)-61 b Fv(\037)30 b Fw(and)3238 4517 y(^)3218 4543 y Fv( )t Fw(,)g(w)m(e)h(ma)m(y)0 4672 y(supp)s(ose)37 b(that)e(the)h(quan)m(tit)m(y)h(in)e(\(44\))g(is)h(p)s(ositiv)m(e.)53 b(Then,)38 b(tak)m(e)47 b(^)-60 b Fv(\037)2542 4687 y Fs(j)2611 4672 y Fw(:=)32 b Fv(j)18 b Fw(^)-61 b Fv(\037)36 b Fw(and)3094 4646 y(^)3081 4672 y Fv(\036)3139 4687 y Fs(j)3208 4672 y Fw(:=)c Fv(j)3402 4646 y Fw(^)3389 4672 y Fv(\036)p Fw(.)52 b(W)-8 b(e)0 4788 y(get)33 b(that)892 4958 y(lim)846 5019 y Fs(j)t Fl(!)p Fr(+)p Fl(1)1091 4822 y Fp(Z)1218 4958 y Fw(^)-60 b Fv(\037)1268 4973 y Fs(j)1304 4958 y Fv(d)p Fw(\()p Fv(\026)1452 4973 y Fl(\003)1513 4958 y Fu(\000)23 b Fv(\026)p Fw(\))f Fu(\000)1831 4822 y Fp(Z)1960 4932 y Fw(^)1948 4958 y Fv(\036)2006 4973 y Fs(j)2042 4958 y Fv(d)p Fw(\()p Fv(\027)2179 4973 y Fl(\003)2240 4958 y Fu(\000)h Fv(\027)6 b Fw(\))45 b(=)f(+)p Fu(1)17 b Fv(:)0 5148 y Fw(The)34 b(ab)s(o)m(v)m(e)f (considerations)h(sho)m(w)g(that)273 5378 y Fv(g)324 5337 y Fl(\003)363 5378 y Fw(\()p Fv(\026;)17 b(\027)6 b Fw(\))27 b(=)h(inf)733 5441 y Fs(\037;\036)861 5378 y Fu(\000)955 5243 y Fp(Z)1072 5378 y Fv(\037d)p Fw(\()p Fv(\026)1281 5393 y Fl(\003)1342 5378 y Fu(\000)22 b Fv(\026)p Fw(\))g(+)1658 5243 y Fp(Z)1774 5378 y Fv(\036d)p Fw(\()p Fv(\027)1969 5393 y Fl(\003)2030 5378 y Fu(\000)h Fv(\027)6 b Fw(\))28 b(=)2354 5208 y Fp(\()2434 5311 y Fw(0)292 b(if)33 b(\()p Fv(\026;)17 b(\027)6 b Fw(\))27 b Fu(2)h(M)3339 5326 y Fr(2)2434 5451 y Fu(\0001)164 b Fw(otherwise)r Fv(;)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 168 251 a Fq(20)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y Fw(whic)m(h)34 b(giv)m(es)g(\(42\))o(.)2866 b Fi(\003)168 616 y Fw(W)-8 b(e)35 b(no)m(w)h(complete)g(the)f(pro)s(of)g(of)f(Prop)s (osition)h(6)g(b)m(y)h(arguing)f(as)g(follo)m(ws.)51 b(W)-8 b(e)36 b(note)f(that)g(the)168 732 y(function)43 b Fv(g)i Fw(\(resp.,)f Fv(h)p Fw(\),)h(as)d(de\014ned)i(ab)s(o)m(v)m (e,)i(is)d(conca)m(v)m(e)h(\(resp.,)i(con)m(v)m(ex\))f(and)d(upp)s(er)h (\(resp.,)168 849 y(lo)m(w)m(er\))33 b(semicon)m(tin)m(uous)j(\(w)m(e)d (remark)h(indeed)g(that)e Fu(C)39 b Fw(is)33 b(a)f(closed)i(con)m(v)m (ex)h(set\).)168 965 y(Then,)f(b)m(y)f(the)g(Legendre-F)-8 b(enc)m(hel-Ro)s(c)m(k)j(afellar)34 b(Theorem)g(\(see,)g(e.g.,)e ([Vil03])h(or)f([Gom06]\),)1418 1122 y(sup)q(\()p Fv(g)26 b Fu(\000)c Fv(h)p Fw(\))45 b(=)f(inf)7 b(\()p Fv(h)2246 1081 y Fl(\003)2307 1122 y Fu(\000)23 b Fv(g)2458 1081 y Fl(\003)2497 1122 y Fw(\))17 b Fv(:)168 1279 y Fw(Consequen)m(tly)-8 b(,)35 b(\(29\),)e(\(36\))e(and)i(Lemma)g(7)g(yield:)522 1488 y(~)-55 b Fv(u)572 1447 y Fs(\017)687 1488 y Fw(=)268 b(inf)846 1555 y Fr(\()p Fs(\026;\027)t Fr(\))p Fl(2M)1132 1564 y Fk(1)1168 1555 y Fl(\\M)1299 1564 y Fk(2)1350 1353 y Fp(Z)1466 1488 y Fv(L)17 b(d\026)22 b Fw(+)1779 1353 y Fp(Z)1895 1488 y Fv( 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2528 y Fj(0)1441 2546 y Fs(;X)q(;v)r(;s)p Fr(\))1655 2461 y Fw(\()p Fu(\000)1797 2436 y Fw(~)1770 2461 y Fv(A)q(\036)22 b Fu(\000)h Fv(@)2075 2476 y Fs(t)2105 2461 y Fv(\036)f Fu(\000)g Fv(L)p Fw(\))h Fu(\000)49 b Fw(sup)2510 2546 y Fr(\()p Fs(x)2577 2528 y Fj(0)2600 2546 y Fs(;X)5 b Fr(\))2710 2461 y Fw(\()p Fv( )26 b Fu(\000)d Fv(\036)p Fw(\))f(+)3153 2325 y Fp(Z)3269 2461 y Fv(\036)17 b(d\027)3449 2420 y Fs(\017)3443 2485 y(t)687 2758 y Fw(=)102 b(sup)846 2849 y Fs(\036)p Fl(2)p Fs(C)990 2826 y Fk(2)985 2872 y(\003)1031 2758 y Fw(\()p Fv(t)22 b Fu(\000)h Fv(T)14 b Fw(\))67 b(sup)1352 2843 y Fr(\()p Fs(x)1419 2825 y Fj(0)1441 2843 y Fs(;X)q(;s)p Fr(\))1599 2758 y Fw(\()1662 2733 y(~)1637 2758 y Fv(H)30 b Fu(\000)22 b Fv(@)1898 2773 y Fs(t)1929 2758 y Fv(\036)p Fw(\))f Fu(\000)49 b Fw(sup)2146 2843 y Fr(\()p Fs(x)2213 2825 y Fj(0)2235 2843 y Fs(;X)5 b Fr(\))2346 2758 y Fw(\()p Fv( )26 b Fu(\000)c Fv(\036)p Fw(\))g(+)2788 2622 y Fp(Z)2904 2758 y Fv(\036)17 b(d\027)3084 2717 y 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Fw(\()p Fv(T)36 b Fu(\000)23 b Fv(t)p Fw(\))1547 3588 y Fp(")1982 3758 y Fw(inf)1605 3844 y Fr(\()p Fs(x)1672 3825 y Fj(0)1694 3844 y Fs(;X)5 b Fr(=)1842 3817 y Fm(x)1877 3803 y Fj(0)p 1842 3829 58 3 v 1857 3870 a Fm(\017)1909 3844 y Fs(;s)p Fr(\))p Fl(2)p Fn(R)2084 3825 y Fm(n)2126 3844 y Fl(\002)p Fn(R)2229 3825 y Fm(n)2271 3844 y Fl(\002)p Fr(\()p Fs(t;T)10 b Fr(\))2494 3758 y Fv(@)2545 3773 y Fs(t)2575 3758 y Fv(\036)22 b Fu(\000)2780 3733 y Fw(~)2754 3758 y Fv(H)2860 3677 y Fp(\000)2905 3758 y Fv(D)2989 3717 y Fr(2)3028 3758 y Fv(\036;)17 b(D)s(\036;)g(x)3371 3717 y Fl(0)3394 3758 y Fv(;)g(X)r(;)g(\017;)g(!)3713 3677 y Fp(\001)3757 3588 y(#)403 4064 y Fw(+)22 b(inf)529 4127 y Fs(x)569 4109 y Fj(0)635 3924 y Fp(\024)688 4064 y Fv( )t Fw(\()p Fv(x)848 4023 y Fl(0)872 4064 y Fw(\))g Fu(\000)g Fv(\036)p Fw(\()p Fv(x)1182 4023 y Fl(0)1206 4064 y Fv(;)1259 3997 y(x)1314 3961 y Fl(0)p 1259 4041 79 4 v 1279 4133 a Fv(\017)1348 4064 y(;)17 b(T)d Fw(\))1501 3924 y Fp(\025)1570 4064 y Fw(+)403 4332 y(+)501 4196 y Fp(Z)556 4422 y Fn(R)604 4403 y Fm(n)646 4422 y Fl(\002)p Fn(R)749 4403 y Fm(n)813 4332 y Fv(\036)p Fw(\()p Fv(x)964 4291 y Fl(0)987 4332 y Fv(;)j(X)r(;)g(t)p Fw(\))p Fv(d\027)1336 4291 y Fs(\017)1330 4357 y(t)209 4611 y Fw(=)301 b(sup)313 4702 y Fs(')p Fr(\()p Fs(x)426 4683 y Fj(0)448 4702 y Fs(;s)p Fr(\))p Fl(2)p Fs(C)630 4679 y Fk(2)625 4725 y(\003)671 4702 y Fr(\()p Fn(R)746 4683 y Fm(n)789 4702 y Fl(\002)p Fr([)p Fs(t;T)10 b Fr(]\))1006 4611 y Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(t)p Fw(\))1326 4470 y Fp(\024)1576 4611 y Fw(inf)1379 4677 y Fr(\()p Fs(x)1446 4658 y Fj(0)1468 4677 y Fs(;s)p Fr(\))p Fl(2)p Fn(R)1643 4658 y Fm(n)1686 4677 y Fl(\002)p Fr(\()p Fs(t;T)10 b Fr(\))1908 4611 y Fv(@)1959 4626 y Fs(t)1989 4611 y Fv(')407 4916 y Fu(\000)p Fv(H)590 4776 y Fp(\022)663 4916 y Fv(D)747 4875 y Fr(2)744 4941 y Fs(x)788 4916 y Fv(';)17 b(D)977 4931 y Fs(x)1020 4916 y Fv(';)g(x)1183 4875 y Fl(0)1207 4916 y Fv(;)1260 4849 y(x)1315 4813 y Fl(0)p 1260 4894 V 1280 4985 a Fv(\017)1349 4916 y(;)g(\017;)g(!)1541 4776 y Fp(\023\025)403 5126 y Fw(+)22 b(inf)529 5190 y Fs(x)569 5171 y Fj(0)635 5126 y Fw([)p Fv( )t Fw(\()p Fv(x)822 5085 y Fl(0)846 5126 y Fw(\))g Fu(\000)h Fv(')p Fw(\()p Fv(x)1163 5085 y Fl(0)1186 5126 y Fv(;)17 b(T)d Fw(\)])403 5373 y(+)501 5237 y Fp(Z)556 5463 y Fn(R)604 5444 y Fm(n)646 5463 y Fl(\002)p Fn(R)749 5444 y Fm(n)813 5373 y Fv(')p Fw(\()p Fv(x)970 5331 y Fl(0)993 5373 y Fv(;)j(t)p Fw(\))p Fv(d\027)1215 5331 y Fs(\017)1209 5397 y(t)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(21)0 450 y Fo(Pr)-5 b(o)g(of.)41 b Fw(Let)36 b Ft({)k Fw(b)s(e)c(as)g(in)g(the)g(pro)s(of)f(of)g(Prop)s(osition)h(3,)h (satisfying)f(the)h(additional)e(uniformit)m(y)0 566 y(h)m(yp)s(othesis:)46 b(for)32 b(an)m(y)h Fv(\016)f(>)27 b Fw(0)32 b(there)i(exists)g Fv(\024)p Fw(\()p Fv(\016)t Fw(\))28 b Fv(>)f Fw(0)33 b(suc)m(h)h(that)1495 735 y(inf)1472 802 y Fl(j)p Fs(y)r Fl(j\025)p Fs(\016)1653 735 y Ft({)t Fw(\()p Fv(y)t Fw(\))27 b Fu(\025)i Fv(\024)p Fw(\()p Fv(\016)t Fw(\))p Fv(:)0 958 y Fw(Then)0 1163 y(\(45\))1231 1028 y Fp(Z)1286 1253 y Fn(R)1334 1234 y Fm(n)1377 1253 y Fl(\002)p Fn(R)1480 1234 y Fm(n)1543 1163 y Ft({)1630 1023 y Fp(\022)1703 1163 y Fv(X)h Fu(\000)1924 1096 y Fv(x)1979 1060 y Fl(0)p 1924 1141 79 4 v 1943 1232 a Fv(\017)2012 1023 y Fp(\023)2102 1163 y Fv(d\027)2207 1122 y Fs(\017)2201 1188 y(t)2268 1163 y Fw(=)d(0)p Fv(:)0 1390 y Fw(By)f(con)m(tradiction,)i(assume)f(that)e(the)h(\014rst)g (equalit)m(y)h(claimed)g(in)e(Prop)s(osition)h(8)f(is)h(false,)i (hence,)0 1507 y(b)m(y)35 b(\(38\))o(,)e(that)275 1684 y(~)-56 b Fv(u)324 1643 y Fs(\017)357 1684 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))82 b Fv(<)154 b Fw(sup)917 1769 y Fs(\036)p Fr(\()p Fs(x)1026 1751 y Fj(0)1048 1769 y Fs(;X)q(;s)p Fr(\))1601 1684 y Fw(inf)1223 1769 y Fr(\()p Fs(x)1290 1750 y Fj(0)1313 1769 y Fs(;X)5 b Fr(=)1461 1742 y Fm(x)1496 1728 y Fj(0)p 1461 1754 58 3 v 1476 1795 a Fm(\017)1528 1769 y Fs(;s)p Fr(\))p Fl(2)p Fn(R)1703 1750 y Fm(n)1745 1769 y Fl(\002)p Fn(R)1848 1750 y Fm(n)1890 1769 y Fl(\002)p Fr(\()p Fs(t;T)10 b Fr(\))2113 1684 y Fv(@)2164 1699 y Fs(t)2194 1684 y Fv(\036)22 b Fu(\000)2399 1658 y Fw(~)2373 1684 y Fv(H)2478 1603 y Fp(\000)2524 1684 y Fv(D)2608 1643 y Fr(2)2647 1684 y Fv(\036;)17 b(D)s(\036;)g(x)2990 1643 y Fl(0)3013 1684 y Fv(;)g(X)r(;)g(\017;)g(!) 3332 1603 y Fp(\001)917 1987 y Fw(+)30 b(inf)1010 2050 y Fs(x)1050 2031 y Fj(0)1071 2050 y Fs(;X)1171 1847 y Fp(\024)1223 1987 y Fv( )t Fw(\()p Fv(x)1383 1946 y Fl(0)1407 1987 y Fw(\))22 b Fu(\000)h Fv(\036)p Fw(\()p Fv(x)1718 1946 y Fl(0)1741 1987 y Fv(;)1795 1920 y(x)1850 1884 y Fl(0)p 1795 1964 79 4 v 1814 2056 a Fv(\017)1883 1987 y(;)17 b(T)d Fw(\))2036 1847 y Fp(\025)917 2255 y Fw(+)1010 2119 y Fp(Z)1065 2345 y Fn(R)1113 2326 y Fm(n)1155 2345 y Fl(\002)p Fn(R)1258 2326 y Fm(n)1321 2255 y Fv(\036)p Fw(\()p Fv(x)1472 2214 y Fl(0)1496 2255 y Fv(;)j(X)r(;)g(t)p Fw(\))p Fv(d\027)1845 2214 y Fs(\017)1839 2279 y(t)1894 2255 y Fv(;)0 2482 y Fw(for)32 b(some)h Fv(x)c Fu(2)f Ft(R)643 2446 y Fs(n)690 2482 y Fw(.)0 2598 y(This)34 b(w)m(ould)f(mean)g(that)g(for)f(some)h Fv(\016)f(>)27 b Fw(0,)33 b(p)s(ossibly)h(dep)s(ending)g(on)e Fv(x)p Fw(,)h(that)600 2767 y(~)-55 b Fv(u)650 2726 y Fs(\017)682 2767 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))435 2920 y(=)155 b(sup)594 3006 y Fs(\036)p Fr(\()p Fs(x)703 2987 y Fj(0)725 3006 y Fs(;X)q(;s)p Fr(\))1398 2920 y Fw(inf)901 3006 y Fr(\()p Fs(x)968 2987 y Fj(0)990 3006 y Fs(;X)q(;s)p Fr(\))p Fl(2)p Fn(R)1244 2987 y Fm(n)1286 3006 y Fl(\002)p Fn(R)1389 2987 y Fm(n)1432 3006 y Fl(\002)p Fr(\()p Fs(t;T)10 b Fr(\))p Fs(;)32 b Fl(j)p Fs(X)5 b Fl(\000)1837 2979 y Fm(x)1872 2965 y Fj(0)p 1837 2991 58 3 v 1853 3032 a Fm(\017)1904 3006 y Fl(j)p Fs(>\016)2029 2920 y Fv(@)2080 2935 y Fs(t)2110 2920 y Fv(\036)22 b Fu(\000)2315 2895 y Fw(~)2290 2920 y Fv(H)2395 2840 y Fp(\000)2440 2920 y Fv(D)2524 2879 y Fr(2)2564 2920 y Fv(\036;)17 b(D)s(\036;)g(x)2907 2879 y Fl(0)2929 2920 y Fv(;)g(X)r(;)g(\017;)g(!)3248 2840 y Fp(\001)594 3157 y Fw(+)215 b(inf)687 3243 y Fs(x)727 3224 y Fj(0)749 3243 y Fs(;X)q(;)32 b Fl(j)p Fs(X)5 b Fl(\000)1028 3216 y Fm(x)1063 3202 y Fj(0)p 1027 3228 V 1043 3269 a Fm(\017)1094 3243 y Fl(j)p Fs(>\016)1219 3157 y Fw([)p Fv( )t Fw(\()p Fv(x)1406 3116 y Fl(0)1430 3157 y Fw(\))22 b Fu(\000)h Fv(\036)p Fw(\()p Fv(x)1741 3116 y Fl(0)1764 3157 y Fv(;)17 b(X)r(;)g(T)d Fw(\)])22 b(+)-2169 b(\(46\))594 3452 y(+)687 3317 y Fp(Z)742 3542 y Fn(R)790 3523 y Fm(n)832 3542 y Fl(\002)p Fn(R)935 3523 y Fm(n)998 3452 y Fv(\036)p Fw(\()p Fv(x)1149 3411 y Fl(0)1173 3452 y Fv(;)17 b(X)r(;)g(t)p Fw(\))p Fv(d\027)1522 3411 y Fs(\017)1516 3477 y(t)1555 3452 y Fv(:)0 3700 y Fw(W)-8 b(e)36 b(replace)g Fv(\036)p Fw(\()p Fv(x)655 3664 y Fl(0)679 3700 y Fv(;)17 b(X)r(;)g(t)p Fw(\))35 b(b)m(y)h Fv(\036)p Fw(\()p Fv(x)1247 3664 y Fl(0)1270 3700 y Fv(;)17 b(X)r(;)g(t)p Fw(\))24 b(+)g Fv(\025)p Fw(\()p Fv(t)g Fu(\000)h Fv(T)14 b Fw(\))p Ft({)2090 3619 y Fp(\000)2135 3700 y Fv(X)30 b Fu(\000)2356 3661 y Fs(x)2396 3638 y Fj(0)p 2356 3677 63 4 v 2373 3735 a Fs(\017)2428 3619 y Fp(\001)2509 3700 y Fw(in)35 b(\(46\),)h(with)g Fv(\025)c(>)h Fw(0.)51 b(Since,)0 3835 y(due)41 b(to)f(the)h(observ)-5 b(ation)41 b(next)g(to)f(\(37\),)i(this)f(do)s(es)g(not)f(c)m(hange) 2581 3809 y(~)2556 3835 y Fv(H)7 b Fw(,)43 b(w)m(e)e(ha)m(v)m(e)h(that) e(the)h(\014rst)0 3951 y(in\014m)m(um)34 b(in)f(\(46\))f(app)s(ears)h (as)737 4181 y Fv(\025)p Ft({)881 4041 y Fp(\022)955 4181 y Fv(X)d Fu(\000)1175 4114 y Fv(x)1230 4078 y Fl(0)p 1175 4158 79 4 v 1195 4249 a Fv(\017)1264 4041 y Fp(\023)1359 4181 y Fw(+)22 b Fv(@)1508 4196 y Fs(t)1539 4181 y Fv(\036)f Fu(\000)1743 4156 y Fw(~)1718 4181 y Fv(H)1823 4100 y Fp(\000)1869 4181 y Fv(D)1953 4140 y Fr(2)1992 4181 y Fv(\036;)c(D)s(\036;)g(x)2335 4140 y Fl(0)2357 4181 y Fv(;)g(X)r(;)g(\017;)g(!)2676 4100 y Fp(\001)1155 4404 y Fu(\025)29 b Fv(\025\024)p Fw(\()p Fv(\016)t Fw(\))22 b(+)g Fv(@)1668 4419 y Fs(t)1698 4404 y Fv(\036)g Fu(\000)1902 4379 y Fw(~)1877 4404 y Fv(H)1982 4324 y Fp(\000)2028 4404 y Fv(D)2112 4363 y Fr(2)2151 4404 y Fv(\036;)17 b(D)s(\036;)g(x)2494 4363 y Fl(0)2516 4404 y Fv(;)g(X)r(;)g(\017;)g(!) 2835 4324 y Fp(\001)2897 4404 y Fv(;)0 4579 y Fw(and)37 b(the)h(other)f(in\014m)m(um,)j(as)e(w)m(ell)g(as)g(the)f(in)m(tegral)h (are)f(unc)m(hanged,)j(due)e(to)f(\(45\))o(.)58 b(By)38 b(send-)0 4695 y(ing)31 b Fv(\025)c Fu(!)h(1)i Fw(w)m(e)i(w)m(ould)f (ha)m(v)m(e)h(a)f(con)m(tradiction,)h(and)f(th)m(us)g(that)g(the)g (in\014m)m(um)h(in)f(\(38\))f(m)m(ust)i(b)s(e)0 4819 y(ac)m(hiev)m(ed)j(when)f Fv(X)h Fw(=)878 4780 y Fs(x)918 4756 y Fj(0)p 878 4796 63 4 v 894 4853 a Fs(\017)950 4819 y Fw(.)0 4935 y(This)f(sho)m(ws)g(the)f(\014rst)g(equalit)m(y)h (in)f(Prop)s(osition)g(8.)0 5053 y(The)43 b(second)g(one)g(follo)m(ws)g (b)m(y)g(taking)f Fv(')p Fw(\()p Fv(x)1665 5017 y Fl(0)1689 5053 y Fv(;)17 b(t)p Fw(\))43 b(:=)h Fv(\036)p Fw(\()p Fv(x)2147 5017 y Fl(0)2171 5053 y Fv(;)2224 5014 y Fs(x)2264 4990 y Fj(0)p 2224 5030 V 2241 5087 a Fs(\017)2296 5053 y Fv(;)17 b(t)p Fw(\))42 b(and)g(b)m(y)h(observing)h(that)e Fv(\027)3521 5017 y Fs(\017)3515 5078 y(t)3596 5053 y Fw(is)0 5186 y(supp)s(orted)33 b(on)g Fu(f)p Fv(X)i Fw(=)874 5147 y Fs(x)914 5123 y Fj(0)p 874 5163 V 891 5220 a Fs(\017)946 5186 y Fu(g)p Fw(,)d(thanks)i(to)e(\(45\))o(,)h(so)g(that)1115 5281 y Fp(Z)1231 5417 y Fv(\036)p Fw(\()p Fv(x)1382 5376 y Fl(0)1405 5417 y Fv(;)17 b(X)r(;)g(t)p Fw(\))g Fv(d\027)1771 5376 y Fs(\017)1765 5441 y(t)1831 5417 y Fw(=)1935 5281 y Fp(Z)2051 5417 y Fv(')p Fw(\()p Fv(x)2208 5376 y Fl(0)2232 5417 y Fv(;)g(t)p Fw(\))g Fv(d\027)2471 5376 y Fs(\017)2465 5441 y(t)2520 5417 y Fv(:)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 168 251 a Fq(22)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)3752 450 y Fi(\003)168 650 y Fw(From)23 b(this)h(p)s(oin)m(t)g(on,)h(w)m(e)g (will)f(assume)h(Assumption)g(6.)41 b(With)24 b(this,)i(w)m(e)e(are)g (no)m(w)g(in)g(the)g(p)s(osition)168 767 y(to)32 b(impro)m(v)m(e)i (Prop)s(osition)f(4:)168 957 y FA(Prop)s(osition)k(9.)48 b Fw(~)-55 b Fv(u)958 921 y Fs(\017)1018 957 y Fw(=)27 b Fv(u)1177 921 y Fs(\017)1244 957 y Fo(in)35 b Ft(R)1436 921 y Fs(n)1505 957 y Fu(\002)23 b Fw([)p Fv(t;)17 b(T)d Fw(])22 b Fu(\002)g Fw(\012)p Fo(.)168 1154 y(Pr)-5 b(o)g(of.)41 b Fw(By)e(\(22\))o(,)e Fv(\027)916 1118 y Fs(\017)910 1179 y(t)986 1154 y Fw(is)h(supp)s(orted)g(in)f Fv(X)43 b Fw(=)1915 1115 y Fs(x)1955 1092 y Fj(0)p 1915 1132 63 4 v 1931 1189 a Fs(\017)1987 1154 y Fw(.)56 b(Th)m(us,)40 b(b)m(y)e(Prop)s(osition)f(8)g(and)g(\(9\),)h(w)m(e)g(ha)m(v)m(e)168 1271 y(that)857 1492 y(~)-55 b Fv(u)907 1451 y Fs(\017)1022 1492 y Fu(\025)84 b Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(t)p Fw(\))17 b(inf)1505 1555 y Fs(x)1545 1536 y Fj(0)1567 1555 y Fs(;s)1670 1492 y Fv(@)1721 1507 y Fs(t)1751 1492 y Fv(v)1802 1451 y Fs(\017)1798 1517 y(\016)1858 1492 y Fu(\000)23 b Fv(H)2063 1352 y Fp(\022)2137 1492 y Fv(D)2221 1451 y Fr(2)2218 1517 y Fs(x)2261 1492 y Fv(v)2312 1451 y Fs(\017)2308 1517 y(\016)2346 1492 y Fv(;)17 b(D)2471 1507 y Fs(x)2515 1492 y Fv(v)2566 1451 y Fs(\017)2562 1517 y(\016)2600 1492 y Fv(;)g(x)2699 1451 y Fl(0)2722 1492 y Fv(;)2776 1425 y(x)2831 1389 y Fl(0)p 2776 1469 79 4 v 2796 1560 a Fv(\017)2865 1492 y(;)g(\017;)g(!)3057 1352 y Fp(\023)1183 1702 y Fw(+)g(inf)1303 1765 y Fs(x)1343 1746 y Fj(0)1410 1702 y Fw([)p Fv( )t Fw(\()p Fv(x)1597 1661 y Fl(0)1621 1702 y Fw(\))22 b Fu(\000)g Fv(v)1831 1661 y Fs(\017)1827 1727 y(\016)1882 1702 y Fw(\()p Fv(x)1975 1661 y Fl(0)1999 1702 y Fv(;)17 b(T)d Fw(\))o(])22 b(+)g Fv(v)2349 1661 y Fs(\017)2345 1727 y(\016)2400 1702 y Fw(\()p Fv(x;)17 b(t)p Fw(\))1022 1887 y Fu(\025)84 b(\000)p Fv(\016)26 b Fw(+)c(0)g(+)g Fv(v)1647 1846 y Fs(\017)1643 1912 y(\016)1698 1887 y Fw(\()p Fv(x;)17 b(t)p Fw(\))33 b Fv(:)168 2072 y Fw(Sending)j Fv(\016)h Fu(!)32 b Fw(0,)k(and)f(using) j(\(10\))o(,)e(w)m(e)h(conclude)g(that)k(~)-55 b Fv(u)2372 2036 y Fs(\017)2437 2072 y Fu(\025)33 b Fv(u)2603 2036 y Fs(\017)2635 2072 y Fw(.)52 b(The)36 b(con)m(v)m(erse)i(inequalit)m (y)g(is)168 2188 y(assured)c(b)m(y)f(Prop)s(osition)g(4.)2500 b Fi(\003)168 2389 y Fw(Finally)-8 b(,)28 b(w)m(e)g(presen)m(t)g(a)e (result)i(that)e(con)m(v)m(erts)j(the)e(constrained)h(minimization)g (problem)g(of)33 b(\(29\))168 2505 y(and)h(\(30\))g(in)m(to)g(an)g (unconstrained)i(problem.)50 b(The)35 b(idea,)g(quite)g(standard)g(in)f (linear)h(program-)168 2621 y(ming,)c(consists)i(in)e(using)g(the)h (appro)m(ximate)g(solution)f Fv(v)2307 2585 y Fs(\017)2303 2647 y(\016)2372 2621 y Fw(of)f(the)h(dual)g(problem)h(as)f(a)g (Lagrange)168 2737 y(m)m(ultiplier)j(so)f(to)f(remo)m(v)m(e)i(the)f (constrain)m(ts.)168 2928 y FA(Prop)s(osition)47 b(10)h Fw(\(Unconstrained)43 b(minimization\))p FA(.)k Fo(Fix)42 b Fv(\016)47 b(>)42 b Fw(0)p Fo(,)j(and)d(let)h Fv(v)3217 2891 y Fs(\017)3213 2954 y(\016)3294 2928 y Fo(b)-5 b(e)43 b(as)f(in)h(As-)168 3044 y(sumption)34 b(6.)45 b(Then,)604 3140 y Fp(\014)604 3200 y(\014)604 3260 y(\014)604 3319 y(\014)637 3285 y Fv(u)693 3243 y Fs(\017)725 3285 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))k Fu(\000)1165 3144 y Fp(\024)1218 3285 y Fw(inf)1255 3344 y Fs(\026)1352 3149 y Fp(Z)1408 3375 y Fn(R)1456 3356 y Fm(n)1498 3375 y Fl(\002)p Fs(U)7 b Fl(\002)p Fr([)p Fs(t;T)j Fr(])1819 3285 y Fv(L)p Fw(\()p Fv(x)1978 3243 y Fl(0)2002 3285 y Fv(;)2055 3217 y(x)2110 3181 y Fl(0)p 2055 3262 V 2075 3353 a Fv(\017)2144 3285 y(;)17 b(v)t(;)g(!)t Fw(\))k(+)h Fv(Av)2629 3243 y Fs(\017)2625 3309 y(\016)2685 3285 y Fw(+)g Fv(@)2834 3300 y Fs(t)2864 3285 y Fv(v)2915 3243 y Fs(\017)2911 3309 y(\016)2966 3285 y Fv(d\026)f Fw(+)3195 3149 y Fp(Z)3311 3285 y Fv(v)3362 3243 y Fs(\017)3358 3309 y(\016)3396 3285 y Fv(d\027)3501 3243 y Fs(\017)3495 3309 y(t)3534 3144 y Fp(\025)3587 3140 y(\014)3587 3200 y(\014)3587 3260 y(\014)3587 3319 y(\014)443 3503 y Fu(\024)84 b Fv(\016)t Fw(\()p Fv(T)35 b Fu(\000)23 b Fv(t)p Fw(\))f(+)56 b(sup)1074 3589 y Fs(s)p Fl(2)p Fr([)p Fs(t;T)10 b Fr(])1306 3503 y Fu(j)p Fv(u)1390 3462 y Fs(\017)1422 3503 y Fw(\()p Fv(x;)17 b(s;)g(!)t Fw(\))k Fu(\000)h Fv(v)1923 3462 y Fs(\017)1919 3528 y(\016)1957 3503 y Fw(\()p Fv(x;)17 b(s;)g(!)t Fw(\))p Fu(j)g Fv(;)-2191 b Fw(\(47\))168 3767 y Fo(wher)-5 b(e)33 b(the)h(ab)-5 b(ove)33 b(in\014mum)g(is)h (taken)f(over)h(al)5 b(l)33 b(me)-5 b(asur)g(es)33 b Fv(\026)p Fw(\()p Fv(x)2536 3731 y Fl(0)2560 3767 y Fv(;)17 b(v)t(;)g(s)p Fw(\))33 b Fo(on)g Ft(R)3026 3731 y Fs(n)3093 3767 y Fu(\002)20 b Fv(U)31 b Fu(\002)20 b Fw([)p Fv(t;)d(T)d Fw(])p Fo(,)34 b(with)168 3883 y(total)h(mass)f Fv(T)i Fu(\000)22 b Fv(t)p Fo(.)168 4074 y(Pr)-5 b(o)g(of.)41 b Fw(By)34 b(\(6\),)916 4316 y Fv(L)p Fw(\()p Fv(x)1075 4275 y Fl(0)1099 4316 y Fv(;)1152 4249 y(x)1207 4212 y Fl(0)p 1152 4293 V 1172 4384 a Fv(\017)1241 4316 y(;)17 b(v)t(;)g(!)t Fw(\))k(+)h Fv(Av)1726 4275 y Fs(\017)1722 4341 y(\016)1788 4316 y Fu(\025)28 b(\000)p Fv(H)8 b Fw(\()p Fv(D)2181 4275 y Fr(2)2178 4341 y Fs(x)2222 4316 y Fv(v)2273 4275 y Fs(\017)2269 4341 y(\016)2307 4316 y Fv(;)17 b(D)2432 4331 y Fs(x)2475 4316 y Fv(v)2526 4275 y Fs(\017)2522 4341 y(\016)2560 4316 y Fv(;)g(x)2659 4275 y Fl(0)2683 4316 y Fv(;)2736 4249 y(x)2791 4212 y Fl(0)p 2736 4293 V 2756 4384 a Fv(\017)2825 4316 y(;)g(\017;)g(!)t Fw(\))p Fv(;)168 4530 y Fw(and)32 b(so,)h(in)m(tegrating)g(with)g(resp) s(ect)h(to)e(an)m(y)i(measure)g Fv(\026)e Fw(with)h(total)f(mass)i Fv(T)h Fu(\000)23 b Fv(t)p Fw(,)168 4632 y Fp(Z)284 4767 y Fv(L)p Fw(\()p Fv(x)443 4726 y Fl(0)467 4767 y Fv(;)521 4700 y(x)576 4664 y Fl(0)p 521 4744 V 540 4835 a Fv(\017)609 4767 y(;)17 b(v)t(;)g(!)t Fw(\))12 b(+)g Fv(Av)1075 4726 y Fs(\017)1071 4792 y(\016)1121 4767 y Fw(+)g Fv(@)1260 4782 y Fs(t)1290 4767 y Fv(v)1341 4726 y Fs(\017)1337 4792 y(\016)1391 4767 y Fv(d\026)27 b Fu(\025)1634 4632 y Fp(Z)1750 4767 y Fv(@)1801 4782 y Fs(t)1831 4767 y Fv(v)1882 4726 y Fs(\017)1878 4792 y(\016)1928 4767 y Fu(\000)12 b Fv(H)c Fw(\()p Fv(D)2228 4726 y Fr(2)2225 4792 y Fs(x)2269 4767 y Fv(v)2320 4726 y Fs(\017)2316 4792 y(\016)2354 4767 y Fv(;)17 b(D)2479 4782 y Fs(x)2523 4767 y Fv(v)2574 4726 y Fs(\017)2570 4792 y(\016)2608 4767 y Fv(;)g(x)2707 4726 y Fl(0)2730 4767 y Fv(;)2784 4700 y(x)2839 4664 y Fl(0)p 2784 4744 V 2804 4835 a Fv(\017)2873 4767 y(;)g(\017;)g(!)t Fw(\))g Fv(d\026)25 b Fu(\025)k(\000)p Fv(\016)21 b Fw(\()p Fv(T)26 b Fu(\000)12 b Fv(t)p Fw(\))17 b Fv(;)168 5005 y Fw(thanks)33 b(to)g(\(9\))o(.)44 b(This)33 b(and)g(the)g(de\014nition)h(of)e Fv(\027)1979 4968 y Fs(\017)1973 5029 y(t)2012 5005 y Fw(,)g(equation)i(\(22\))o(,)f(imply) h(that)637 5251 y Fv(u)693 5210 y Fs(\017)725 5251 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))k Fu(\000)1165 5110 y Fp(\024)1218 5251 y Fw(inf)1255 5311 y Fs(\026)1352 5115 y Fp(Z)1408 5341 y Fn(R)1456 5322 y Fm(n)1498 5341 y Fl(\002)p Fs(U)7 b Fl(\002)p Fr([)p Fs(t;T)j Fr(])1819 5251 y Fv(L)p Fw(\()p Fv(x)1978 5210 y Fl(0)2002 5251 y Fv(;)2055 5183 y(x)2110 5147 y Fl(0)p 2055 5228 V 2075 5319 a Fv(\017)2144 5251 y(;)17 b(v)t(;)g(!)t Fw(\))k(+)h Fv(Av)2629 5210 y Fs(\017)2625 5275 y(\016)2685 5251 y Fw(+)g Fv(@)2834 5266 y Fs(t)2864 5251 y Fv(v)2915 5210 y Fs(\017)2911 5275 y(\016)2966 5251 y Fv(d\026)f Fw(+)3195 5115 y Fp(Z)3311 5251 y Fv(v)3362 5210 y Fs(\017)3358 5275 y(\016)3396 5251 y Fv(d\027)3501 5210 y Fs(\017)3495 5275 y(t)3534 5110 y Fp(\025)476 5469 y Fu(\024)84 b Fv(\016)t Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(t)p Fw(\))h(+)f Fv(u)1164 5428 y Fs(\017)1196 5469 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))k Fu(\000)i Fv(v)1687 5428 y Fs(\017)1683 5494 y(\016)1721 5469 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))g Fv(:)-1916 b Fw(\(48\))p eop end %%Page: 23 23 TeXDict begin 23 22 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(23)0 450 y Fw(On)45 b(the)h(other)f(hand,)j(let)e Fv(\024)j(>)g Fw(0,)f(and)d(supp)s(ose)h(that)f(\()p FA(x)p Fw(\()p Fv(s)p Fw(\))p Fv(;)17 b FA(v)r Fw(\()p Fv(s)p Fw(\)\))44 b(is)i(an)f(almost)g(optimal)0 566 y(tra)5 b(jectory)33 b(for)f(\(5\),)h(for)f Fv(s)27 b Fu(2)h Fw([)p Fv(t;)17 b(T)d Fw(],)33 b(so)g(that)0 804 y(\(49\))482 b Fv(u)712 762 y Fs(\017)772 804 y Fu(\025)877 668 y Fp(Z)12 b(Z)1060 804 y Fv(L)p Fw(\()p Fv(x)1219 762 y Fl(0)1243 804 y Fv(;)1297 736 y(x)1352 700 y Fl(0)p 1297 781 79 4 v 1316 872 a Fv(\017)1385 804 y(;)17 b(v)t(;)g(!)t Fw(\))p Fv(d\026)1737 762 y Fs(\017)1737 828 y(o)1773 804 y Fw(\()p Fv(x)1866 762 y Fl(0)1890 804 y Fv(;)g(v)t(;)g(s)p Fw(;)g Fv(!)t Fw(\))j(+)i Fv(E)6 b( )21 b Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\)\))22 b Fu(\000)g Fv(\024)17 b(;)0 1036 y Fw(where)34 b(the)f(measure)h Fv(\026)889 1000 y Fs(\017)889 1061 y(o)959 1036 y Fw(on)f Ft(R)1167 1000 y Fs(n)1236 1036 y Fu(\002)22 b Fv(U)33 b Fu(\002)23 b Fw([)p Fv(t;)17 b(T)d Fw(])32 b(is)h(de\014ned)h(b)m(y)0 1287 y(\(50\))895 1152 y Fp(Z)1011 1287 y Fv(\036)p Fw(\()p Fv(x)1162 1246 y Fl(0)1185 1287 y Fv(;)17 b(v)t(;)g(s)p Fw(\))g Fv(d\026)1535 1246 y Fs(\017)1535 1312 y(o)1599 1287 y Fw(:=)28 b Fv(E)1825 1152 y Fp(Z)1924 1178 y Fs(T)1880 1377 y(t)1996 1287 y Fv(\036)p Fw(\()p FA(x)p Fw(\()p Fv(s)p Fw(\))p Fv(;)17 b FA(v)r Fw(\()p Fv(s)p Fw(\))p Fv(;)g(s)p Fw(\))g Fv(ds)g(;)0 1524 y Fw(for)32 b(an)m(y)h Fv(\036)28 b Fu(2)g Fv(C)583 1539 y Fl(\003)622 1524 y Fw(\()p Ft(R)732 1488 y Fs(n)801 1524 y Fu(\002)23 b Fv(U)32 b Fu(\002)23 b Fw([)p Fv(t;)17 b(T)d Fw(]\).)0 1640 y(Notice)33 b(that)g Fv(\026)577 1604 y Fs(\017)577 1665 y(o)614 1640 y Fw(\()p Ft(R)724 1604 y Fs(n)793 1640 y Fu(\002)23 b Fv(U)33 b Fu(\002)23 b Fw([)p Fv(t;)17 b(T)d Fw(]\))27 b(=)h Fv(T)35 b Fu(\000)23 b Fv(t)p Fw(,)33 b(therefore)1199 1877 y(inf)1237 1937 y Fs(\026)1333 1742 y Fp(Z)1450 1877 y Fv(L)p Fw(\()p Fv(x)1609 1836 y Fl(0)1633 1877 y Fv(;)1686 1810 y(x)1741 1774 y Fl(0)p 1686 1854 V 1706 1946 a Fv(\017)1775 1877 y(;)17 b(v)t(;)g(!)t Fw(\))k(+)h Fv(Av)2260 1836 y Fs(\017)2256 1902 y(\016)2316 1877 y Fw(+)g Fv(@)2465 1892 y Fs(t)2495 1877 y Fv(v)2546 1836 y Fs(\017)2542 1902 y(\016)2597 1877 y Fv(d\026)1038 2141 y Fu(\024)1199 2005 y Fp(Z)1315 2141 y Fv(L)p Fw(\()p Fv(x)1474 2100 y Fl(0)1498 2141 y Fv(;)1552 2074 y(x)1607 2037 y Fl(0)p 1552 2118 V 1571 2209 a Fv(\017)1640 2141 y(;)17 b(v)t(;)g(!)t Fw(\))k(+)h Fv(Av)2125 2100 y Fs(\017)2121 2166 y(\016)2181 2141 y Fw(+)g Fv(@)2330 2156 y Fs(t)2360 2141 y Fv(v)2411 2100 y Fs(\017)2407 2166 y(\016)2462 2141 y Fv(d\026)2572 2100 y Fs(\017)2572 2166 y(o)2610 2141 y Fv(:)0 2373 y Fw(From)32 b(this,)h(\(49\))f(and)h(\(22\),)f(w)m(e)i(get)e(that)381 2615 y Fv(\024)22 b Fw(+)g Fv(u)613 2573 y Fs(\017)646 2615 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))k Fu(\000)1086 2474 y Fp(\024)1138 2615 y Fw(inf)1176 2674 y Fs(\026)1273 2479 y Fp(Z)1328 2705 y Fn(R)1376 2686 y Fm(n)1419 2705 y Fl(\002)p Fs(U)7 b Fl(\002)p Fr([)p Fs(t;T)j Fr(])1739 2615 y Fv(L)p Fw(\()p Fv(x)1898 2573 y Fl(0)1922 2615 y Fv(;)1976 2547 y(x)2031 2511 y Fl(0)p 1976 2592 V 1995 2683 a Fv(\017)2064 2615 y(;)17 b(v)t(;)g(!)t Fw(\))k(+)h Fv(Av)2549 2573 y Fs(\017)2545 2639 y(\016)2605 2615 y Fw(+)g Fv(@)2754 2630 y Fs(t)2784 2615 y Fv(v)2835 2573 y Fs(\017)2831 2639 y(\016)2886 2615 y Fv(d\026)g Fw(+)3116 2479 y Fp(Z)3232 2615 y Fv(v)3283 2573 y Fs(\017)3279 2639 y(\016)3317 2615 y Fv(d\027)3422 2573 y Fs(\017)3416 2639 y(t)3455 2474 y Fp(\025)221 2903 y Fu(\025)83 b Fv(E)6 b( )543 2762 y Fp(\022)616 2903 y FA(x)p Fw(\()p Fv(T)14 b Fw(\))p Fv(;)875 2836 y FA(x)p Fw(\()p Fv(T)g Fw(\))p 875 2880 206 4 v 959 2971 a Fv(\017)1091 2903 y(;)j(T)1206 2762 y Fp(\023)1301 2903 y Fu(\000)1401 2767 y Fp(Z)1517 2903 y Fv(Av)1641 2862 y Fs(\017)1637 2928 y(\016)1697 2903 y Fw(+)22 b Fv(@)1846 2918 y Fs(t)1876 2903 y Fv(v)1927 2862 y Fs(\017)1923 2928 y(\016)1978 2903 y Fv(d\026)2088 2862 y Fs(\017)2088 2928 y(o)2148 2903 y Fu(\000)2247 2767 y Fp(Z)2364 2903 y Fv(v)2415 2862 y Fs(\017)2411 2928 y(\016)2465 2903 y Fv(d\027)2570 2862 y Fs(\017)2564 2928 y(t)0 2903 y Fw(\(51\))221 3179 y(=)84 b Fv(E)6 b( )543 3039 y Fp(\022)616 3179 y FA(x)p Fw(\()p Fv(T)14 b Fw(\))p Fv(;)875 3112 y FA(x)p Fw(\()p Fv(T)g Fw(\))p 875 3156 V 959 3248 a Fv(\017)1091 3179 y(;)j(T)1206 3039 y Fp(\023)1301 3179 y Fu(\000)1401 3044 y Fp(Z)1517 3179 y Fv(Av)1641 3138 y Fs(\017)1637 3204 y(\016)1697 3179 y Fw(+)22 b Fv(@)1846 3194 y Fs(t)1876 3179 y Fv(v)1927 3138 y Fs(\017)1923 3204 y(\016)1978 3179 y Fv(d\026)2088 3138 y Fs(\017)2088 3204 y(o)2148 3179 y Fu(\000)g Fv(v)2298 3138 y Fs(\017)2294 3204 y(\016)2332 3179 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))g Fv(:)0 3421 y Fw(Since,)34 b(b)m(y)h(\(50\))d(and)g(Dynkin's)i(F)-8 b(orm)m(ula,)848 3517 y Fp(Z)964 3653 y Fv(Av)1088 3612 y Fs(\017)1084 3678 y(\016)1144 3653 y Fw(+)22 b Fv(@)1293 3668 y Fs(t)1323 3653 y Fv(v)1374 3612 y Fs(\017)1370 3678 y(\016)1425 3653 y Fv(d\026)1535 3612 y Fs(\017)1535 3678 y(o)1595 3653 y Fw(+)g Fv(v)1744 3612 y Fs(\017)1740 3678 y(\016)1778 3653 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))27 b(=)g Fv(E)6 b(v)2356 3612 y Fs(\017)2352 3678 y(\016)2390 3653 y Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\))p Fv(;)j(T)d Fw(\))p Fv(;)0 3885 y Fw(w)m(e)34 b(deduce)g(from)e(\(51\))g(that)381 4127 y Fv(\024)22 b Fw(+)g Fv(u)613 4085 y Fs(\017)646 4127 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))k Fu(\000)1086 3986 y Fp(\024)1138 4127 y Fw(inf)1176 4186 y Fs(\026)1273 3991 y Fp(Z)1328 4217 y Fn(R)1376 4198 y Fm(n)1419 4217 y Fl(\002)p Fs(U)7 b Fl(\002)p Fr([)p Fs(t;T)j Fr(])1739 4127 y Fv(L)p Fw(\()p Fv(x)1898 4085 y Fl(0)1922 4127 y Fv(;)1976 4059 y(x)2031 4023 y Fl(0)p 1976 4104 79 4 v 1995 4195 a Fv(\017)2064 4127 y(;)17 b(v)t(;)g(!)t Fw(\))k(+)h Fv(Av)2549 4085 y Fs(\017)2545 4151 y(\016)2605 4127 y Fw(+)g Fv(@)2754 4142 y Fs(t)2784 4127 y Fv(v)2835 4085 y Fs(\017)2831 4151 y(\016)2886 4127 y Fv(d\026)g Fw(+)3116 3991 y Fp(Z)3232 4127 y Fv(v)3283 4085 y Fs(\017)3279 4151 y(\016)3317 4127 y Fv(d\027)3422 4085 y Fs(\017)3416 4151 y(t)3455 3986 y Fp(\025)221 4345 y Fu(\025)83 b Fv(E)23 b Fw([)p Fv( )d Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\)\))22 b Fu(\000)h Fv(v)1041 4304 y Fs(\017)1037 4370 y(\016)1075 4345 y Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\))p Fv(;)j(T)8 b(;)17 b(!)t Fw(\)])26 b(=)i Fv(E)22 b Fw([)q Fv(u)1910 4304 y Fs(\017)1942 4345 y Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\))p Fv(;)j(T)8 b(;)17 b(!)t Fw(\))j Fu(\000)j Fv(v)2613 4304 y Fs(\017)2609 4370 y(\016)2647 4345 y Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\))p Fv(;)j(T)d Fw(\))32 b(])h Fv(:)0 4525 y Fw(From)f(this)h(estimates,)i (sending)f Fv(\024)27 b Fu(!)h Fw(0,)k(and)h(\(48\))o(,)g(w)m(e)g (obtain)g(\(47\))o(.)931 b Fi(\003)0 4719 y Fw(W)-8 b(e)33 b(remark)g(that)g(Prop)s(osition)f(10)h(and)f(Assumption)j(6)d(imply)h (that)674 4955 y(lim)672 5018 y Fs(\016)r Fl(!)p Fr(0)828 4955 y Fw(inf)866 5015 y Fs(\026)962 4819 y Fp(Z)1018 5045 y Fn(R)1066 5026 y Fm(n)1108 5045 y Fl(\002)p Fs(U)7 b Fl(\002)p Fr([)p Fs(t;T)j Fr(])1429 4955 y Fv(L)1512 4814 y Fp(\022)1585 4955 y Fv(x)1640 4914 y Fl(0)1664 4955 y Fv(;)1717 4887 y(x)1772 4851 y Fl(0)p 1717 4932 V 1737 5023 a Fv(\017)1806 4955 y(;)17 b(v)t(;)g(!)2010 4814 y Fp(\023)2104 4955 y Fw(+)22 b Fv(Av)2326 4914 y Fs(\017)2322 4979 y(\016)2382 4955 y Fw(+)g Fv(@)2531 4970 y Fs(t)2561 4955 y Fv(v)2612 4914 y Fs(\017)2608 4979 y(\016)2663 4955 y Fv(d\026)g Fw(+)2893 4819 y Fp(Z)3009 4955 y Fv(v)3060 4914 y Fs(\017)3056 4979 y(\016)3094 4955 y Fv(d\027)3199 4914 y Fs(\017)3193 4979 y(t)513 5173 y Fw(=)83 b Fv(u)728 5132 y Fs(\017)760 5173 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))-1079 b(\(52\))0 5353 y(uniformly)36 b(in)f(\()p Fv(x;)17 b(t;)g(!)t Fw(\))31 b Fu(2)h Ft(R)1083 5317 y Fs(n)1153 5353 y Fu(\002)24 b Fw([0)p Fv(;)17 b(T)d Fw(])23 b Fu(\002)i Fw(\012,)36 b(where)g(the)f(the)g(in\014m)m (um)i(is)e(tak)m(en)h(o)m(v)m(er)g(all)e(mea-)0 5469 y(sures)g Fv(\026)p Fw(\()p Fv(x)397 5433 y Fl(0)420 5469 y Fv(;)17 b(v)t(;)g(s)p Fw(\))32 b(on)g Ft(R)882 5433 y Fs(n)951 5469 y Fu(\002)23 b Fv(U)33 b Fu(\002)22 b Fw([)p Fv(t;)17 b(T)d Fw(],)33 b(with)g(total)f(mass)i Fv(T)i Fu(\000)22 b Fv(t)p Fw(.)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 168 251 a Fq(24)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)916 450 y Fw(7.)49 b Fx(St)-7 b(a)g(tionarity)39 b(and)f(Effective)f(La)n (grangian)168 624 y Fw(From)28 b(this)h(section)h(on)f(w)m(e)g(will)h (assume)g(Assumption)g(7.)42 b(W)-8 b(e)29 b(observ)m(e)i(that,)e(as)g (a)f(consequence)168 741 y(of)k(Assumption)i(7)e(and)h(equation)g (\(6\),)g(the)g(Hamiltonian)g(is)g(also)f(stationary)-8 b(,)33 b(that)g(is)168 916 y(\(53\))650 b Fv(H)8 b Fw(\()p Fv(M)d(;)17 b(p;)g(x;)g(X)29 b Fw(+)22 b Fv(Y)5 b(;)17 b(\017;)g(!)t Fw(\))27 b(=)g Fv(H)8 b Fw(\()p Fv(M)d(;)17 b(p;)g(x;)g(X)r(;)g(\017;)g(\034)2798 931 y Fs(Y)2859 916 y Fv(!)t Fw(\))g Fv(:)168 1092 y Fw(Under)34 b(reasonable)h(h)m(yp) s(othesis)h(one)f(can)f(sho)m(w)h(that)f(the)g(minimizing)i(measures)g Fv(\026)3396 1056 y Fs(\017)3462 1092 y Fw(are)e(tigh)m(t)168 1208 y(in)g(the)h(v)-5 b(ariable)35 b Fv(x)p Fw(,)h(ho)m(w)m(ev)m(er,)h (the)e(same)g(is)g(false)g(for)f(the)h(v)-5 b(ariable)35 b Fv(X)8 b Fw(.)49 b(Therefore,)36 b(w)m(e)g(need)f(a)168 1324 y(pro)s(cedure)40 b(to)f(compactify)i(the)f(space)g(in)g(suc)m(h)h (a)e(w)m(a)m(y)i(that)e(w)m(e)h(can)g(use)h(w)m(eak)f(con)m(v)m (ergence)168 1441 y(to)s(ols.)55 b(The)38 b(stationary)f(setting)g(pro) m(vides)i(a)d(con)m(v)m(enien)m(t)k(w)m(a)m(y)e(to)e(ac)m(hiev)m(e)j (this,)f(and)f(for)f(that)168 1557 y(w)m(e)28 b(in)m(tro)s(duce)h(the)f (follo)m(wing)g(construction:)42 b(giv)m(en)28 b(a)g(measure)g Fv(\026)2658 1521 y Fs(\017)2691 1557 y Fw(\()p Fv(x)2784 1521 y Fl(0)2808 1557 y Fv(;)17 b(X)r(;)g(v)t(;)g(s)p Fw(;)g Fv(!)t Fw(\))25 b(in)j Ft(R)3473 1521 y Fs(n)3532 1557 y Fu(\002)12 b Ft(R)3693 1521 y Fs(n)3752 1557 y Fu(\002)168 1673 y Fv(U)34 b Fu(\002)24 b Fw([)p Fv(t;)17 b(T)d Fw(],)34 b(w)m(e)i(construct)f(a)f(measure)44 b(^)-57 b Fv(\026)1734 1637 y Fs(\017)1767 1673 y Fw(\()p Fv(x)1860 1637 y Fl(0)1883 1673 y Fv(;)17 b(v)t(;)g(s;)g(!)t Fw(\))33 b(in)h Ft(R)2435 1637 y Fs(n)2506 1673 y Fu(\002)23 b Fv(U)34 b Fu(\002)24 b Fw([)p Fv(t;)17 b(T)d Fw(])24 b Fu(\002)f Fw(\012)35 b(de\014ned)h(in)f(the)168 1789 y(follo)m(wing)e(w)m(a)m(y:)168 2012 y(\(54\))981 1876 y Fp(Z)1097 2012 y Fv(\036)p Fw(\()p Fv(x)1248 1971 y Fl(0)1271 2012 y Fv(;)17 b(v)t(;)g(s;)g(!)t Fw(\))p Fv(d)8 b Fw(^)-57 b Fv(\026)1713 1971 y Fs(\017)1788 2012 y Fw(:=)1935 1876 y Fp(Z)11 b(Z)2118 2012 y Fv(\036)p Fw(\()p Fv(x)2269 1971 y Fl(0)2292 2012 y Fv(;)17 b(v)t(;)g(s;)g(\034)2563 2027 y Fs(X)2630 2012 y Fv(!)t Fw(\))p Fv(d\026)2843 1971 y Fs(\017)2874 2012 y Fv(d!)j(;)168 2240 y Fw(for)32 b(an)m(y)h(b)s(ounded)g(measurable)h(function)f Fv(\036)g Fw(in)f Ft(R)2067 2204 y Fs(n)2137 2240 y Fu(\002)22 b Fv(U)33 b Fu(\002)23 b Fw([)p Fv(t;)17 b(T)d Fw(])22 b Fu(\002)g Fw(\012.)168 2356 y(Con)m(v)m(ersely)-8 b(,)48 b(giv)m(en)43 b(a)f(measure)51 b(^)-56 b Fv(\026)1497 2320 y Fs(\017)1529 2356 y Fw(\()p Fv(x)1622 2320 y Fl(0)1646 2356 y Fv(;)17 b(v)t(;)g(s;)g(!)t Fw(\),)42 b(one)h(can)f(build)h(a)f (measure)3172 2330 y(\026)3172 2356 y(^)-57 b Fv(\026)3223 2320 y Fs(\017)3255 2356 y Fw(\()p Fv(x)3348 2320 y Fl(0)3372 2356 y Fv(;)17 b(X)r(;)g(v)t(;)g(s;)g(!)t Fw(\))168 2473 y(in)33 b Ft(R)354 2437 y Fs(n)423 2473 y Fu(\002)22 b Ft(R)594 2437 y Fs(n)663 2473 y Fu(\002)h Fv(U)33 b Fu(\002)22 b Fw([)p Fv(t;)17 b(T)d Fw(])22 b Fu(\002)h Fw(\012)33 b(in)g(the)g(follo)m(wing)g(w)m(a)m(y:)168 2701 y(\(55\))1017 2565 y Fp(Z)1133 2701 y Fv(\036)p Fw(\()p Fv(x)1284 2660 y Fl(0)1307 2701 y Fv(;)17 b(X)r(;)g(v)t(;)g(s;) g(!)t Fw(\))p Fv(d)1825 2675 y Fw(\026)1825 2701 y(^)-57 b Fv(\026)1876 2660 y Fs(\017)1951 2701 y Fw(:=)2098 2565 y Fp(Z)2214 2701 y Fv(\036)p Fw(\()p Fv(x)2365 2660 y Fl(0)2389 2701 y Fv(;)17 b Fw(0)p Fv(;)g(v)t(;)g(s;)g(!)t Fw(\))p Fv(d)8 b Fw(^)-57 b Fv(\026)2924 2660 y Fs(\017)2970 2701 y Fv(;)168 2929 y Fw(for)32 b(an)m(y)h(b)s(ounded)g(measurable)h Fv(\036)f Fw(in)f Ft(R)1685 2893 y Fs(n)1754 2929 y Fu(\002)23 b Ft(R)1926 2893 y Fs(n)1995 2929 y Fu(\002)g Fv(U)33 b Fu(\002)22 b Fw([)p Fv(t;)17 b(T)d Fw(])22 b Fu(\002)h Fw(\012.)168 3045 y(Of)30 b(course,)j(with)e(de\014nitions)i(similar)e (to)g(\(54\))f(and)i(\(55\))e(\(b)m(y)i(omitting)f(the)g(dep)s(endence) j(on)c Fv(v)t Fw(\))168 3162 y(one)j(can)f(construct)i(from)e Fv(\027)1238 3125 y Fs(\017)1232 3187 y(T)1320 3162 y Fw(measures)40 b(^)-54 b Fv(\027)1793 3125 y Fs(\017)1787 3187 y(T)1874 3162 y Fw(and)2070 3135 y(\026)2070 3162 y(^)f Fv(\027)2118 3125 y Fs(\017)2112 3187 y(T)2167 3162 y Fw(.)168 3278 y(Note)32 b(that)h(if)f Fv(\036)g Fw(is)i(stationary)-8 b(,)32 b(i.e.,)i(if)1118 3453 y Fv(\036)p Fw(\()p Fv(x)1269 3412 y Fl(0)1292 3453 y Fv(;)17 b(X)r(;)g(v)t(;)g(s;)g(\034)1690 3468 y Fs(Y)1750 3453 y Fv(!)t Fw(\))43 b(=)i Fv(\036)p Fw(\()p Fv(x)2168 3412 y Fl(0)2191 3453 y Fv(;)17 b(X)30 b Fw(+)22 b Fv(Y)5 b(;)17 b(v)t(;)g(s;)g(!)t Fw(\))g Fv(;)168 3629 y Fw(then)1521 3656 y Fp(Z)1637 3792 y Fv(\036)g(d)1771 3765 y Fw(\026)1771 3792 y(^)-57 b Fv(\026)1822 3751 y Fs(\017)1881 3792 y Fw(=)1985 3656 y Fp(Z)2101 3792 y Fv(\036)17 b(d\026)2286 3751 y Fs(\017)2317 3792 y Fv(d!)j(:)168 4029 y FA(Prop)s(osition)46 b(11.)f Fo(\(i\))d(L)-5 b(et)42 b Fv(\026)1356 3993 y Fs(\017)1430 4029 y Fo(and)f Fv(\027)1680 3993 y Fs(\017)1674 4054 y(T)1771 4029 y Fo(b)-5 b(e)42 b(a)f(se)-5 b(quenc)g(e)41 b(of)g(minimizing)f(me)-5 b(asur)g(es)42 b(for)52 b Fw(\(29\))o Fo(,)168 4146 y(c)-5 b(orr)g(esp)g(onding)37 b(to)h(the)h(initial)f(tr) -5 b(ac)g(e)39 b Fv(\027)1659 4109 y Fs(\017)1653 4170 y(t)1692 4146 y Fo(,)g(ac)-5 b(c)g(or)g(ding)37 b(to)45 b Fw(\(30\))p Fo(.)55 b(Then,)39 b(the)f(quantity)i(in)45 b Fw(\(29\))38 b Fo(is)168 4262 y(uniformly)c(b)-5 b(ounde)g(d.)168 4378 y(\(ii\))47 b(L)-5 b(et)48 b Fv(\026)595 4342 y Fs(\017)675 4378 y Fo(and)f Fv(\027)931 4342 y Fs(\017)925 4403 y(T)1028 4378 y Fo(b)-5 b(e)48 b(a)f(se)-5 b(quenc)g(e)47 b(of)g(minimizing)f(me)-5 b(asur)g(es)47 b(satisfying)56 b Fw(\(30\))48 b Fo(for)f(which)168 4494 y(the)d(quantity)g(in)51 b Fw(\(29\))43 b Fo(is)h(uniformly)f(b)-5 b(ounde)g(d.)71 b(L)-5 b(et)52 b Fw(^)-56 b Fv(\026)2321 4458 y Fs(\017)2397 4494 y Fo(b)-5 b(e)43 b(as)h(de\014ne)-5 b(d)42 b(in)51 b Fw(\(54\))p Fo(,)46 b(and)d(de\014ne)168 4610 y(similarly)h Fw(^)-54 b Fv(\027)629 4574 y Fs(\017)623 4636 y(T)678 4610 y Fo(.)59 b(Then,)40 b(ther)-5 b(e)39 b(exist)h(subnets,)g(which,) g(by)g(abuse)f(of)h(notation)f(ar)-5 b(e)39 b(stil)5 b(l)40 b(denote)-5 b(d)168 4727 y(as)42 b Fw(^)-57 b Fv(\026)351 4691 y Fs(\017)418 4727 y Fo(and)40 b Fw(^)-54 b Fv(\027)662 4691 y Fs(\017)656 4752 y(T)711 4727 y Fo(,)35 b(and)f(me)-5 b(asur)g(es)34 b Fv(\032)p Fw(\()p Fv(x)1525 4691 y Fl(0)1549 4727 y Fv(;)17 b(s)p Fw(\))35 b Fo(and)f Fv(\022)s Fw(\()p Fv(v)t(;)17 b(!)t Fw(;)g Fv(x)2246 4691 y Fl(0)2268 4727 y Fv(;)g(s)p Fw(\))35 b Fo(such)f(that)168 4902 y Fw(\(56\))894 b(^)-57 b Fv(\026)1287 4861 y Fs(\017)1319 4902 y Fw(\()p Fv(x)1412 4861 y Fl(0)1436 4902 y Fv(;)17 b(v)t(;)g(s;)g(!)t Fw(\))26 b Fv(*)h(\022)s Fw(\()p Fv(v)t(;)17 b(!)t Fw(;)g Fv(x)2264 4861 y Fl(0)2286 4902 y Fv(;)g(s)p Fw(\))g Fv(\032)p Fw(\()p Fv(x)2574 4861 y Fl(0)2597 4902 y Fv(;)g(s)p Fw(\))g Fv(:)168 5078 y Fo(F)-7 b(urthermor)i(e,)36 b(for)h(e)-5 b(ach)36 b(\014xe)-5 b(d)36 b Fw(\()p Fv(x)1461 5042 y Fl(0)1485 5078 y Fv(;)17 b(s)p Fw(\))p Fo(,)36 b Fv(\022)41 b Fo(is)36 b(a)h(pr)-5 b(ob)g(ability)37 b(me)-5 b(asur)g(e)36 b(in)h Fv(U)d Fu(\002)24 b Fw(\012)p Fo(.)51 b(A)-5 b(dditional)5 b(ly,)168 5194 y(we)34 b(have)g(me)-5 b(asur)g(es)40 b Fw(\026)-54 b Fv(\027)1002 5209 y Fs(T)1092 5194 y Fo(and)39 b Fw(\026)-54 b Fv(\027)1329 5209 y Fs(t)1394 5194 y Fo(on)34 b Ft(R)1605 5158 y Fs(n)1652 5194 y Fo(,)h(de\014ne)-5 b(d)34 b(in)g(the)h(fol)5 b(lowing)33 b(way:)168 5417 y Fw(\(57\))1058 5281 y Fp(Z)1174 5417 y Fv(')p Fw(\()p Fv(x)1331 5376 y Fl(0)1354 5417 y Fw(\))17 b Fv(d)6 b Fw(\026)-55 b Fv(\027)1508 5432 y Fs(T)1563 5417 y Fw(\()p Fv(x)1656 5376 y Fl(0)1679 5417 y Fw(\))45 b(:=)f(lim)1909 5477 y Fs(\017)p Fl(!)p Fr(0)2061 5281 y Fp(Z)2177 5417 y Fv(')p Fw(\()p Fv(x)2334 5376 y Fl(0)2358 5417 y Fw(\))17 b Fv(d)6 b Fw(^)-55 b Fv(\027)2518 5376 y Fs(\017)2512 5441 y(T)2566 5417 y Fw(\()p Fv(x)2659 5376 y Fl(0)2683 5417 y Fv(;)17 b(X)r(;)g(!)t Fw(\))p eop end %%Page: 25 25 TeXDict begin 25 24 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(25)0 450 y Fo(and)0 646 y Fw(\(58\))889 510 y Fp(Z)1005 646 y Fv(')p Fw(\()p Fv(x)1162 604 y Fl(0)1185 646 y Fw(\))17 b Fv(d)6 b Fw(\026)-55 b Fv(\027)1339 661 y Fs(t)1369 646 y Fw(\()p Fv(x)1462 604 y Fl(0)1485 646 y Fw(\))44 b(:=)h(lim)1715 705 y Fs(\017)p Fl(!)p Fr(0)1867 510 y Fp(Z)1983 646 y Fv(')p Fw(\()p Fv(x)2140 604 y Fl(0)2164 646 y Fw(\))17 b Fv(d)6 b Fw(^)-55 b Fv(\027)2324 604 y Fs(\017)2318 670 y(t)2356 646 y Fw(\()p Fv(x)2449 604 y Fl(0)2473 646 y Fv(;)17 b(X)r(;)g(!)t Fw(\))g Fv(;)0 873 y Fo(for)121 834 y Fr(2)191 873 y Fo(any)35 b Fv(')27 b Fu(2)h Fv(C)633 888 y Fs(c)668 873 y Fw(\()p Ft(R)778 836 y Fs(n)825 873 y Fw(\))p Fo(.)0 1053 y(Pr)-5 b(o)g(of.)41 b Fw(The)34 b(claim)f(in)g(\(i\))f(easily)i(follo)m(ws)f(b)m(y)h (comparing)f(with)g(the)g(con)m(trol)g FA(v)c Fw(:=)f(0.)0 1169 y(W)-8 b(e)40 b(no)m(w)g(pro)m(v)m(e)h(the)f(claim)h(in)e(\(ii\).) 65 b(Consider)41 b(the)f(con)m(trol)g FA(v)i Fw(:=)d(0)h(and)f(the)h (corresp)s(onding)0 1286 y(tra)5 b(jectory)34 b FA(x)515 1301 y Fr(0)554 1286 y Fw(\()p Fv(t)p Fw(;)17 b Fv(!)t Fw(\).)45 b(By)33 b(comparing)h(with)f(the)h(measures)h(de\014ned)f(in) g(\(23\))e(and)i(\(24\))o(,)f(and)g(b)m(y)0 1402 y(exploiting)i(\(4\),) d(w)m(e)i(get)e(that)h(there)g(exists)h(a)f(constan)m(t)g Fv(C)40 b Fw(suc)m(h)34 b(that)532 1641 y Fv(C)90 b Fu(\025)853 1506 y Fp(Z)969 1501 y(\024)1022 1506 y(Z)1121 1532 y Fs(T)1077 1731 y(t)1193 1641 y Fv(L)1276 1501 y Fp(\022)1349 1641 y FA(x)1408 1656 y Fr(0)1448 1641 y Fw(\()p Fv(s)p Fw(;)17 b Fv(!)t Fw(\))p Fv(;)1731 1574 y FA(x)1790 1589 y Fr(0)1830 1574 y Fw(\()p Fv(s)p Fw(;)g Fv(!)t Fw(\))p 1731 1618 329 4 v 1876 1710 a Fv(\017)2070 1641 y(;)g Fw(0)p Fv(;)g(!)2272 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Fr(0)2409 2040 y Fw(\)\()p Fv(x)2540 1999 y Fl(0)2564 2040 y Fw(0)p Fv(;)c(v)t(;)g(!)t Fw(\))26 b(:)i Fv(D)3020 1999 y Fr(2)3017 2065 y Fs(X)3084 2040 y Fv(\036)p Fw(\(0)p Fv(;)17 b(!)t Fw(\))3376 1930 y Fp(i)1115 2241 y Fv(d\022)s Fw(\()p Fv(v)t(;)g(!)t Fw(;)g Fv(x)1511 2200 y Fl(0)1533 2241 y Fv(;)g(s)p Fw(\))g Fv(\013)q Fw(\()p Fv(x)1834 2200 y Fl(0)1857 2241 y Fv(;)g(s)p Fw(\))g Fv(d\032)p Fw(\()p Fv(x)2196 2200 y Fl(0)2219 2241 y Fv(;)g(s)p Fw(\))54 b(=)i(0)17 b Fv(;)168 2407 y Fw(where)34 b Fv(c)27 b Fw(=)h(0)k(in)h(case)g(A,)g(and)g Fv(c)27 b Fw(=)h(1)k(in)h(case)h (B.)168 2523 y(Since)f Fv(\013)h Fw(is)f(arbitrary)-8 b(,)32 b(w)m(e)i(conclude)g(that)276 2600 y Fp(Z)393 2625 y(h)440 2735 y Fv(f)11 b Fw(\()p Fv(x)592 2694 y Fl(0)615 2735 y Fv(;)17 b Fw(0)p Fv(;)g(v)t(;)g(!)t Fw(\))j Fu(\001)i Fv(D)1101 2750 y Fs(X)1168 2735 y Fv(\036)p Fw(\(0)p Fv(;)17 b(!)t Fw(\))k(+)1592 2668 y Fv(c)p 1589 2712 V 1589 2804 a Fw(2)1647 2735 y(\()p Fv(\033)1740 2750 y Fr(0)1780 2735 y Fv(\033)1839 2694 y Fs(T)1835 2760 y Fr(0)1894 2735 y Fw(\)\()p Fv(x)2025 2694 y Fl(0)2049 2735 y Fw(0)p Fv(;)c(v)t(;)g(!)t Fw(\))26 b(:)i Fv(D)2505 2694 y Fr(2)2502 2760 y Fs(X)2569 2735 y Fv(\036)p Fw(\(0)p Fv(;)17 b(!)t Fw(\))2861 2625 y Fp(i)2940 2735 y Fv(d\022)s Fw(\()p Fv(v)t(;)g(!)t Fw(;)g Fv(x)3336 2694 y Fl(0)3358 2735 y Fv(;)g(s)p Fw(\))54 b(=)i(0)168 2953 y(for)41 b(almost)i(an)m(y)g(\()p Fv(x)938 2917 y Fl(0)962 2953 y Fv(;)17 b(s)p Fw(\))41 b(with)i(resp)s(ect)h(to)e Fv(\032)p Fw(,)j(thence)e(conditions)h(\(61\))d(or)h(\(63\))g(are)g(ful\014lled,) 168 3070 y(according)33 b(to)f(cases)i(A)f(and)f(B.)168 3186 y(Multiplying)j(\(70\))d(b)m(y)i Fv(\017)1081 3150 y Fr(2)1153 3186 y Fw(and)f(arguing)f(as)h(ab)s(o)m(v)m(e,)g(one)g(sho) m(ws)h(that)e(\(65\))g(holds)i(in)e(case)i(C.)168 3302 y(The)g(fact)e(that)h Fv(\022)j Fw(satis\014es)f(\(60\))o(,)e(\(62\))f (or)h(\(64\))f(with)i Fv($)h Fw(replaced)f(b)m(y)51 b(\026)-66 b Fv($)r Fw(\()p Fv(x)3015 3266 y Fl(0)3039 3302 y Fv(;)17 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5257 y Fv(;)g(s)p Fw(\))g Fv(;)168 5469 y Fw(as)32 b(w)m(e)i(wished.)3008 b Fi(\003)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(29)0 450 y Fw(W)-8 b(e)44 b(remark)h(that,)h(for)d(some)i(c)m(hoices) h(of)d Fv($)j Fw(and)e(\004,)j(the)d(constrained)h(minimization)g (prob-)0 566 y(lem)f(giv)m(en)h(in)f(\(59\))f(ma)m(y)h(b)s(e)g(ill)f(p) s(osed,)k(since)e(there)f(migh)m(t)h(b)s(e)e(no)h(measure)h (ful\014lling)f(con-)0 683 y(ditions)c(\(61\))o(,)g(\(63\))e(or)h (\(65\))o(,)i(and)f(conditions)g(\(60\))o(,)f(\(62\))g(or)g(\(64\))o(,) i(according)f(to)f(cases)h(A-C.)0 799 y(Ho)m(w)m(ev)m(er,)g(one)d(of)g (the)g(consequences)j(of)d(Prop)s(osition)g(12)f(is)h(that)g(suc)m(h)h (a)f(problem)g(is)h(alw)m(a)m(ys)0 915 y(w)m(ell)45 b(de\014ned)h(for)d (some)i(particular)f(c)m(hoice)h(of)f Fv($)i Fw(and)e(\004,)j(i.e.,)g (at)d(least)g(for)g Fv($)49 b Fw(:=)64 b(\026)-66 b Fv($)r Fw(\()p Fv(x)3510 879 y 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Fw(as)33 b(desired.)3140 b Fi(\003)p eop end %%Page: 30 30 TeXDict begin 30 29 bop 168 251 a Fq(30)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)1356 450 y Fw(8.)49 b Fx(Homogenized)38 b(pr)n(oblems)168 624 y Fw(In)30 b(this)h(section,)h(w)m(e)g(discuss)g(the)f(homogenized)g (problem,)h(whic)m(h)g(is)f(built)f(using)h(the)g(E\013ectiv)m(e)168 741 y(Lagrangian)g(constructed)k(in)e(the)g(previous)h(section.)168 857 y(Let)27 b Fv(\027)385 872 y Fs(t)442 857 y Fw(b)s(e)g(as)g(in)g (\(22\))g(and)g(let)p 1311 777 67 4 v 27 w Fv(L)q Fw(\()p Fv(x)1471 821 y Fl(0)1494 857 y Fv(;)17 b($)s(;)g Fw(\004\))26 b(or)p 1909 777 V 27 w Fv(L)p Fw(\()p Fv(x)2068 821 y Fl(0)2092 857 y Fv(;)17 b($)s Fw(\))26 b(b)s(e)h(de\014ned)h(as)g(in)f (the)g(previous)i(section.)168 973 y(W)-8 b(e)33 b(consider)h(the)f (follo)m(wing)f(problems:)120 1110 y(A)g(or)h(B.)41 b(W)-8 b(e)39 b(lo)s(ok)f(for)g(measures)i Fv(\045)p Fw(\()p Fv(x)1595 1074 y Fl(0)1619 1110 y Fv(;)17 b($)s(;)g(s)p Fw(\))37 b(on)h Ft(R)2125 1074 y Fs(n)2198 1110 y Fu(\002)27 b Ft(R)2374 1074 y Fs(n)2447 1110 y Fu(\002)f Fw([)p Fv(t;)17 b(T)d Fw(])38 b(and)45 b(\026)-55 b Fv(\027)3036 1125 y Fs(T)3091 1110 y Fw(\()p Fv(x)3184 1074 y Fl(0)3208 1110 y Fw(\))38 b(on)g Ft(R)3497 1074 y Fs(n)3583 1110 y Fw(whic)m(h)482 1226 y(minimize)168 1411 y(\(72\))926 b(\026)-55 b Fv(u)o Fw(\()p Fv(x;)17 b(t)p Fw(\))28 b(:=)1686 1276 y Fp(Z)p 1802 1331 V 1802 1411 a Fv(L)p Fw(\()p Fv(x)1961 1370 y Fl(0)1985 1411 y Fv(;)17 b($)s Fw(\))p Fv(d\045)k Fw(+)2371 1276 y Fp(Z)2487 1411 y Fv( )t(d)6 b Fw(\026)-55 b Fv(\027)2653 1426 y Fs(T)2708 1411 y Fv(;)482 1621 y Fw(under)34 b(the)f(constrain)m(t)g(in)g(\(68\))o(.)343 1758 y(C.)42 b(W)-8 b(e)30 b(lo)s(ok)f(for)g(measures)j Fv(\045)p Fw(\()p Fv(x)1560 1722 y Fl(0)1584 1758 y Fv(;)17 b($)s(;)g Fw(\004)p Fv(;)g(s)p Fw(\))27 b(on)j Ft(R)2181 1722 y Fs(n)2243 1758 y Fu(\002)16 b Ft(R)2408 1722 y Fs(n)2472 1758 y Fu(\002)g Fw(Mat\()p Fv(n)g Fu(\002)g Fv(n)p Fw(\))g Fu(\002)g Fw([)p Fv(t;)h(T)d Fw(])30 b(and)35 b(\026)-55 b Fv(\027)3619 1773 y Fs(T)3675 1758 y Fw(\()p Fv(x)3768 1722 y Fl(0)3791 1758 y Fw(\))482 1875 y(on)33 b Ft(R)690 1838 y Fs(n)769 1875 y Fw(whic)m(h)h(minimize)168 2060 y(\(73\))871 b(\026)-55 b Fv(u)p Fw(\()p Fv(x;)17 b(t)p Fw(\))28 b(:=)1631 1924 y Fp(Z)p 1748 1980 V 1748 2060 a Fv(L)p Fw(\()p Fv(x)1907 2019 y Fl(0)1931 2060 y Fv(;)17 b($)s(;)g Fw(\004\))p Fv(d\045)j Fw(+)2425 1924 y Fp(Z)2542 2060 y Fv( )t(d)6 b Fw(\026)-55 b Fv(\027)2708 2075 y Fs(T)2763 2060 y Fv(;)482 2270 y Fw(under)34 b(the)f(constrain)m (t)g(in)g(\(69\))o(.)168 2407 y(This)g(v)-5 b(ariational)32 b(problem)i(is)f(w)m(ell)h(p)s(osed,)f(according)g(to)f(the)h(result)h (to)e(come:)168 2581 y FA(Prop)s(osition)56 b(16.)49 b Fo(Ther)-5 b(e)49 b(exist)h(minimizing)d(me)-5 b(asur)g(es)49 b Fv(\045)p Fw(\()p Fv(x)2618 2545 y Fl(0)2642 2581 y Fv(;)17 b($)s(;)g(s)p Fw(\))48 b Fo(on)h Ft(R)3172 2545 y Fs(n)3252 2581 y Fu(\002)34 b Ft(R)3435 2545 y Fs(n)3515 2581 y Fu(\002)f Fw([)p Fv(t;)17 b(T)d Fw(])168 2698 y Fo(and)40 b Fw(\026)-55 b Fv(\027)405 2713 y Fs(T)460 2698 y Fw(\()p Fv(x)553 2661 y Fl(0)577 2698 y Fw(\))35 b Fo(on)f Ft(R)861 2661 y Fs(n)908 2698 y Fo(,)h(as)f(r)-5 b(e)g(queste)g(d)35 b(ab)-5 b(ove.)44 b(Mor)-5 b(e)g(over,)168 2855 y Fw(\(74\))974 b Fu(j)6 b Fw(\026)-55 b Fv(u)o Fw(\()p Fv(x;)17 b(t)p Fw(\))p Fu(j)28 b(\024)g Fv(C)7 b Fw(\(1)22 b(+)g Fu(k)p Fv( )t Fu(k)2221 2870 y Fl(1)2317 2855 y Fw(+)g Fv(T)36 b Fu(\000)23 b Fv(t)p Fw(\))168 3013 y Fo(for)34 b(any)h Fv(x)28 b Fu(2)g Ft(R)759 2976 y Fs(n)841 3013 y Fo(and)34 b Fv(t)28 b Fu(\024)h Fv(T)168 3187 y Fo(Pr)-5 b(o)g(of.)41 b Fw(The)33 b(idea)g(of)f(the)h(pro)s(of)f (is)h(that)f(the)h(measures)h(built)f(in)g(Prop)s(ositions)g(11)f(and)h (12)f(giv)m(e)168 3303 y(an)38 b(upp)s(er)h(b)s(ound)g(on)g(the)g (action)f(and)h(a)f(natural)h(candidate)g(for)f(the)h(minimization)h (pro)s(cess)168 3420 y(\(though)27 b(ma)m(yb)s(e)h(not)f(the)g(minim)m (um)i(itself)7 b(\).)42 b(The)28 b(minimization)g(pro)s(cess)g(then)g (con)m(v)m(erges)h(due)168 3541 y(to)k(the)i(momen)m(t)g(estimates)h (of)d(Prop)s(osition)h(5)g(and)g(the)h(lo)m(w)m(er)g(semicon)m(tin)m (uit)m(y)i(of)p 3370 3461 V 34 w Fv(L)d Fw(giv)m(en)h(b)m(y)168 3657 y(Prop)s(osition)e(14.)168 3773 y(F)-8 b(ollo)m(wing)32 b(are)h(the)g(details)g(of)f(the)h(pro)s(of.)43 b(First,)33 b(w)m(e)g(consider)h(the)f(pro)s(cess)1005 3974 y Fv(d)p FA(x)1115 3933 y Fs(\017)1175 3974 y Fw(=)28 b Fv(f)1338 3864 y Fp(\020)1397 3974 y FA(x)1456 3933 y Fs(\017)1489 3974 y Fv(;)1543 3907 y FA(x)1602 3871 y Fs(\017)p 1543 3952 92 4 v 1569 4043 a Fv(\017)1644 3974 y(;)17 b Fw(0)p Fv(;)g(!)1846 3864 y Fp(\021)1904 3974 y Fv(dt)22 b Fw(+)g Fv(\033)2169 3864 y Fp(\020)2229 3974 y FA(x)2288 3933 y Fs(\017)2321 3974 y Fv(;)2374 3907 y FA(x)2433 3871 y Fs(\017)p 2374 3952 V 2401 4043 a Fv(\017)2476 3974 y(;)17 b(\017;)g Fw(0)p Fv(;)g(!)2761 3864 y Fp(\021)2819 3974 y Fv(dW)2962 3989 y Fs(t)168 4167 y Fw(with)42 b FA(x)458 4131 y Fs(\017)491 4167 y Fw(\()p Fv(t)p Fw(\))i(=)f Fv(x)g Fw(and)f(let)g Fv(\026)1271 4182 y Fs(\017)1345 4167 y Fw(and)g Fv(\027)1592 4182 y Fs(\017)1667 4167 y Fw(b)s(e)g(as)g(in)g(\(23\))g(and)g(\(24\))f(with)i Fv(v)t Fw(\()p Fv(s)p Fw(\))g(:=)g(0.)72 b(Then,)45 b(b)m(y)168 4283 y(Assumption)34 b(14,)684 4352 y Fp(Z)801 4488 y Fv(Ld\026)977 4503 y Fs(\017)1031 4488 y Fw(+)1129 4352 y Fp(Z)1246 4488 y Fv( )t(d\027)1412 4503 y Fs(\017)526 4766 y Fw(=)82 b Fv(E)779 4630 y Fp(Z)879 4656 y Fs(T)834 4856 y(t)950 4766 y Fv(L)1016 4655 y Fp(\020)1076 4766 y FA(x)1135 4725 y Fs(\017)1168 4766 y Fw(\()p Fv(s)p Fw(\))p Fv(;)1344 4698 y FA(x)1403 4662 y Fs(\017)1435 4698 y Fw(\()p Fv(s)p Fw(\))p 1344 4743 214 4 v 1431 4834 a Fv(\017)1567 4766 y(;)17 b Fw(0)p Fv(;)g(!)1769 4655 y Fp(\021)1827 4766 y Fv(ds)22 b Fw(+)g Fv( )2111 4655 y Fp(\020)2170 4766 y FA(x)2229 4725 y Fs(\017)2262 4766 y Fv(;)2316 4698 y FA(x)2375 4662 y Fs(\017)p 2316 4743 92 4 v 2342 4834 a Fv(\017)2418 4655 y Fp(\021)2522 4766 y Fu(\024)44 b Fv(C)7 b Fw(\(1)22 b(+)g Fu(k)p Fv( )t Fu(k)3094 4781 y Fl(1)3191 4766 y Fw(+)g Fv(T)35 b Fu(\000)23 b Fv(t)p Fw(\))-3386 b(\(75\))168 4974 y(and)46 b(so)g(the)h(second)h (claim)e(in)h(Prop)s(osition)f(11)g(holds)h(true.)85 b(Consequen)m(tly)-8 b(,)52 b(the)47 b(claim)g(in)168 5091 y(Prop)s(osition)23 b(12)f(holds)i(true.)40 b(In)24 b(particular,)h(the)e(measure)h Fv(\026)2458 5106 y Fs(\017)2514 5091 y Fw(induces)g(in)f(the)h(limit)f(a)g(measure)h Fv(\045)168 5207 y Fw(satisfying)35 b(\(68\))d(and)g(\(69\).)43 b(By)35 b(\(71\))d(and)g(\(75\),)g(w)m(e)i(get)e(that)168 5417 y(\(76\))1114 5281 y Fp(Z)p 1231 5337 67 4 v 1231 5417 a Fv(Ld\045)22 b Fw(+)1518 5281 y Fp(Z)1634 5417 y Fv( )t(d)6 b Fw(\026)-55 b Fv(\027)1800 5432 y Fs(T)1883 5417 y Fu(\024)28 b Fv(C)7 b Fw(\(1)22 b(+)g Fu(k)p Fv( )t Fu(k)2439 5432 y Fl(1)2535 5417 y Fw(+)g Fv(T)36 b Fu(\000)23 b Fv(t)p Fw(\))p eop end %%Page: 31 31 TeXDict begin 31 30 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(31)0 450 y Fw(that)32 b(is,)i(the)f(quan)m(tities)h(in)f(\(72\))f (and)g(\(73\))g(are)h(not)f(alw)m(a)m(ys)i(in\014nite.)0 566 y(This)39 b(allo)m(w)e(us)h(to)f(use)i(the)e(direct)h(metho)s(d)g (in)g(the)g(calculus)g(of)f(v)-5 b(ariations,)39 b(that)e(is,)i(w)m(e)g (tak)m(e)0 683 y(a)34 b(minimizing)j(sequence)g(of)e(measures,)i(sa)m (y)f Fv(\045)1779 698 y Fs(n)1861 683 y Fw(and)k(\026)-55 b Fv(\027)2100 698 y Fs(T)6 b(;n)2214 683 y Fw(.)50 b(W)-8 b(e)35 b(apply)h(to)e(these)i(measures)h(the)0 799 y(momen)m(t)d (estimates)h(of)e(Prop)s(osition)g(5)g(\(with)h(resp)s(ect)h(to)e(the)g (notation)g(in)h(Prop)s(osition)f(5,)g(w)m(e)0 915 y(replace)43 b(what)f(there)h(is)g(called)g Fv(\026)1305 879 y Fs(\017)1337 915 y Fw(,)h Fv(\027)1462 879 y Fs(\017)1456 940 y(T)1512 915 y Fw(,)g Fv(f)11 b Fw(,)44 b Fv(\033)t(\033)1831 879 y Fs(T)1928 915 y Fw(and)e Fv(L)h Fw(with)f(the)h(quan)m(tities)h (denoted)f(here)0 1037 y(b)m(y)37 b Fv(\045)189 1052 y Fs(n)236 1037 y Fw(,)43 b(\026)-54 b Fv(\027)349 1052 y Fs(T)6 b(;n)462 1037 y Fw(,)38 b Fv($)s Fw(,)e(\004)g(and)p 969 956 67 4 v 37 w Fv(L)p Fw(\).)55 b(Note)36 b(that)g(the)h(measures) h Fv(\045)2253 1052 y Fs(n)2336 1037 y Fw(and)k(\026)-54 b Fv(\027)2578 1052 y Fs(T)6 b(;n)2727 1037 y Fw(satisfy)38 b(a)e(constrain)m(t)h(of)0 1153 y(the)e(t)m(yp)s(e)h(in)g(\(30\))o(,)g (thanks)g(to)f(\(68\))f(or)h(\(69\))o(,)h(that)f(the)g(uniform)h(b)s (ound)f(on)g(the)g(action)g(of)g(the)0 1269 y(t)m(yp)s(e)e(in)g(\(32\)) e(is)i(assured)h(here)f(b)m(y)i(\(76\))c(and)i(that)f(the)h(gro)m(wth)f (condition)h(in)g(Assumption)h(4)e(is)0 1385 y(giv)m(en)i(here)f(b)m(y) h(Prop)s(osition)e(15.)0 1501 y(Then,)43 b(the)d(tigh)m(tness)h(giv)m (en)g(b)m(y)f(Prop)s(osition)g(5)f(and)h(Assumption)h(8)f(giv)m(es)h (the)f(existence)i(of)0 1618 y(the)33 b(desired)h(minimal)f(measures,)i (since,)f(b)m(y)g(Prop)s(osition)e(14,)p 2408 1538 V 33 w Fv(L)g Fw(is)i(lo)m(w)m(er)f(semicon)m(tin)m(uous.)0 1734 y(Also,)g(the)g(uniform)g(b)s(ound)g(in)f(\(76\))g(easily)i (implies)g(\(74\).)1382 b Fi(\003)0 1912 y FA(Prop)s(osition)50 b(17.)d Fo(The)d(me)-5 b(asur)g(e)44 b Fv(\045)p Fo(,)k(as)c(intr)-5 b(o)g(duc)g(e)g(d)44 b(in)h(Pr)-5 b(op)g(osition)44 b(12,)i (satis\014es)53 b Fw(\(68\))44 b Fo(in)0 2029 y(c)-5 b(ases)34 b(A)h(and)g(B,)f(and)45 b Fw(\(69\))34 b Fo(in)g(c)-5 b(ase)35 b(C.)0 2207 y(Pr)-5 b(o)g(of.)41 b Fw(Let)33 b Fv(')p Fw(\()p Fv(x)630 2171 y Fl(0)653 2207 y Fv(;)17 b(t)p Fw(\))33 b(b)s(e)g(a)f(smo)s(oth)h(function.)44 b(W)-8 b(e)33 b(use)g(\(21\),)f(\(12\))g(and)h(\(13\))f(to)g(gather)g (that)372 2301 y Fp(Z)489 2437 y Fv(@)540 2452 y Fs(t)570 2437 y Fv(')p Fw(\()p Fv(x)727 2396 y Fl(0)750 2437 y Fv(;)17 b(s)p Fw(\))22 b(+)g Fv(f)11 b Fw(\()p Fv(x)1150 2396 y Fl(0)1173 2437 y Fv(;)17 b Fw(0)p Fv(;)g(v)t(;)g(!)t Fw(\))j Fu(\001)i Fv(D)1659 2452 y Fs(x)1703 2437 y Fv(')p Fw(\()p Fv(x)1860 2396 y Fl(0)1884 2437 y Fv(;)17 b(s)p Fw(\))k(+)2141 2370 y Fv(\033)t(\033)2259 2334 y Fs(T)2314 2370 y Fw(\()p Fv(x)2407 2334 y Fl(0)2431 2370 y Fv(;)c Fw(0)p Fv(;)g(\017;)g(v)t(;)g(!)t Fw(\))p 2141 2414 706 4 v 2470 2505 a(2)2885 2437 y(:)27 b Fv(D)3023 2396 y Fr(2)3020 2462 y Fs(x)3064 2437 y Fv(')p Fw(\()p Fv(x)3221 2396 y Fl(0)3245 2437 y Fv(;)17 b(s)p Fw(\))g Fv(d)8 b Fw(^)-57 b Fv(\026)3500 2396 y Fs(\017)213 2696 y Fw(=)372 2560 y Fp(Z)12 b(Z)555 2696 y Fv(')p Fw(\()p Fv(x)712 2655 y Fl(0)735 2696 y Fv(;)17 b(T)d Fw(\))j Fv(d\027)1010 2655 y Fs(\017)1004 2720 y(T)1075 2696 y Fv(d!)25 b Fu(\000)1312 2560 y Fp(Z)11 b(Z)1494 2696 y Fv(')p Fw(\()p Fv(x)1651 2655 y Fl(0)1675 2696 y Fv(;)17 b(t)p Fw(\))g Fv(d\027)1914 2655 y Fs(\017)1908 2720 y(t)1963 2696 y Fv(d!)j(:)0 2907 y Fw(Therefore,)34 b(w)m(e)f(use)h(Prop)s(osition)f(11)f(to)g (send)i Fv(\017)28 b Fu(!)f Fw(0,)33 b(and)f(w)m(e)i(obtain)849 2983 y Fp(Z)12 b(Z)1032 3118 y Fv(@)1083 3133 y Fs(t)1113 3118 y Fv(')p Fw(\()p Fv(x)1270 3077 y Fl(0)1294 3118 y Fv(;)17 b(s)p Fw(\))k(+)i Fv(f)11 b Fw(\()p Fv(x)1694 3077 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Fv(;)17 b(t)p Fw(\))g Fv(d)6 b Fw(\026)-55 b Fv(\027)2127 3601 y Fs(\017)2121 3667 y(t)2176 3642 y Fv(;)0 3859 y Fw(where)34 b Fv(\033)341 3823 y Fl(0)392 3859 y Fw(:=)27 b(0)33 b(in)g(cases)g(A)g(and)g(B,)g(and)f Fv(\033)1635 3823 y Fl(0)1686 3859 y Fw(:=)c Fv(\033)1872 3874 y Fr(0)1912 3859 y Fw(\()p Fv(x)2005 3823 y Fl(0)2028 3859 y Fv(;)17 b Fw(0)p Fv(;)g(v)t(;)g(!)t Fw(\))31 b(in)i(case)g(C.)0 3975 y(By)g(means)h(of)e(Prop)s(osition)h(12,)f(w)m(e)h(kno)m(w)h(that) 1230 4119 y(1)p 1230 4163 49 4 v 1230 4255 a(2)1322 4051 y Fp(Z)1438 4186 y Fv(\033)1493 4201 y Fr(0)1533 4186 y Fv(\033)1592 4145 y Fs(T)1588 4211 y Fr(0)1664 4186 y Fv(d\022)s Fw(\()p Fv(v)t(;)17 b(!)t Fw(;)g Fv(x)2060 4145 y Fl(0)2082 4186 y Fv(s)p Fw(\))27 b(=)2305 4161 y(\026)2297 4186 y(\004\()p Fv(x)2455 4145 y Fl(0)2478 4186 y Fv(;)17 b(s)p Fw(\))835 4445 y(and)1237 4309 y Fp(Z)1353 4445 y Fv(f)11 b Fw(\()p Fv(x;)17 b Fw(0)p Fv(;)g(v)t(;)g(!)t Fw(\))g Fv(d\022)s Fw(\()p Fv(v)t(;)g(!)t Fw(;)g Fv(x)2253 4404 y 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b(in)h(case)h(C.)0 5206 y(The)g(latter)e(quan)m (tit)m(y)i(equals)772 5281 y Fp(Z)888 5417 y Fv(@)939 5432 y Fs(t)969 5417 y Fv(')22 b Fw(+)g Fv($)j Fu(\001)c Fv(D)1389 5432 y Fs(x)1433 5417 y Fv(')p Fw(\()p Fv(x)1590 5376 y Fl(0)1614 5417 y Fv(;)c(s)p Fw(\))22 b(+)g Fv(c)p Fw(\004\()p Fv(x)2062 5376 y Fl(0)2085 5417 y Fv(;)17 b(s)p Fw(\))28 b(:)f Fv(D)2379 5376 y Fr(2)2376 5441 y Fs(x)2420 5417 y Fv(')p Fw(\()p Fv(x)2577 5376 y Fl(0)2601 5417 y Fv(;)17 b(s)p Fw(\))g Fv(d\045)g(;)p eop end %%Page: 32 32 TeXDict begin 32 31 bop 168 251 a Fq(32)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y Fw(th)m(us)33 b(yielding)h(the)f(desired)h(result.)2240 b Fi(\003)1467 683 y Fw(9.)49 b Fx(Unif)n(orm)39 b(estima)-7 b(tes)168 858 y Fw(This)34 b(section)g(is)g(dedicated)g(to)f(pro)m (ving)g(certain)h(uniform)f(estimates)i(that)d(allo)m(w)i(us)g(to)e (obtain)168 974 y(con)m(v)m(ergence)37 b(of)e Fv(u)879 938 y Fs(\017)946 974 y Fw(to)f(a)h(non-random)g(limit.)51 b(F)-8 b(or)34 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1671 y Fw(is)i(non-random.)168 1853 y FA(Prop)s(osition)j(18.)42 b Fo(We)35 b(have)168 2048 y Fw(\(78\))665 b Fu(k)p Fv(u)1113 2063 y Fs(t)1142 2048 y Fu(k)1192 2063 y Fl(1)1294 2048 y Fu(\024)28 b Fw(lim)17 b(sup)1482 2128 y Fs(\017)p Fl(!)p Fr(0)1715 2048 y Fw(sup)1768 2128 y Fs(x)1878 1934 y Fp(\014)1878 1994 y(\014)1878 2053 y(\014)1912 2048 y Fv(H)2001 1938 y Fp(\020)2060 2048 y Fv(D)2144 2007 y Fr(2)2183 2048 y Fv( )t(;)g(D)s( )t(;)g(x;)2597 1981 y(x)p 2597 2025 56 4 v 2605 2117 a(\017)2662 2048 y(;)g(\017;)g(!)2854 1938 y Fp(\021)2913 1934 y(\014)2913 1994 y(\014)2913 2053 y(\014)2963 2048 y Fv(:)168 2280 y Fo(Pr)-5 b(o)g(of.)41 b Fw(By)46 b(Assumption)h(10,)i(w)m(e)e(ma)m(y) f(assume)h(that)f(b)s(oth)f(the)i(terminal)f(data)f Fv( )50 b Fw(and)c(the)168 2396 y(solution)35 b Fv(u)595 2360 y Fs(\017)662 2396 y Fw(are)g(smo)s(oth.)51 b(De\014ne)36 b Fv(w)s Fw(\()p Fv(x;)17 b(t)p Fw(\))31 b(=)h Fv(u)2000 2411 y Fs(t)2029 2396 y Fw(\()p Fv(x;)17 b(T)37 b Fu(\000)25 b Fv(t)p Fw(\).)51 b(Then,)36 b Fv(w)h Fw(satis\014es)g(the)e(parab)s (olic)168 2513 y(equation)168 2682 y(\(79\))1374 b Fv(w)1786 2697 y Fs(t)1837 2682 y Fw(+)22 b Fv(Lw)31 b Fw(=)c(0)p Fv(;)168 2852 y Fw(where)34 b Fv(L)e Fw(is)i(the)f(elliptic)g(op)s (erator)1308 3021 y Fv(Lw)d Fw(=)e Fv(D)1659 3036 y Fs(M)1738 3021 y Fv(H)35 b Fw(:)28 b Fv(D)1993 2980 y Fr(2)2032 3021 y Fv(w)d Fw(+)d Fv(D)2306 3036 y Fs(p)2345 3021 y Fv(H)30 b Fu(\001)22 b Fv(D)s(w)s(;)168 3190 y Fw(and)36 b Fv(w)s Fw(\()p Fv(x;)17 b Fw(0\))34 b(=)g Fv(H)8 b Fw(\()p Fv(D)1013 3154 y Fr(2)1052 3190 y Fv( )t(;)17 b(D)s( )t(;)g(x;)1466 3151 y Fs(x)p 1466 3167 40 4 v 1472 3225 a(\017)1516 3190 y Fw(\).)55 b(Note)36 b(that)h(the)g(fact)f (that)g Fv(L)h Fw(is)g(elliptic)h(\(thence)g(\(79\))e(is)168 3308 y(parab)s(olic\))c(is)h(a)f(consequence)k(of)j(\(14\).)168 3424 y(Since)47 b(equation)g(\(79\))e(has)h(maxim)m(um)i(principle)f (\(see,)k(for)45 b(instance,)51 b(Prop)s(osition)46 b(4.1)g(and)168 3540 y(Theorem)34 b(4.2)e(in)h([DiB95)o(])g(or)f(page)g(165)g(in)h([Sh) m(u92)q(]\),)f(w)m(e)i(obtain)e(the)h(required)h(estimate.)117 b Fi(\003)168 3722 y FA(Remark)38 b(19.)j Fw(W)-8 b(e)33 b(observ)m(e)i(that)d(the)h(righ)m(t)g(hand)g(side)g(of)39 b(\(78\))32 b(is)h(\014nite,)h(due)f(to)f(\(8\).)168 3903 y FA(Prop)s(osition)37 b(20.)42 b Fu(k)p Fv(D)s(u)1148 3867 y Fs(\017)1180 3903 y Fu(k)1230 3918 y Fl(1)1339 3903 y Fo(is)35 b(uniformly)f(b)-5 b(ounde)g(d)34 b(in)h Fv(\017)p Fo(.)168 4085 y(Pr)-5 b(o)g(of.)41 b Fw(W)-8 b(e)40 b(will)g(pro)m(v)m(e)h(separately)g(the)g(cases)g(in)f(whic)m(h) h Fv(H)47 b Fw(satis\014es)42 b(P1)d(or)h(P2)g(of)f(Assump-)168 4201 y(tion)32 b(11.)168 4317 y FA(Case)38 b(1.)43 b Fw(If)33 b Fv(H)40 b Fw(satis\014es)34 b(P1)f(then)g(w)m(e)h(ha)m(v)m (e,)g(from)e(Prop)s(osition)h(18)1599 4487 y Fv(H)8 b Fw(\()p Fv(D)s(u)1866 4445 y Fs(\017)1897 4487 y Fv(;)17 b(x;)g(X)8 b Fw(\))27 b Fu(\024)h Fv(C)r(;)168 4656 y Fw(whic)m(h)34 b(b)m(y)h(\(16\))d(implies)i(that)e Fu(k)p Fv(D)s(u)1522 4620 y Fs(\017)1554 4656 y Fu(k)1604 4671 y Fl(1)1711 4656 y Fw(is)h(b)s(ounded.)168 4772 y FA(Case)j(2.)43 b Fw(By)32 b(Assumption)h(10,)e(w)m(e)h(ma)m(y)g(supp)s(ose)h(that)e Fv(u)2395 4736 y Fs(\017)2458 4772 y Fw(and)g Fv( )36 b Fw(are)31 b(smo)s(oth.)43 b(Consider)33 b(the)168 4889 y(follo)m(wing)g(rescaled)h(v)m(ersion)g(of)e(the)h(solution)1544 5101 y Fv(U)10 b Fw(\()p Fv(\034)c(;)17 b(y)t Fw(\))27 b(=)1981 5034 y(1)p 1981 5078 49 4 v 1986 5169 a Fv(\017)2040 5101 y(u)2096 5060 y Fs(\017)2128 5101 y Fw(\()p Fv(\017\034)6 b(;)17 b(\017y)t Fw(\))p Fv(:)168 5300 y Fw(Then)168 5469 y(\(80\))1378 b Fu(k)p Fv(U)10 b Fu(k)1896 5484 y Fl(1)1998 5469 y Fv(<)28 b Fw(+)p Fu(1)p eop end %%Page: 33 33 TeXDict begin 33 32 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(33)0 450 y Fw(b)s(ecause)34 b(of)e(Prop)s(osition)h(18)f(and)h (Remark)g(19,)f(and)h Fv(U)43 b Fw(satis\014es)34 b(the)f(equation)1034 688 y Fu(\000)p Fv(U)1177 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1595 y Fs(\034)215 1580 y Fw(=)27 b Fv(D)s(U)33 b Fu(\001)22 b Fv(D)s(U)701 1595 y Fs(\034)215 1788 y Fw(=)328 1721 y(1)p 328 1765 49 4 v 333 1856 a Fv(\017)387 1788 y(D)468 1803 y Fs(i)496 1788 y Fv(U)10 b(D)653 1803 y Fs(M)721 1815 y Fm(j)s(k)792 1788 y Fv(H)e(D)965 1747 y Fr(3)962 1813 y Fs(ij)t(k)1061 1788 y Fv(U)33 b Fw(+)22 b Fv(D)1339 1803 y Fs(i)1367 1788 y Fv(U)10 b(D)1524 1803 y Fs(p)1560 1813 y Fm(j)1597 1788 y Fv(H)e(D)1770 1747 y Fr(2)1767 1813 y Fs(ij)1827 1788 y Fv(U)33 b Fw(+)22 b Fv(\017D)2144 1803 y Fs(i)2172 1788 y Fv(U)10 b(D)2329 1803 y Fs(x)2369 1813 y Fm(i)2400 1788 y Fv(H)30 b Fw(+)22 b Fv(D)2690 1803 y Fs(i)2718 1788 y Fv(U)10 b(D)2875 1803 y Fs(X)2933 1813 y Fm(i)2964 1788 y Fv(H)215 2025 y Fw(=)328 1958 y(1)p 328 2002 V 333 2094 a Fv(\017)387 2025 y(D)468 2040 y Fs(M)536 2052 y Fm(j)s(k)607 2025 y Fv(H)e(D)780 1984 y Fr(2)777 2050 y Fs(j)t(k)851 2025 y Fv(w)25 b Fu(\000)1055 1958 y Fw(1)p 1055 2002 V 1060 2094 a Fv(\017)1114 2025 y(D)1195 2040 y Fs(M)1263 2052 y Fm(j)s(k)1333 2025 y Fv(H)8 b(D)1506 1984 y Fr(2)1503 2050 y Fs(ij)1563 2025 y Fv(U)i(D)1723 1984 y Fr(2)1720 2050 y Fs(ik)1788 2025 y Fv(U)33 b Fw(+)22 b Fv(D)2066 2040 y Fs(p)2102 2050 y Fm(j)2138 2025 y Fv(H)8 b(D)2308 2040 y Fs(j)2344 2025 y Fv(w)24 b Fw(+)e Fv(\017D)2656 2040 y Fs(i)2685 2025 y Fv(U)10 b(D)2842 2040 y Fs(x)2882 2050 y Fm(i)2913 2025 y Fv(H)29 b Fw(+)22 b Fv(D)3202 2040 y Fs(i)3231 2025 y Fv(U)10 b(D)3388 2040 y Fs(X)3446 2050 y Fm(i)3477 2025 y Fv(H)r(:)0 1464 y Fw(\(81\))0 2250 y(If)33 b Fv(w)h Fw(admits)g(an)e(in)m(terior)h(maxim)m(um,)i(w)m(e)e(ha)m(v)m(e)h Fv(w)1923 2265 y Fs(t)1980 2250 y Fw(=)28 b(0)k(and)h Fv(D)s(w)d Fw(=)d(0)32 b(there,)i(and)1450 2445 y Fv(D)1531 2460 y Fs(M)1599 2472 y Fm(j)s(k)1669 2445 y Fv(H)8 b(D)1842 2404 y Fr(2)1839 2470 y Fs(j)t(k)1914 2445 y Fv(w)30 b Fu(\025)e Fw(0)17 b Fv(;)0 2644 y Fw(due)33 b(to)f(\(14\).)43 b(Th)m(us,)703 2888 y Fu(\000)790 2821 y Fw(1)p 790 2865 V 795 2956 a Fv(\017)850 2888 y(D)931 2903 y Fs(M)999 2915 y Fm(j)s(k)1069 2888 y Fv(H)8 b(D)1242 2847 y Fr(2)1239 2913 y Fs(ij)1299 2888 y Fv(U)i(D)1459 2847 y Fr(2)1456 2913 y Fs(ik)1523 2888 y Fv(U)33 b Fw(+)22 b Fv(\017D)1840 2903 y Fs(i)1869 2888 y Fv(U)10 b(D)2026 2903 y Fs(x)2066 2913 y Fm(i)2097 2888 y Fv(H)29 b Fw(+)22 b Fv(D)2386 2903 y Fs(i)2415 2888 y Fv(U)10 b(D)2572 2903 y Fs(X)2630 2913 y Fm(i)2661 2888 y Fv(H)35 b Fu(\024)28 b Fw(0)p Fv(:)0 3118 y Fw(But)35 b(then)g(P2)g(of)f(Assumption)i(11)e(and)h (\(80\))f(imply)i(that)e Fv(D)s(U)45 b Fw(is)35 b(uniformly)h(b)s (ounded)f(in)g(this)0 3234 y(case.)0 3351 y(W)-8 b(e)32 b(no)m(w)g(mo)s(dify)g(the)g(ab)s(o)m(v)m(e)h(argumen)m(t)f(in)g(case)h Fv(w)h Fw(do)s(es)e(not)f(attain)h(its)g(maxim)m(um.)45 b(F)-8 b(or)31 b(this,)0 3467 y(w)m(e)39 b(observ)m(e)g(that)f Fu(k)p Fv(w)s Fu(k)891 3482 y Fl(1)1001 3467 y Fv(<)e Fw(+)p Fu(1)p Fw(,)i(thanks)h(to)e(\(15\).)59 b(Hence,)40 b(if,)f(for)e(a)g(small)i Fv(q)i Fw(\(ev)m(en)e(smaller)0 3586 y(than)h Fv(\017)p Fw(\),)j(w)m(e)e(de\014ne)g Fv(w)895 3550 y Fr(\()p Fs(q)r Fr(\))988 3586 y Fw(\()p Fv(y)t Fw(\))e(:=)i Fv(w)s Fw(\()p Fv(y)t Fw(\))26 b Fu(\000)i Fv(q)20 b Fw(ln\(1)28 b(+)f Fu(j)p Fv(y)t Fu(j)2101 3550 y Fr(2)2139 3586 y Fw(\))40 b(w)m(e)h(ha)m(v)m(e)h(that)e Fv(w)2893 3550 y Fr(\()p Fs(q)r Fr(\))3025 3586 y Fw(has)h(an)f(in)m (terior)0 3720 y(maxim)m(um,)f(sa)m(y)e(at)f Fv(y)830 3735 y Fs(q)867 3720 y Fw(.)55 b(Then,)37 b Fv(w)1307 3669 y Fr(\()p Fs(q)r Fr(\))1304 3743 y Fs(t)1400 3720 y Fw(\()p Fv(y)1486 3735 y Fs(q)1523 3720 y Fw(\))d(=)g(0,)i Fv(D)s(w)1974 3669 y Fr(\()p Fs(q)r Fr(\))1971 3743 y Fs(t)2066 3720 y Fw(\()p Fv(y)2152 3735 y Fs(q)2190 3720 y Fw(\))e(=)f(0)j(and)h Fv(D)2731 3735 y Fs(M)2799 3747 y Fm(j)s(k)2869 3720 y Fv(H)8 b(D)3042 3684 y Fr(2)3039 3746 y Fs(j)t(k)3114 3720 y Fv(w)3187 3669 y Fr(\()p Fs(q)r Fr(\))3184 3743 y Fs(t)3279 3720 y Fw(\()p Fv(y)3365 3735 y Fs(q)3402 3720 y Fw(\))34 b Fu(\025)h Fw(0,)0 3868 y(b)m(y)j(\(14\).)54 b(As)36 b(a)g(result,)h Fv(w)999 3883 y Fs(t)1029 3868 y Fw(\()p Fv(y)1115 3883 y Fs(q)1152 3868 y Fw(\))d(=)f(0,)k Fu(j)p Fv(D)s(w)1628 3883 y Fs(t)1657 3868 y Fw(\()p Fv(y)1743 3883 y Fs(q)1781 3868 y Fw(\))p Fu(j)c(\024)h Fv(C)7 b(q)40 b Fw(and)c Fv(D)2425 3883 y Fs(M)2493 3895 y Fm(j)s(k)2563 3868 y Fv(H)8 b(D)2736 3832 y Fr(2)2733 3894 y Fs(j)t(k)2808 3868 y Fv(w)2881 3817 y Fr(\()p Fs(q)r Fr(\))2878 3890 y Fs(t)2973 3868 y Fw(\()p Fv(y)3059 3883 y Fs(q)3097 3868 y Fw(\))33 b Fu(\025)i(\000)p Fv(C)7 b(q)t Fw(,)37 b(for)0 3990 y(some)c Fv(C)i(>)28 b Fw(0.)43 b(Substituting)34 b(in)e(\(81\),)g(w)m (e)i(obtain)f(that)667 4233 y Fu(\000)754 4166 y Fw(1)p 754 4210 V 759 4302 a Fv(\017)813 4233 y(D)894 4248 y Fs(M)962 4260 y Fm(j)s(k)1032 4233 y Fv(H)8 b(D)1205 4192 y Fr(2)1202 4258 y Fs(ij)1262 4233 y Fv(U)i(D)1422 4192 y Fr(2)1419 4258 y Fs(ik)1487 4233 y Fv(U)33 b Fw(+)22 b Fv(\017D)1804 4248 y Fs(i)1832 4233 y Fv(U)10 b(D)1989 4248 y Fs(x)2029 4258 y Fm(i)2060 4233 y Fv(H)30 b Fw(+)22 b Fv(D)2350 4248 y Fs(i)2378 4233 y Fv(U)10 b(D)2535 4248 y Fs(X)2593 4258 y Fm(i)2624 4233 y Fv(H)35 b Fu(\024)28 b Fv(C)2915 4248 y Fs(\017)2948 4233 y Fv(q)0 4464 y Fw(at)k(the)h(p)s(oin)m(t)g Fv(y)590 4479 y Fs(q)628 4464 y Fw(,)f(for)g(some)i Fv(C)1151 4479 y Fs(\017)1211 4464 y Fv(>)27 b Fw(0,)33 b(due)g(to)f(\(8\).)0 4580 y(Then,)i(P2)e(of)h(Assumption)h(11)e(and)h(\(80\))f(yield)h(that)648 4775 y(1)27 b Fu(\025)h Fv(C)899 4790 y Fs(\017)932 4775 y Fv(q)j Fu(\025)d Fv(F)14 b Fw(\()p Fv(D)s(U)c Fw(\()p Fv(y)1472 4790 y Fs(q)1510 4775 y Fw(\))p Fv(;)17 b(U)1658 4790 y Fs(\034)1701 4775 y Fw(\()p Fv(y)1787 4790 y Fs(q)1825 4775 y Fw(\)\))27 b Fu(\025)145 b Fw(inf)2033 4841 y Fl(j)p Fs(\020)5 b Fl(j\024k)p Fs(U)2247 4849 y Fm(\034)2285 4841 y Fl(k)2320 4849 y Fj(1)2402 4775 y Fv(F)14 b Fw(\()p Fv(D)s(U)c Fw(\()p Fv(y)2763 4790 y Fs(q)2800 4775 y Fw(\))p Fv(;)17 b(\020)8 b Fw(\))17 b Fv(;)0 5024 y Fw(as)33 b(long)f(as)h Fv(q)j Fw(is)d(small)h(enough.)0 5140 y(From)27 b(this)g(and)g(\(17\),)h(w)m(e)g(infer)g(that)e Fu(j)p Fv(D)s(U)10 b Fw(\()p Fv(y)1693 5155 y Fs(q)1731 5140 y Fw(\))p Fu(j)27 b Fw(is)g(uniformly)h(b)s(ounded)g(for)e(small)i Fv(q)t Fw(.)41 b(But)28 b(then,)0 5257 y Fv(w)i Fu(\024)e Fv(w)s Fw(\()p Fv(y)364 5272 y Fs(q)401 5257 y Fw(\))33 b(is)g(uniformly)g(b)s(ounded,)h(and)e(so)h(is)g Fv(D)s(U)10 b Fw(,)33 b(as)g(desired.)1073 b Fi(\003)0 5469 y Fw(As)33 b(a)f(consequence,)k(w)m(e)e(ha)m(v)m(e)p eop end %%Page: 34 34 TeXDict begin 34 33 bop 168 251 a Fq(34)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y FA(Prop)s(osition)55 b(21.)49 b Fo(Fixe)-5 b(d)48 b(any)h Fv(!)57 b Fu(2)d Fw(\012)p Fo(,)f(thr)-5 b(ough)48 b(an)h(appr)-5 b(opriate)48 b(subse)-5 b(quenc)g(e,)51 b(we)e(have)168 566 y(that)f Fv(u)436 530 y Fs(\017)469 566 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))47 b Fo(c)-5 b(onver)g(ges)47 b(to)h Fv(u)1469 530 y Fs(])1501 566 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))47 b Fo(lo)-5 b(c)g(al)5 b(ly)48 b(uniformly)g(in)g Fw(\()p Fv(x;)17 b(t)p Fw(\))52 b Fu(2)h Ft(R)3221 530 y Fs(n)3300 566 y Fu(\002)32 b Fw([0)p Fv(;)17 b(T)d Fw(])p Fo(,)51 b(for)168 683 y(some)34 b Fv(u)473 646 y Fs(])532 683 y Fw(:)27 b Ft(R)658 646 y Fs(n)727 683 y Fu(\002)c Fw([0)p Fv(;)17 b(T)d Fw(])27 b Fu(!)g Ft(R)p Fo(.)45 b(A)n(lso,)1333 892 y Fw(lim)1334 952 y Fs(\017)p Fl(!)p Fr(0)1485 756 y Fp(Z)1541 982 y Fr(\012)1612 892 y Fv(u)1668 851 y Fs(\017)1701 892 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))g Fv(d!)46 b Fw(=)e Fv(u)2371 851 y Fr(0)2410 892 y Fw(\()p Fv(x;)17 b(t)p Fw(\))g Fv(;)168 1110 y Fo(for)34 b(some)g Fv(u)628 1074 y Fr(0)695 1110 y Fw(:)28 b Ft(R)822 1074 y Fs(n)891 1110 y Fu(\002)23 b Fw([0)p Fv(;)17 b(T)d Fw(])27 b Fu(!)g Ft(R)p Fo(.)168 1271 y(Pr)-5 b(o)g(of.)41 b Fw(This)34 b(follo)m(ws)g(from)f(the)g (Ascoli-Arzel\022)-49 b(a)34 b(Theorem,)h(since)2686 1191 y Fp(R)2733 1306 y Fr(\012)2804 1271 y Fv(u)2860 1235 y Fs(\017)2893 1271 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))g Fv(d!)35 b Fw(is)e(equicon-)168 1387 y(tin)m(uous,)52 b(thanks)d(to)e(Prop)s(ositions)i(18)e(and)h(20,)j(and)c(equib)s (ounded,)54 b(due)48 b(to)f(the)h(terminal)168 1504 y(condition.)3162 b Fi(\003)168 1665 y FA(Prop)s(osition)37 b(22.)42 b Fo(We)35 b(have)f(that)168 1822 y Fw(\(82\))920 b Fu(j)p Fv(u)1346 1781 y Fs(\017)1378 1822 y Fw(\()p Fv(x;)17 b(t;)g(\034)1636 1837 y Fs(y)1678 1822 y Fv(!)t Fw(\))k Fu(\000)i Fv(u)1958 1781 y Fs(\017)1990 1822 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))p Fu(j)26 b(\024)i Fv(C)7 b(\017)p Fu(j)p Fv(y)t Fu(j)17 b Fv(:)168 1979 y Fo(Mor)-5 b(e)g(over,)168 2183 y Fw(\(83\))217 b Fu(j)604 2047 y Fp(Z)720 2183 y Fv(u)776 2141 y Fs(\017)808 2183 y Fw(\()p Fv(x)23 b Fu(\000)f Fv(y)t(;)17 b(t;)g(!)t Fw(\))g Fv(d!)23 b Fu(\000)1553 2047 y Fp(Z)1669 2183 y Fv(u)1725 2141 y Fs(\017)1757 2183 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))g Fv(d!)t Fu(j)25 b(\024)j(j)p Fv(y)t Fu(j)2475 2072 y Fp(h)2521 2183 y Fu(k)p Fv(D)s( )t Fu(k)2772 2198 y Fl(1)2868 2183 y Fw(+)22 b Fv(C)7 b Fw(\()p Fv(T)36 b Fu(\000)23 b Fv(t)p Fw(\))3347 2072 y Fp(i)3410 2183 y Fv(:)168 2391 y Fo(Pr)-5 b(o)g(of.)41 b Fw(Notice)33 b(that)585 2574 y Fu(\000)p Fv(u)718 2533 y Fs(\017)718 2598 y(t)751 2574 y Fw(\()p Fv(x;)17 b(t;)g(\034)1009 2589 y Fs(y)1051 2574 y Fv(!)t Fw(\))k(+)h Fv(H)1362 2463 y Fp(\020)1421 2574 y Fv(D)1505 2533 y Fr(2)1544 2574 y Fv(u)1600 2533 y Fs(\017)1632 2574 y Fw(\()p Fv(x;)17 b(t;)g(\034)1890 2589 y Fs(y)1932 2574 y Fv(!)t Fw(\))p Fv(;)g(D)s(u)2219 2533 y Fs(\017)2250 2574 y Fw(\()p Fv(x;)g(t;)g(\034)2508 2589 y Fs(y)2550 2574 y Fv(!)t Fw(\))p Fv(;)g(x;)2805 2506 y(x)p 2805 2551 56 4 v 2813 2642 a(\017)2870 2574 y(;)g(\017;)g(\034)3039 2589 y Fs(y)3081 2574 y Fv(!)3146 2463 y Fp(\021)3232 2574 y Fw(=)28 b(0)p Fv(:)168 2766 y Fw(By)34 b(\(53\),)563 2950 y Fu(\000)p Fv(u)696 2909 y Fs(\017)696 2975 y(t)729 2950 y Fw(\()p Fv(x;)17 b(t;)g(\034)987 2965 y Fs(y)1028 2950 y Fv(!)t Fw(\))22 b(+)g Fv(H)8 b Fw(\()p Fv(D)1462 2909 y Fr(2)1500 2950 y Fv(u)1556 2909 y Fs(\017)1589 2950 y Fw(\()p Fv(x;)17 b(t;)g(\034)1847 2965 y Fs(y)1888 2950 y Fv(!)t Fw(\))p Fv(;)g(D)s(u)2175 2909 y Fs(\017)2206 2950 y Fw(\()p Fv(x;)g(t;)g(\034)2464 2965 y Fs(y)2506 2950 y Fv(!)t Fw(\))p Fv(;)g(x;)g(y)24 b Fw(+)2932 2883 y Fv(x)p 2932 2927 V 2940 3018 a(\017)2998 2950 y(;)17 b(\017;)g(!)t Fw(\))26 b(=)i(0)p Fv(:)168 3142 y Fw(Set)33 b Fv(v)t Fw(\()p Fv(x;)17 b(y)t(;)g(t)p Fw(;)g Fv(!)t Fw(\))25 b(=)j Fv(u)987 3106 y Fs(\017)1019 3142 y Fw(\()p Fv(x)22 b Fu(\000)h Fv(\017y)t(;)17 b(t;)g(\034)1490 3157 y Fs(y)1531 3142 y Fv(!)t Fw(\).)43 b(Then,)34 b Fv(v)i Fw(solv)m(es)1189 3329 y Fu(\000)p Fv(v)1313 3344 y Fs(t)1365 3329 y Fw(+)22 b Fv(H)8 b Fw(\()p Fv(D)1674 3288 y Fr(2)1713 3329 y Fv(v)t(;)17 b(D)s(v)t(;)g(x)k Fu(\000)h Fv(\017y)t(;)2307 3262 y(x)p 2307 3306 V 2315 3397 a(\017)2372 3329 y(;)17 b(\017;)g(!)t Fw(\))27 b(=)g(0)p Fv(:)168 3516 y Fw(As)33 b(b)s(efore,)g(b)m(y)g(Assumption)h(10,)f(w)m (e)g(ma)m(y)g(assume)i(that)d Fv(v)k Fw(is)d(smo)s(oth.)44 b(Then)34 b(set)1791 3721 y Fv(w)c Fw(=)2025 3653 y Fv(@)5 b(v)p 2005 3698 148 4 v 2005 3789 a(@)g(y)2109 3804 y Fs(k)2179 3721 y Fv(:)168 3926 y Fw(Th)m(us,)998 4043 y Fu(\000)p Fv(w)1145 4058 y Fs(t)1197 4043 y Fw(+)22 b Fv(D)1376 4058 y Fs(M)1455 4043 y Fv(H)35 b Fw(:)28 b Fv(D)1710 4002 y Fr(2)1749 4043 y Fv(w)d Fw(+)d Fv(D)2023 4058 y Fs(p)2062 4043 y Fv(H)30 b Fu(\001)22 b Fv(D)s(w)i Fu(\000)f Fv(\017D)2621 4058 y Fs(x)2661 4070 y Fm(k)2703 4043 y Fv(H)35 b Fw(=)28 b(0)p Fv(:)168 4179 y Fw(By)37 b(Assumption)i(12,)f(the)g(ellipticit)m(y)h(condition)f(in)g(\(14\))e (and)i(the)f(maxim)m(um)i(principle)g(\(see,)168 4296 y(e.g.,)33 b(Prop)s(osition)g(4.1)f(and)g(Theorem)i(4.2)e(in)h ([DiB95]\),)f(w)m(e)i(conclude)g(that)1238 4452 y Fu(k)p Fv(w)s Fw(\()p Fv(x;)17 b(t)p Fw(\))p Fu(k)1621 4467 y Fl(1)1722 4452 y Fu(\024)29 b Fv(\017)p Fu(k)p Fv(D)s( )t Fu(k)2118 4467 y Fl(1)2214 4452 y Fw(+)22 b Fv(\017C)7 b Fw(\()p Fv(T)36 b Fu(\000)23 b Fv(t)p Fw(\))p Fv(;)168 4609 y Fw(whic)m(h)34 b(implies)168 4766 y(\(84\))376 b Fu(j)p Fv(u)802 4725 y Fs(\017)834 4766 y Fw(\()p Fv(x)22 b Fu(\000)h Fv(\017y)t(;)17 b(t;)g(\034)1305 4781 y Fs(y)1346 4766 y Fv(!)t Fw(\))k Fu(\000)i Fv(u)1626 4725 y Fs(\017)1658 4766 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))p Fu(j)27 b(\024)h(j)p Fv(y)t Fu(j)17 b Fw([)n Fv(\017)p Fu(k)p Fv(D)s( )t Fu(k)2577 4781 y Fl(1)2674 4766 y Fw(+)22 b Fv(\017C)7 b Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(t)p Fw(\)])34 b Fv(:)168 4923 y Fw(F)-8 b(urthermore,)27 b(the)f(uniform)g(Lipsc)m(hitz)i(con)m(tin)m (uit)m(y)g(of)d Fv(u)2298 4887 y Fs(\017)2356 4923 y Fw(\(recall)h(Prop)s(osition)g(20\))f(implies)i(that)1122 5080 y Fu(j)p Fv(u)1206 5039 y Fs(\017)1238 5080 y Fw(\()p Fv(x)22 b Fu(\000)h Fv(\017y)t(;)17 b(t;)g(\034)1709 5095 y Fs(y)1750 5080 y Fv(!)t Fw(\))k Fu(\000)i Fv(u)2030 5039 y Fs(\017)2062 5080 y Fw(\()p Fv(x;)17 b(t;)g(\034)2320 5095 y Fs(y)2362 5080 y Fv(!)t Fw(\))p Fu(j)27 b(\024)h Fv(C)7 b(\017)p Fu(j)p Fv(y)t Fu(j)p Fv(:)168 5237 y Fw(The)33 b(latter)g(t)m(w)m(o)g(estimates)h(yield)g(the)f(claim)g(in)g (\(82\))o(.)168 5353 y(Then,)39 b(\(83\))e(follo)m(ws)i(from)f(\(84\))f (b)m(y)i(replacing)g Fv(\017y)j Fw(with)c Fv(y)k Fw(and)c(recalling)h (that)e Fv(\034)3298 5368 y Fs(y)3378 5353 y Fw(is)i(measure)168 5469 y(preserving,)34 b(according)f(to)f(Assumption)j(7.)1913 b Fi(\003)p eop end %%Page: 35 35 TeXDict begin 35 34 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(35)0 450 y FA(Prop)s(osition)38 b(23.)j Fo(We)35 b(have)883 628 y Fw(lim)17 b(sup)965 708 y Fs(\017)p Fl(!)p Fr(0)1260 628 y Fw(sup)1257 700 y Fm(x)p Fj(2)p Fd(R)1370 686 y Fm(n)1237 749 y(t)p Fj(2)p Fk([0)p Fm(;T)8 b Fk(])1209 805 y Fm(!)1248 820 y Fk(0)1282 805 y Fm(;!)1340 820 y Fk(1)1374 805 y Fj(2)p Fk(\012)1486 628 y Fu(j)p Fv(u)1570 587 y Fs(\017)1602 628 y Fw(\()p Fv(x;)17 b(t;)g(!)1879 643 y Fr(0)1918 628 y Fw(\))22 b Fu(\000)h Fv(u)2134 587 y Fs(\017)2166 628 y Fw(\()p Fv(x;)17 b(t;)g(!)2443 643 y Fr(1)2482 628 y Fw(\))p Fu(j)27 b(!)h Fw(0)p Fv(:)0 977 y Fo(Pr)-5 b(o)g(of.)41 b Fw(T)-8 b(o)33 b(pro)m(v)m(e)h(the)f(ab)s (o)m(v)m(e)g(estimate,)h(observ)m(e)g(that)556 1156 y Fu(j)p Fv(u)640 1115 y Fs(\017)672 1156 y Fw(\()p Fv(x;)17 b(t;)g(!)949 1171 y Fr(0)988 1156 y Fw(\))22 b Fu(\000)h Fv(u)1204 1115 y Fs(\017)1236 1156 y Fw(\()p Fv(x;)17 b(t;)g(!)1513 1171 y Fr(1)1552 1156 y Fw(\))p Fu(j)27 b(\024)h(j)p Fv(u)1834 1115 y Fs(\017)1866 1156 y Fw(\()p Fv(x;)17 b(t;)g(!)2143 1171 y Fr(0)2183 1156 y Fw(\))22 b Fu(\000)g Fv(u)2398 1115 y Fs(\017)2430 1156 y Fw(\()p Fv(x;)17 b(t;)g(\034)2688 1171 y Fs(y)2730 1156 y Fv(!)2791 1171 y Fr(0)2830 1156 y Fw(\))p Fu(j)1835 1307 y Fw(+)22 b Fu(j)p Fv(u)2017 1266 y Fs(\017)2049 1307 y Fw(\()p Fv(x;)17 b(t;)g(\034)2307 1322 y Fs(y)2349 1307 y Fv(!)2410 1322 y Fr(0)2449 1307 y Fw(\))22 b Fu(\000)g Fv(u)2664 1266 y Fs(\017)2697 1307 y Fw(\()p Fv(x;)17 b(t;)g(!)2974 1322 y Fr(1)3013 1307 y Fw(\))p Fu(j)p Fv(:)0 1491 y Fw(Let)41 b Fv(\016)226 1506 y Fr(1)306 1491 y Fv(>)g Fw(0)f(b)s(e)h(\014xed,)j(and)c(supp)s(ose)i(that)e(for)g(ev)m(ery)i Fv(\017)f Fw(there)g(exists)i Fv(y)h Fw(=)d Fv(y)t Fw(\()p Fv(x;)17 b(t;)g(\017;)g(\016)3357 1506 y Fr(1)3396 1491 y Fw(\))40 b(suc)m(h)0 1607 y(that)32 b Fu(j)p Fv(y)t Fu(j)27 b(\024)h Fv(M)10 b Fw(,)33 b(with)g Fv(M)44 b Fw(dep)s(ending)33 b(only)g(on)g Fv(\016)1836 1622 y Fr(1)1908 1607 y Fw(\(and)g(so)f(indep)s(enden)m(t)j(of)d Fv(\017)p Fw(\),)h(for)f(whic)m(h)0 1839 y(\(85\))879 b(sup)1049 1911 y Fm(x)p Fj(2)p Fd(R)1162 1897 y Fm(n)1030 1960 y(t)p Fj(2)p Fk([0)p Fm(;T)8 b Fk(])1001 2015 y Fm(!)1040 2030 y Fk(0)1075 2015 y Fm(;!)1133 2030 y Fk(1)1167 2015 y Fj(2)p Fk(\012)1279 1839 y Fu(j)p Fv(u)1363 1798 y Fs(\017)1394 1839 y Fw(\()p Fv(x;)17 b(t;)g(\034)1652 1854 y Fs(y)1694 1839 y Fv(!)1755 1854 y Fr(0)1794 1839 y Fw(\))22 b Fu(\000)h Fv(u)2010 1798 y Fs(\017)2042 1839 y Fw(\()p Fv(x;)17 b(t;)g(!)2319 1854 y Fr(1)2359 1839 y Fw(\))p Fu(j)27 b(\024)2567 1772 y Fv(\016)2610 1787 y Fr(1)p 2567 1816 83 4 v 2584 1908 a Fw(2)2660 1839 y Fv(:)0 2180 y Fw(Then,)34 b(from)e(Prop)s(osition)h(22,)f(for)g Fv(\017)h Fw(su\016cien)m(tly)i(small,)f(w)m(e)f(ha)m(v)m(e)1045 2406 y(sup)1041 2478 y Fm(x)p Fj(2)p Fd(R)1154 2464 y Fm(n)1021 2527 y(t)p Fj(2)p Fk([0)p Fm(;T)8 b Fk(])1060 2583 y Fm(!)1099 2598 y Fk(0)1133 2583 y(\012)1027 2642 y Fj(j)p Fm(y)r Fj(j\024)p Fm(M)1252 2406 y Fu(j)p Fv(u)1336 2365 y Fs(\017)1368 2406 y Fw(\()p Fv(x;)17 b(t;)g(!)1645 2421 y Fr(0)1684 2406 y Fw(\))22 b Fu(\000)h Fv(u)1900 2365 y Fs(\017)1932 2406 y Fw(\()p Fv(x;)17 b(t;)g(\034)2190 2421 y Fs(y)2232 2406 y Fv(!)2293 2421 y Fr(0)2332 2406 y Fw(\))p Fu(j)27 b(\024)2540 2339 y Fv(\016)2583 2354 y Fr(1)p 2540 2384 V 2557 2475 a Fw(2)2633 2406 y Fv(;)0 2811 y Fw(whic)m(h)36 b(then,)h(b)m(y)f(sending)g Fv(\016)1070 2826 y Fr(1)1141 2811 y Fu(!)31 b Fw(0)1321 2775 y Fr(+)1415 2811 y Fw(yields)37 b(the)e(estimate.)52 b(Therefore)36 b(our)e(task)i(no)m(w)g(consists)0 2928 y(in)d(establishing)i(\(85\).)0 3044 y(T)-8 b(o)33 b(this)g(exten)m(t,)h(\014x)f Fv(\016)f(>)27 b Fw(0)33 b(and)f(c)m(ho)s(ose)i Fv(y)h Fw(as)e(in)g(Assumption)h(13.) 43 b(Then)1152 3222 y(~)-55 b Fv(u)1202 3181 y Fs(\017)1234 3222 y Fw(\()p Fv(x;)17 b(t)p Fw(\))28 b(:=)g Fv(u)1659 3181 y Fs(\017)1691 3222 y Fw(\()p Fv(x;)17 b(t;)g(!)1968 3237 y Fr(1)2007 3222 y Fw(\))22 b(+)g Fv(\016)t Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(t)p Fw(\))0 3401 y(satis\014es)527 3586 y(0)27 b(=)h Fu(\000)p Fv(u)840 3544 y Fs(\017)840 3610 y(t)872 3586 y Fw(\()p Fv(x;)17 b(t;)g(!)1149 3601 y Fr(1)1189 3586 y Fw(\))22 b(+)g Fv(H)1436 3475 y Fp(\020)1495 3586 y Fv(D)1579 3544 y Fr(2)1618 3586 y Fv(u)1674 3544 y Fs(\017)1706 3586 y Fw(\()p Fv(x;)17 b(t;)g(!)1983 3601 y Fr(1)2022 3586 y Fw(\))p Fv(;)g(D)9 b Fw(~)-55 b Fv(u)2244 3544 y Fs(\017)2276 3586 y Fw(\()p Fv(x;)17 b(t;)g(!)2553 3601 y Fr(1)2592 3586 y Fw(\))p Fv(;)g(x;)2783 3518 y(x)p 2783 3563 56 4 v 2791 3654 a(\017)2848 3586 y(;)g(\017;)g(!)3036 3601 y Fr(1)3075 3475 y Fp(\021)603 3806 y Fw(=)28 b Fu(\000)p Fv(\016)e Fu(\000)j Fw(~)-55 b Fv(u)1009 3765 y Fs(\017)1009 3831 y(t)1063 3806 y Fw(+)22 b Fv(H)1250 3695 y Fp(\020)1309 3806 y Fv(D)1393 3765 y Fr(2)1439 3806 y Fw(~)-55 b Fv(u)o(;)17 b(D)9 b Fw(~)-55 b Fv(u)o(;)17 b(x;)1824 3738 y(x)p 1824 3783 V 1832 3874 a(\017)1889 3806 y(;)g(\017;)g(!)2077 3821 y Fr(1)2116 3695 y Fp(\021)603 4026 y Fu(\024)28 b(\000)6 b Fw(~)-55 b Fv(u)841 3985 y Fs(\017)841 4051 y(t)896 4026 y Fw(+)22 b Fv(H)1083 3916 y Fp(\020)1142 4026 y Fv(D)1226 3985 y Fr(2)1271 4026 y Fw(~)-55 b Fv(u;)17 b(D)9 b Fw(~)-55 b Fv(u)o(;)17 b(x;)1657 3959 y(x)p 1657 4003 V 1665 4094 a(\017)1722 4026 y(;)g(\017;)g(\034)1891 4041 y Fs(y)1933 4026 y Fv(!)1994 4041 y Fr(0)2033 3916 y Fp(\021)2109 4026 y Fv(;)0 4234 y Fw(that)32 b(is)1026 4378 y Fu(\000)6 b Fw(~)-55 b Fv(u)1159 4337 y Fs(\017)1159 4403 y(t)1214 4378 y Fw(+)22 b Fv(H)1401 4268 y Fp(\020)1460 4378 y Fv(D)1544 4337 y Fr(2)1589 4378 y Fw(~)-55 b Fv(u;)17 b(D)9 b Fw(~)-55 b Fv(u)o(;)17 b(x;)1974 4311 y(x)p 1974 4356 V 1982 4447 a(\017)2040 4378 y(;)g(\017;)g(\034)2209 4393 y Fs(y)2250 4378 y Fv(!)2311 4393 y Fr(0)2350 4268 y Fp(\021)2438 4378 y Fu(\025)28 b Fw(0)17 b Fv(:)0 4572 y Fw(Then,)36 b Fv(u)340 4535 y Fs(\017)396 4572 y Fu(\000)30 b Fw(~)-55 b Fv(u)553 4535 y Fs(\017)620 4572 y Fw(is)35 b(a)f(subsolution)i(of)f(a)f(linear)h(parab)s(olic)g(equation,)h(due)f (to)g(\(14\))o(,)g(whic)m(h)i(then)0 4688 y(implies,)45 b(b)m(y)d(the)g(comparison)g(principle)h(for)e(viscosit)m(y)i (solutions)f(\(see)h(Prop)s(osition)e(4.1)g(and)0 4804 y(Theorem)34 b(4.2)e(in)h([DiB95)o(]\))g(that)1338 4982 y Fv(u)1394 4941 y Fs(\017)1427 4982 y Fw(\()p Fv(x;)17 b(t;)g(\034)1685 4997 y Fs(y)1726 4982 y Fv(!)1787 4997 y Fr(0)1827 4982 y Fw(\))27 b Fu(\024)34 b Fw(~)-55 b Fv(u)2053 4941 y Fs(\017)2086 4982 y Fw(\()p Fv(x;)17 b(t)p Fw(\))p Fv(:)0 5161 y Fw(Accordingly)-8 b(,)29 b(w)m(e)e(obtain)f(\(85\))g(b)m(y)h(p)s(ossibly)h(exc)m(hanging)g(the)e (roles)h(of)f Fv(\034)2716 5176 y Fs(y)2758 5161 y Fv(!)2819 5176 y Fr(0)2884 5161 y Fw(and)g Fv(!)3128 5176 y Fr(1)3168 5161 y Fw(,)h(as)g(desired.)3584 5277 y Fi(\003)0 5469 y Fw(By)33 b(collecting)h(the)f(results)h(in)e(this)i(section,)f(w)m(e) h(get)f(the)g(follo)m(wing)g(result:)p eop end %%Page: 36 36 TeXDict begin 36 35 bop 168 251 a Fq(36)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y FA(Prop)s(osition)37 b(24.)42 b Fo(Thr)-5 b(ough)34 b(an)h(appr)-5 b(opriate)34 b(subse)-5 b(quenc)g(e,)1462 617 y Fw(lim)1463 676 y Fs(\017)p Fl(!)p Fr(0)1615 617 y Fv(u)1671 575 y Fs(\017)1703 617 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))43 b(=)h Fv(u)2241 575 y Fr(0)2280 617 y Fw(\()p Fv(x;)17 b(t)p Fw(\))g Fv(;)168 820 y Fo(uniformly)34 b(in)h Fv(K)29 b Fu(\002)23 b Fw([0)p Fv(;)17 b(T)d Fw(])21 b Fu(\002)i Fw(\012)p Fo(,)35 b(for)g(any)g(c)-5 b(omp)g(act)34 b(set)h Fv(K)f Fu(\032)29 b Ft(R)2584 784 y Fs(n)2631 820 y Fo(.)168 1000 y(Pr)-5 b(o)g(of.)41 b Fw(Fix)23 b Fv(!)692 1015 y Fr(0)759 1000 y Fu(2)28 b Fw(\012.)41 b(Then,)26 b(b)m(y)e(Prop)s (osition)f(21,)g Fv(u)2110 963 y Fs(\017)2139 973 y Fm(j)2175 1000 y Fw(\()p Fv(x;)17 b(t;)g(!)2452 1015 y Fr(0)2491 1000 y Fw(\))23 b(con)m(v)m(erges)i(to)e Fv(u)3145 963 y Fs(])3176 1000 y Fw(\()p Fv(x;)17 b(t;)g(!)3453 1015 y Fr(0)3492 1000 y Fw(\))23 b(lo)s(cally)168 1116 y(uniformly)33 b(in)g(\()p Fv(x;)17 b(t)p Fw(\).)44 b(Let)32 b Fv(u)1237 1080 y Fr(0)1276 1116 y Fw(\()p Fv(x;)17 b(t)p Fw(\))28 b(:=)g Fv(u)1701 1080 y Fs(])1732 1116 y Fw(\()p Fv(x;)17 b(t;)g(!)2009 1131 y Fr(0)2048 1116 y Fw(\).)44 b(Then,)1179 1283 y(lim)1133 1344 y Fs(j)t Fl(!)p Fr(+)p Fl(1)1615 1283 y Fw(sup)1378 1369 y Fr(\()p Fs(x;t;!)r Fr(\))p Fl(2)p Fs(K)5 b Fl(\002)p Fr([0)p Fs(;T)10 b Fr(])p Fl(\002)p Fr(\012)2017 1283 y Fu(j)p Fv(u)2101 1242 y Fs(\017)2130 1252 y Fm(j)2165 1283 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))22 b Fu(\000)g Fv(u)2661 1242 y Fr(0)2700 1283 y Fw(\()p Fv(x;)17 b(t)p Fw(\))p Fu(j)972 1508 y(\024)130 b Fw(lim)1133 1568 y Fs(j)t Fl(!)p Fr(+)p Fl(1)1615 1508 y Fw(sup)1378 1593 y Fr(\()p Fs(x;t;!)r Fr(\))p Fl(2)p Fs(K)5 b Fl(\002)p Fr([0)p Fs(;T)10 b Fr(])p Fl(\002)p Fr(\012)2017 1508 y Fu(j)p Fv(u)2101 1467 y Fs(\017)2130 1477 y Fm(j)2165 1508 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))22 b Fu(\000)g Fv(u)2661 1467 y Fs(\017)2690 1477 y Fm(j)2726 1508 y Fw(\()p Fv(x;)17 b(t;)g(!)3003 1523 y Fr(0)3042 1508 y Fw(\))p Fu(j)973 1745 y Fw(+)130 b(lim)1133 1805 y Fs(j)t Fl(!)p Fr(+)p Fl(1)1530 1745 y Fw(sup)1378 1830 y Fr(\()p Fs(x;t)p Fr(\))p Fl(2)p Fs(K)5 b Fl(\002)p Fr([0)p Fs(;T)10 b Fr(])1845 1745 y Fu(j)p Fv(u)1929 1703 y Fs(\017)1958 1713 y Fm(j)1993 1745 y Fw(\()p Fv(x;)17 b(t;)g(!)2270 1760 y Fr(0)2310 1745 y Fw(\))22 b Fu(\000)g Fv(u)2525 1703 y Fs(])2556 1745 y Fw(\()p Fv(x;)17 b(t;)g(!)2833 1760 y Fr(0)2873 1745 y Fw(\))p Fu(j)972 1959 y(\024)84 b Fw(0)22 b(+)g(0)17 b Fv(;)168 2125 y Fw(where)34 b(Prop)s(osition)e (23)g(has)h(b)s(een)h(exploited.)1830 b Fi(\003)1576 2350 y Fw(10.)48 b Fx(Conver)n(gence)168 2524 y Fw(In)35 b(this)h(section,)h(w)m(e)f(establish)h(that)e(the)g(limit)h(of)f(the)g (viscosit)m(y)j(solutions)e Fv(u)3159 2488 y Fs(\017)3226 2524 y Fw(agrees)g(with)42 b(\026)-55 b Fv(u)o Fw(,)168 2641 y(whic)m(h)36 b(w)m(as)f(de\014ned)h(in)e(\(72\).)48 b(W)-8 b(e)35 b(divide)h(the)e(pro)s(of)g(in)m(to)h(t)m(w)m(o)g(parts,) g(a)f(lo)m(w)m(er)h(b)s(ound)g(and)f(an)168 2757 y(upp)s(er)f(b)s (ound.)168 2937 y FA(Prop)s(osition)k(25)h Fw(\(Lo)m(w)m(er)33 b(Bound\))p FA(.)42 b Fv(u)1696 2901 y Fr(0)1763 2937 y Fu(\025)34 b Fw(\026)-55 b Fv(u)p Fo(.)168 3117 y(Pr)-5 b(o)g(of.)41 b Fw(Consider)35 b(the)e(measures)i Fv(\045)f Fw(and)f Fv(\022)k Fw(constructed)e(in)e(section)h(7.)46 b(Let)33 b(also)g Fv(\026)3336 3080 y Fs(\017)3402 3117 y Fw(and)g Fv(\027)3646 3080 y Fs(\017)3640 3142 y(T)3729 3117 y Fw(b)s(e)168 3233 y(the)g(measures)h(optimizing)h(\(5\))o(.)168 3349 y(Then,)e(w)m(e)g(use)g(Prop)s(osition)f(9,)g(\(29\))f(\(11\))o(,) h(\(54\),)g(Prop)s(osition)g(11,)g(\(71\))o(,)g(Prop)s(osition)g(17,)g (\(72\))168 3465 y(and)g(\(73\))g(to)h(obtain)f(that)707 3684 y Fv(u)763 3643 y Fr(0)802 3684 y Fw(\()p Fv(x;)17 b(t)p Fw(\))28 b(=)f(lim)1144 3744 y Fs(\017)p Fl(!)p Fr(0)1295 3549 y Fp(Z)1351 3774 y Fr(\012)1422 3684 y Fv(u)1478 3643 y Fs(\017)1510 3684 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))g Fv(d!)1040 3956 y Fw(=)27 b(lim)1144 4016 y Fs(\017)p Fl(!)p Fr(0)1295 3816 y Fp(\024)1348 3821 y(Z)11 b(Z)1530 3956 y Fv(L)p Fw(\()p Fv(x)1689 3915 y Fl(0)1713 3956 y Fv(;)17 b(X)r(;)g(v)t(;)g(s;)g(!)t Fw(\))p Fv(d\026)2282 3915 y Fs(\017)2313 3956 y Fv(d!)25 b Fw(+)2548 3821 y Fp(Z)11 b(Z)2730 3956 y Fv( )t Fw(\()p Fv(x)2890 3915 y Fl(0)2914 3956 y Fw(\))p Fv(d\027)3057 3915 y Fs(\017)3051 3981 y(T)3123 3956 y Fv(d!)3239 3816 y Fp(\025)1040 4233 y Fw(=)1143 4092 y Fp(\024)1196 4233 y Fw(lim)1196 4292 y Fs(\017)p Fl(!)p Fr(0)1348 4097 y Fp(Z)1464 4233 y Fv(L)p Fw(\()p Fv(x)1623 4192 y Fl(0)1647 4233 y Fv(;)17 b Fw(0)p Fv(;)g(v)t(;)g(s;)g(!)t Fw(\))p Fv(d)8 b Fw(^)-57 b Fv(\026)2182 4192 y Fs(\017)2212 4092 y Fp(\025)2286 4233 y Fw(+)2384 4097 y Fp(Z)2501 4233 y Fv( )t(d)6 b Fw(\026)-55 b Fv(\027)2667 4248 y Fs(T)1040 4500 y Fw(=)1143 4365 y Fp(Z)12 b(Z)1326 4500 y Fv(L)p Fw(\()p Fv(x)1485 4459 y Fl(0)1509 4500 y Fv(;)17 b Fw(0)p Fv(;)g(v)t(;)g(s;)g(!)t Fw(\))p Fv(d\022)g(d\032)22 b Fw(+)2268 4365 y Fp(Z)2384 4500 y Fv( )t(d)6 b Fw(\026)-55 b Fv(\027)2550 4515 y Fs(T)1040 4759 y Fu(\025)1145 4623 y Fp(Z)p 1261 4679 67 4 v 1261 4759 a Fv(L)p Fw(\()p Fv(x)1420 4718 y Fl(0)1444 4759 y Fv(;)17 b($)s(;)g Fw(\004\))p Fv(d\045)k Fw(+)1939 4623 y Fp(Z)2055 4759 y Fv( )t(d)6 b Fw(\026)-55 b Fv(\027)2221 4774 y Fs(T)2304 4759 y Fw(=)33 b(\026)-55 b Fv(u)p Fw(\()p Fv(x;)17 b(t)p Fw(\))p Fv(:)3752 4971 y Fi(\003)168 5148 y Fw(In)44 b(order)g(to)f(obtain)h (the)g(opp)s(osite)h(inequalit)m(y)g(to)f(Prop)s(osition)g(25,)i(w)m(e) f(need)g(the)f(result)h(in)168 5265 y(Prop)s(osition)33 b(24.)168 5445 y FA(Prop)s(osition)k(26)h Fw(\(Upp)s(er)33 b(b)s(ound\))p FA(.)42 b Fv(u)1694 5408 y Fr(0)1760 5445 y Fw(=)34 b(\026)-55 b Fv(u)o Fo(.)p eop end %%Page: 37 37 TeXDict begin 37 36 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(37)0 450 y Fo(Pr)-5 b(o)g(of.)41 b Fw(Let)34 b Fv(\016)h(>)30 b Fw(0)k(and)g Fv(v)983 414 y Fs(\017)979 476 y(\016)1051 450 y Fw(b)s(e)g(as)h(in)f(Assumption)i(6.)48 b(W)-8 b(e)34 b(denote)h(b)m(y)g Fv(o)2757 465 y Fs(\016)2829 450 y Fw(quan)m(tities)h(\(p)s(ossibly)0 566 y(dep)s(ending)e(on)e(\()p Fv(x;)17 b(t;)g(!)t Fw(\)\))32 b(whic)m(h)i(tend)f(to)f(zero)h(as)g Fv(\016)f Fu(!)27 b Fw(0.)0 683 y(Let)45 b Fv(\024)k(>)f Fw(0.)81 b(W)-8 b(e)45 b(consider)h(almost)f(optimal)g(measures)i(\()p Fv(\045)2361 698 y Fs(\024)2406 683 y Fw(\()p Fv(x)2499 646 y Fl(0)2523 683 y Fv(;)17 b($)s(;)g Fw(\004)p Fv(;)g(s)p Fw(\))p Fv(;)j Fw(\026)-52 b Fv(\027)2980 698 y Fs(T)6 b(;\024)3089 683 y Fw(\()p Fv(x)3182 646 y Fl(0)3206 683 y Fw(\)\))45 b(for)f(\(72\))0 799 y(or)21 b(\(73\),)j(and)e(an)f (almost)h(optimal)g(family)g(of)g(probabilit)m(y)g(measure)h Fv(\021)2579 814 y Fs(\024)2624 799 y Fw(\()p Fv(v)t(;)17 b(!)t Fw(;)g Fv(x)2921 763 y Fl(0)2943 799 y Fv(;)g(s;)g($)s(;)g Fw(\004\))j(for)i(\(59\))o(,)0 915 y(that)32 b(is,)i(w)m(e)f(supp)s (ose)h(that)e Fv(\045)1108 930 y Fs(\024)1154 915 y Fw(\()p Fv(x)1247 879 y Fl(0)1270 915 y Fv(;)17 b($)s(;)g Fw(\004)p Fv(;)g(s)p Fw(\))31 b(almost)i(ev)m(erywhere)p 772 1063 67 4 v 772 1143 a Fv(L)q Fw(\()p Fv(x)932 1101 y Fl(0)955 1143 y Fv(;)17 b($)s(;)g Fw(\004\))27 b Fu(\025)1362 1007 y Fp(Z)1478 1143 y Fv(L)p Fw(\()p Fv(x;)17 b Fw(0)p Fv(;)g(v)t(;)g(!)t Fw(\))p Fv(d\021)2071 1158 y Fs(\024)2114 1143 y Fw(\()p Fv(v)t(;)g(!)t Fw(;)g Fv(x;)g($)s(;)g Fw(\004\))j Fu(\000)j Fv(\024;)0 1365 y Fw(and)33 b(that,)f(in)h(case)g (A)g(or)f(B,)973 1587 y(\026)-55 b Fv(u)p Fw(\()p Fv(x;)17 b(t)p Fw(\))28 b Fu(\025)1366 1451 y Fp(Z)p 1482 1507 V 1482 1587 a Fv(L)p Fw(\()p Fv(x)1641 1546 y Fl(0)1665 1587 y Fv(;)17 b($)s Fw(\))p Fv(d\045)1932 1602 y Fs(\024)1998 1587 y Fw(+)2096 1451 y Fp(Z)2212 1587 y Fv( )t(d)6 b Fw(\026)-55 b Fv(\027)2378 1602 y Fs(T)6 b(;\024)2511 1587 y Fu(\000)23 b Fv(\024;)0 1809 y Fw(whereas,)34 b(in)f(case)g(C,)919 2031 y(\026)-55 b Fv(u)o Fw(\()p Fv(x;)17 b(t)p Fw(\))28 b Fu(\025)1311 1895 y Fp(Z)p 1428 1951 V 1428 2031 a Fv(L)p Fw(\()p Fv(x)1587 1990 y Fl(0)1611 2031 y Fv(;)17 b($)s(;)g Fw(\004\))p Fv(d\045)1987 2046 y Fs(\024)2052 2031 y Fw(+)2150 1895 y Fp(Z)2266 2031 y Fv( )t(d)6 b Fw(\026)-55 b Fv(\027)2432 2046 y Fs(T)6 b(;\024)2566 2031 y Fu(\000)22 b Fv(\024:)0 2258 y Fw(W)-8 b(e)25 b(construct)g(a)f(trial)g(measure)34 b(~)-57 b Fv(\026)1287 2273 y Fs(\024)1332 2258 y Fw(\()p Fv(x)1425 2222 y Fl(0)1448 2258 y Fv(;)17 b(v)t(;)g(s;)g(!)t Fw(\))23 b(on)h Ft(R)2002 2222 y Fs(n)2054 2258 y Fu(\002)5 b Fv(U)16 b Fu(\002)5 b Fw([)p Fv(t;)17 b(T)d Fw(])5 b Fu(\002)g Fw(\012)28 b(de\014ned)e(in)e(the)h(follo)m(wing)0 2375 y(w)m(a)m(y:)0 2597 y(\(86\))528 2461 y Fp(Z)584 2687 y Fn(R)632 2668 y Fm(n)674 2687 y Fl(\002)p Fs(U)7 b Fl(\002)p Fr([)p Fs(t;T)j Fr(])p Fl(\002)p Fr(\012)1101 2597 y Fv(\036)p Fw(\()p Fv(x)1252 2556 y Fl(0)1275 2597 y Fv(;)17 b(v)t(;)g(s;)g(!)t Fw(\))p Fv(d)8 b Fw(~)-57 b Fv(\026)1717 2612 y Fs(\024)1804 2597 y Fw(:=)1951 2461 y Fp(Z)11 b(Z)2133 2597 y Fv(\036)p Fw(\()p Fv(x)2284 2556 y Fl(0)2308 2597 y Fv(;)17 b(v)t(;)g(s;)g(\034)2579 2631 y Fl(\000)2644 2604 y Fm(x)2679 2590 y Fj(0)p 2643 2616 58 3 v 2658 2658 a Fm(\017)2714 2597 y Fv(!)t Fw(\))p Fv(d\021)2916 2612 y Fs(\024)2960 2597 y Fv(d\045)3061 2612 y Fs(\024)3122 2597 y Fv(;)0 2845 y Fw(for)32 b(an)m(y)h Fv(\036)28 b Fu(2)g Fv(C)583 2860 y Fs(c)617 2845 y Fw(\()p Ft(R)727 2809 y Fs(n)796 2845 y Fu(\002)23 b Fv(U)33 b Fu(\002)22 b Fw([)p Fv(t;)17 b(T)d Fw(])22 b Fu(\002)h Fw(\012\).)0 2961 y(Note)33 b(that,)f(b)m(y)i(taking)e Fv(')p Fw(\()p Fv(s)p Fw(\))c(:=)f(\()p Fv(s)22 b Fu(\000)h Fv(T)14 b Fw(\))32 b(in)h(\(68\))f(or)g(\(69\),)g(w)m(e)i(obtain)e (that)1473 3053 y Fp(Z)1606 3189 y Fv(d\045)1707 3204 y Fs(\024)1796 3189 y Fw(=)45 b Fv(T)35 b Fu(\000)23 b Fv(t)17 b(;)0 3411 y Fw(and)33 b(so,)g(since)h Fv(\021)624 3426 y Fs(\024)701 3411 y Fw(is)f(a)f(probabilit)m(y)i(measure,)g(it)f (follo)m(ws)g(that)40 b(~)-57 b Fv(\026)2474 3426 y Fs(\024)2551 3411 y Fw(has)33 b(total)f(mass)i Fv(T)i Fu(\000)22 b Fv(t)p Fw(.)0 3527 y(In)43 b(particular,)j(the)d(total)g(mass)g(of)51 b(~)-57 b Fv(\026)1474 3542 y Fs(\024)1561 3527 y Fw(is)44 b(\014nite)f(and)g(so,)j(recalling)d(Assumption)i(3,)g(w)m(e)f(can)0 3643 y(slice)e(~)-57 b Fv(\026)271 3658 y Fs(\024)348 3643 y Fw(as)0 3818 y(\(87\))866 b(~)-57 b Fv(\026)1091 3833 y Fs(\024)1136 3818 y Fw(\()p Fv(x)1229 3777 y Fl(0)1252 3818 y Fv(;)17 b(v)t(;)g(s;)g(!)t Fw(\))43 b(=:)h Fv(\020)1817 3833 y Fs(\024)1861 3818 y Fw(\()p Fv(x)1954 3777 y Fl(0)1978 3818 y Fv(;)17 b(v)t(;)g(s)p Fw(;)g Fv(!)t Fw(\))g(\002)2403 3833 y Fs(\024)2445 3818 y Fw(\()p Fv(!)t Fw(\))g Fv(;)0 3993 y Fw(where)43 b(\002)367 4008 y Fs(\024)412 3993 y Fw(\()p Fv(!)t Fw(\))d(is)i(a)g(probabilit)m(y)g(measure,)k(and)c Fv(\020)1959 4008 y Fs(\024)2045 3993 y Fw(is,)i(for)d(\002)2415 4008 y Fs(\024)2460 3993 y Fw(-almost)g(an)m(y)i Fv(!)j Fu(2)d Fw(\012)f(\014xed,)k(a)0 4110 y(measure)g(of)e(\014nite)h(total) f(mass.)81 b(Let)45 b Fv(m)p Fw(\()p Fv(!)t Fw(\))f(b)s(e)g(the)h (total)f(mass)i(of)e Fv(\020)2779 4125 y Fs(\024)2868 4110 y Fw(for)g(a)g(\014xed)i Fv(!)t Fw(.)79 b(By)0 4226 y(construction,)0 4454 y(\(88\))397 b(1)28 b(=)851 4387 y(1)p 761 4431 228 4 v 761 4523 a Fv(T)36 b Fu(\000)23 b Fv(t)1015 4319 y Fp(Z)1148 4454 y Fv(d)8 b Fw(~)-57 b Fv(\026)1258 4469 y Fs(\024)1330 4454 y Fw(=)1533 4387 y(1)p 1444 4431 V 1444 4523 a Fv(T)36 b Fu(\000)22 b Fv(t)1698 4319 y Fp(Z)11 b(Z)1897 4454 y Fv(d\020)1991 4469 y Fs(\024)2052 4454 y Fv(d)p Fw(\002)2179 4469 y Fs(\024)2251 4454 y Fw(=)2354 4319 y Fp(Z)2482 4387 y Fv(m)p Fw(\()p Fv(!)t Fw(\))p 2481 4431 V 2481 4523 a Fv(T)36 b Fu(\000)22 b Fv(t)2735 4454 y(d)p Fw(\002)2862 4469 y Fs(\024)2906 4454 y Fw(\()p Fv(!)t Fw(\))17 b Fv(:)0 4676 y Fw(Let)33 b(also)1502 4812 y(^)1494 4838 y Fv(\020)1537 4853 y Fs(\024)1625 4838 y Fw(:=)1782 4771 y Fv(T)j Fu(\000)23 b Fv(t)p 1782 4815 V 1783 4907 a(m)p Fw(\()p Fv(!)t Fw(\))2037 4838 y Fv(\020)2080 4853 y Fs(\024)2141 4838 y Fv(;)0 5074 y Fw(and)33 b(note)f(that)627 5047 y(^)618 5074 y Fv(\020)39 b Fw(has)33 b(total)f(mass)i Fv(T)i Fu(\000)22 b Fv(t)33 b Fw(for)f(an)m(y)i(\014xed)f Fv(!)t Fw(.)0 5190 y(Then,)h(b)m(y)f(Prop)s(osition)g(10,)619 5417 y Fv(o)666 5432 y Fs(\016)725 5417 y Fw(+)22 b Fv(u)879 5376 y Fs(\017)995 5417 y Fu(\024)1155 5281 y Fp(Z)1271 5417 y Fv(L)p Fw(\()p Fv(x)1430 5376 y Fl(0)1454 5417 y Fv(;)1508 5349 y(x)1563 5313 y Fl(0)p 1508 5394 79 4 v 1528 5485 a Fv(\017)1597 5417 y(;)17 b(v)t(;)g(!)t Fw(\))j(+)i Fv(Av)2081 5376 y Fs(\017)2077 5441 y(\016)2138 5417 y Fw(+)g Fv(@)2287 5432 y Fs(t)2317 5417 y Fv(v)2368 5376 y Fs(\017)2364 5441 y(\016)2402 5417 y Fv(d)2462 5391 y Fw(^)2453 5417 y Fv(\020)2496 5432 y Fs(\024)2562 5417 y Fw(+)2660 5281 y Fp(Z)2776 5417 y Fv(v)2827 5376 y Fs(\017)2823 5441 y(\016)2861 5417 y Fv(d\027)2966 5376 y Fs(\017)2960 5441 y(t)3016 5417 y Fv(;)p eop end %%Page: 38 38 TeXDict begin 38 37 bop 168 251 a Fq(38)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y Fw(and)32 b(so,)h(b)m(y)i(\(87\))d(and)h(\(88\))o(,)527 685 y Fv(o)574 700 y Fs(\016)634 685 y Fw(+)732 549 y Fp(Z)859 618 y Fv(m)p Fw(\()p Fv(!)t Fw(\))p 858 662 228 4 v 858 753 a Fv(T)j Fu(\000)23 b Fv(t)1112 685 y(u)1168 644 y Fs(\017)1201 685 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))p Fv(d)p Fw(\002)1647 700 y Fs(\024)1690 685 y Fw(\()p Fv(!)t Fw(\))750 950 y Fu(\024)855 815 y Fp(Z)971 950 y Fv(L)p Fw(\()p Fv(x)1130 909 y Fl(0)1154 950 y Fv(;)1208 883 y(x)1263 847 y Fl(0)p 1208 927 79 4 v 1228 1019 a Fv(\017)1297 950 y(;)g(v)t(;)g(!)t Fw(\))p Fv(d)8 b Fw(~)-57 b Fv(\026)1649 965 y Fs(\024)1713 950 y Fw(+)1811 815 y Fp(Z)1928 950 y Fv(Av)2052 909 y Fs(\017)2048 975 y(\016)2108 950 y Fw(+)22 b Fv(@)2257 965 y Fs(t)2287 950 y Fv(v)2338 909 y Fs(\017)2334 975 y(\016)2372 950 y Fv(d)8 b Fw(~)-57 b Fv(\026)22 b Fw(+)2602 815 y Fp(Z)11 b(Z)2795 883 y Fv(m)p Fw(\()p Fv(!)t Fw(\))p 2794 927 228 4 v 2794 1019 a Fv(T)36 b Fu(\000)23 b Fv(t)3048 950 y(v)3099 909 y Fs(\017)3095 975 y(\016)3133 950 y Fv(d\027)3238 909 y Fs(\017)3232 975 y(t)3271 950 y Fv(d)p Fw(\002)3398 965 y Fs(\024)3443 950 y Fv(:)168 1179 y Fw(Making)33 b(use)g(of)40 b(\(11\))31 b(and)i(\(59\),)f(w)m(e)i(get)e (that)940 1277 y Fp(Z)1056 1412 y Fv(L)p Fw(\()p Fv(x)1215 1371 y Fl(0)1239 1412 y Fv(;)1293 1345 y(x)1348 1309 y Fl(0)p 1293 1389 79 4 v 1312 1480 a Fv(\017)1381 1412 y(;)17 b(v)t(;)g(!)t Fw(\))p Fv(d)8 b Fw(~)-57 b Fv(\026)1733 1427 y Fs(\024)1804 1412 y Fw(=)1907 1277 y Fp(Z)2023 1412 y Fv(L)p Fw(\()p Fv(x)2182 1371 y Fl(0)2206 1412 y Fv(;)2260 1345 y(x)2315 1309 y Fl(0)p 2260 1389 V 2280 1480 a Fv(\017)2349 1412 y(;)17 b(v)t(;)g(\034)2530 1447 y Fl(\000)2595 1420 y Fm(x)2630 1406 y Fj(0)p 2594 1432 58 3 v 2610 1473 a Fm(\017)2665 1412 y Fv(!)t Fw(\))p Fv(d\021)2867 1427 y Fs(\024)2911 1412 y Fv(d\045)3012 1427 y Fs(\024)1804 1671 y Fw(=)1907 1535 y Fp(Z)12 b(Z)2090 1671 y Fv(L)p Fw(\()p Fv(x)2249 1630 y Fl(0)2273 1671 y Fv(;)17 b Fw(0)p Fv(;)g(v)t(;)g(!)t Fw(\))p Fv(d\021)2707 1686 y Fs(\024)2749 1671 y Fv(d\045)2850 1686 y Fs(\024)1804 1930 y Fu(\024)1909 1794 y Fp(Z)p 2025 1850 67 4 v 2025 1930 a Fv(L)p Fw(\()p Fv(x)2184 1888 y Fl(0)2208 1930 y Fv(;)g($)s(;)g Fw(\004\))p Fv(d\045)2584 1945 y Fs(\024)2650 1930 y Fw(+)22 b Fv(C)7 b(\024;)17 b(:)168 2158 y Fw(for)47 b(some)i(suitable)g(constan)m(t)g Fv(C)60 b(>)54 b Fw(0.)89 b(Similarly)-8 b(,)53 b(b)m(y)d(\(19\))o(,)e(\(86\),)g(\(12\))o(,)k (\(13\))o(,)c(\(60\),)g(\(62\))168 2274 y(and)32 b(\(64\),)511 2380 y Fp(Z)628 2516 y Fv(A')17 b(d)8 b Fw(~)-57 b Fv(\026)892 2531 y Fs(\024)1018 2516 y Fw(=)1177 2380 y Fp(Z)1294 2516 y Fv(f)11 b Fw(\()p Fv(x)1446 2475 y Fl(0)1469 2516 y Fv(;)1523 2449 y(x)1578 2413 y Fl(0)p 1523 2493 79 4 v 1542 2584 a Fv(\017)1611 2516 y(;)17 b(v)t(;)g(!)t Fw(\))k Fu(\001)h Fv(D)2005 2531 y Fs(x)2049 2516 y Fv(')g Fw(+)2243 2449 y Fv(\033)t(\033)2361 2413 y Fs(T)p 2243 2493 174 4 v 2305 2584 a Fw(2)2426 2516 y(\()p Fv(x)2519 2475 y Fl(0)2542 2516 y Fv(;)2596 2449 y(x)2651 2413 y Fl(0)p 2596 2493 79 4 v 2616 2584 a Fv(\017)2685 2516 y(;)17 b(v)t(;)g(!)t Fw(\))26 b(:)h Fv(D)3091 2475 y Fr(2)3088 2541 y Fs(x)3132 2516 y Fv(')17 b(d)8 b Fw(~)-57 b Fv(\026)3323 2531 y Fs(\024)1018 2780 y Fw(=)1177 2644 y Fp(Z)12 b(Z)1360 2780 y Fv(f)f Fw(\()p Fv(x)1512 2739 y Fl(0)1535 2780 y Fv(;)1589 2712 y(x)1644 2676 y Fl(0)p 1589 2757 V 1609 2848 a Fv(\017)1678 2780 y(;)17 b(v)t(;)g(\034)1859 2815 y Fl(\000)1924 2787 y Fm(x)1959 2773 y Fj(0)p 1923 2799 58 3 v 1939 2841 a Fm(\017)1994 2780 y Fv(!)t Fw(\))22 b Fu(\001)g Fv(D)2250 2795 y Fs(x)2293 2780 y Fv(')1177 3052 y Fw(+)1263 2984 y Fv(\033)t(\033)1381 2948 y Fs(T)p 1263 3029 174 4 v 1325 3120 a Fw(2)1446 3052 y(\()p Fv(x)1539 3011 y Fl(0)1563 3052 y Fv(;)1617 2984 y(x)1672 2948 y Fl(0)p 1617 3029 79 4 v 1636 3120 a Fv(\017)1705 3052 y(;)17 b(v)t(;)g(\034)1886 3087 y Fl(\000)1951 3059 y Fm(x)1986 3045 y Fj(0)p 1951 3071 58 3 v 1966 3113 a Fm(\017)2022 3052 y Fv(!)t Fw(\))27 b(:)h Fv(D)2291 3011 y Fr(2)2288 3076 y Fs(x)2331 3052 y Fv(')17 b(d\021)2511 3067 y Fs(\024)2572 3052 y Fv(d\045)2673 3067 y Fs(\024)1018 3306 y Fw(=)1177 3171 y Fp(Z)12 b(Z)1360 3306 y Fv(f)f Fw(\()p Fv(x)1512 3265 y Fl(0)1535 3306 y Fv(;)17 b Fw(0)p Fv(;)g(v)t(;)g(!)t Fw(\))j Fu(\001)i Fv(D)2021 3321 y Fs(x)2065 3306 y Fv(')g Fw(+)2259 3239 y Fv(\033)t(\033)2377 3203 y Fs(T)p 2259 3284 174 4 v 2321 3375 a Fw(2)2442 3306 y(\()p Fv(x)2535 3265 y Fl(0)2559 3306 y Fv(;)17 b Fw(0)p Fv(;)g(v)t(;)g(!)t Fw(\))25 b(:)j Fv(D)3058 3265 y Fr(2)3055 3331 y Fs(x)3099 3306 y Fv(')17 b(d\021)3279 3321 y Fs(\024)3340 3306 y Fv(d\045)3441 3321 y Fs(\024)1018 3565 y Fw(=)1177 3430 y Fp(Z)1294 3565 y Fv($)24 b Fu(\001)e Fv(D)1530 3580 y Fs(x)1574 3565 y Fv(')g Fw(+)g(\004)1823 3524 y Fs(\017)1883 3565 y Fw(:)28 b Fv(D)2022 3524 y Fr(2)2061 3565 y Fv(')17 b(d\045)2243 3580 y Fs(\024)2304 3565 y Fv(;)168 3798 y Fw(for)32 b(an)m(y)h Fv(')p Fw(\()p Fv(x)658 3762 y Fl(0)682 3798 y Fw(\))27 b Fu(2)h Fv(C)918 3762 y Fr(2)958 3798 y Fw(\()p Ft(R)1068 3762 y Fs(n)1115 3798 y Fw(\),)k(with)1262 4026 y(\004)1327 3985 y Fs(\017)1360 4026 y Fw(\()p Fv(x)1453 3985 y Fl(0)1477 4026 y Fw(\))44 b(:=)1716 3959 y(1)p 1716 4003 49 4 v 1716 4095 a(2)1792 3891 y Fp(Z)1908 4026 y Fv(\033)t(\033)2026 3985 y Fs(T)2081 4026 y Fw(\()p Fv(x)2174 3985 y Fl(0)2198 4026 y Fv(;)17 b Fw(0)p Fv(;)g(v)t(;)g(!)t Fw(\))g Fv(d\021)2649 4041 y Fs(\024)2707 4026 y Fv(:)168 4255 y Fw(Th)m(us,)34 b(exploiting)h(\(68\))d(and)g(\(69\),)621 4348 y Fp(Z)738 4483 y Fv(Av)862 4442 y Fs(\017)858 4508 y(\016)918 4483 y Fw(+)22 b Fv(@)1067 4498 y Fs(t)1097 4483 y Fv(v)1148 4442 y Fs(\017)1144 4508 y(\016)1182 4483 y Fv(d)8 b Fw(~)-57 b Fv(\026)1292 4498 y Fs(\024)1364 4483 y Fw(=)1467 4348 y Fp(Z)1584 4483 y Fv($)24 b Fu(\001)e Fv(D)1820 4498 y Fs(x)1864 4483 y Fv(v)1915 4442 y Fs(\017)1911 4508 y(\016)1971 4483 y Fw(+)g(\004)2134 4442 y Fs(\017)2194 4483 y Fw(:)28 b Fv(D)2333 4442 y Fr(2)2330 4508 y Fs(x)2374 4483 y Fv(v)2425 4442 y Fs(\017)2421 4508 y(\016)2481 4483 y Fw(+)22 b Fv(@)2630 4498 y Fs(t)2660 4483 y Fv(v)2711 4442 y Fs(\017)2707 4508 y(\016)2745 4483 y Fv(d\045)2846 4498 y Fs(\024)1364 4742 y Fw(=)1467 4606 y Fp(Z)1584 4742 y Fv(v)1635 4701 y Fs(\017)1631 4767 y(\016)1669 4742 y Fv(d)6 b Fw(\026)-55 b Fv(\027)1768 4757 y Fs(T)1845 4742 y Fu(\000)1944 4606 y Fp(Z)2061 4742 y Fv(v)2112 4701 y Fs(\017)2108 4767 y(\016)2146 4742 y Fv(d)6 b Fw(\026)-55 b Fv(\027)2245 4757 y Fs(t)2296 4742 y Fw(+)2394 4606 y Fp(Z)2494 4742 y Fw(\(\004)2597 4701 y Fs(\017)2652 4742 y Fu(\000)22 b Fw(\004)2816 4701 y Fl(\003)2856 4742 y Fw(\))27 b(:)h Fv(D)3060 4701 y Fr(2)3057 4767 y Fs(x)3101 4742 y Fv(v)3152 4701 y Fs(\017)3148 4767 y(\016)3186 4742 y Fv(d\045)3287 4757 y Fs(\024)3348 4742 y Fv(;)168 4965 y Fw(with)33 b(\004)455 4929 y Fl(\003)522 4965 y Fw(:=)28 b(0)k(in)h(cases)h(A)e(and)h(B)g(and)f(\004)1744 4929 y Fl(\003)1812 4965 y Fw(=)27 b(\004)h(=)f(\004)2176 4929 y Fs(\017)2242 4965 y Fw(in)32 b(case)i(C.)168 5081 y(Note)e(that)168 5257 y(\(89\))1320 b(lim)1663 5317 y Fs(\017)p Fl(!)p Fr(0)1814 5257 y Fw(\004)1879 5216 y Fs(\017)1934 5257 y Fu(\000)23 b Fw(\004)2099 5216 y Fl(\003)2182 5257 y Fw(=)45 b(0)168 5469 y(uniformly)33 b(in)g(\()p Fv(x;)17 b(v)t(;)g(!)t Fw(\))31 b(in)i(all)f(the)h(cases)h (A-C.)p eop end %%Page: 39 39 TeXDict begin 39 38 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(39)0 450 y Fw(Also,)33 b(b)m(y)i(\(72\))d(and)h(\(73\))o(,)902 676 y(\026)-55 b Fv(u)p Fw(\()p Fv(x;)17 b(t)p Fw(\))28 b Fu(\025)1311 541 y Fp(Z)p 1428 596 67 4 v 1428 676 a Fv(L)p Fw(\()p Fv(x)1587 635 y Fl(0)1611 676 y Fv(;)17 b($)s(;)g Fw(\004\))p Fv(d\045)1987 691 y Fs(\024)2052 676 y Fw(+)2150 541 y Fp(Z)2266 676 y Fv( )t(d)6 b Fw(\026)-55 b Fv(\027)2432 691 y Fs(T)6 b(;\024)2566 676 y Fu(\000)22 b Fv(\024)17 b(:)0 897 y Fw(By)33 b(collecting)h(the)f(ab)s(o)m(v)m(e)g (estimates,)h(w)m(e)g(conclude)g(that)480 1124 y Fv(o)527 1139 y Fs(\016)587 1124 y Fw(+)685 988 y Fp(Z)812 1057 y Fv(m)p Fw(\()p Fv(!)t Fw(\))p 811 1101 228 4 v 811 1192 a Fv(T)i Fu(\000)22 b Fv(t)1065 1124 y(u)1121 1083 y Fs(\017)1153 1124 y Fw(\()p Fv(x;)17 b(t;)g(!)t Fw(\))p Fv(d)p Fw(\002\()p Fv(!)t Fw(\))703 1389 y Fu(\024)34 b Fw(\026)-55 b Fv(u)o Fw(\()p Fv(x;)17 b(t)p Fw(\))23 b(+)1194 1254 y Fp(Z)1293 1389 y Fw(\()p Fv(v)1382 1348 y Fs(\017)1378 1414 y(\016)1438 1389 y Fu(\000)g Fv( )t Fw(\))p Fv(d)6 b Fw(\026)-55 b Fv(\027)1742 1404 y Fs(T)1819 1389 y Fw(+)1917 1254 y Fp(Z)11 b(Z)2111 1322 y Fv(m)p Fw(\()p Fv(!)t Fw(\))p 2110 1366 V 2110 1458 a Fv(T)36 b Fu(\000)22 b Fv(t)2364 1389 y(v)2415 1348 y Fs(\017)2411 1414 y(\016)2449 1389 y Fv(d\027)2554 1348 y Fs(\017)2548 1414 y(t)2603 1389 y Fv(d)p Fw(\002)g Fu(\000)2852 1254 y Fp(Z)2968 1389 y Fv(v)3019 1348 y Fs(\017)3015 1414 y(\016)3053 1389 y Fv(d)6 b Fw(\026)-55 b Fv(\027)3152 1404 y Fs(t)892 1648 y Fw(+)990 1512 y Fp(Z)1090 1648 y Fw(\(\004)1193 1607 y Fs(\017)1248 1648 y Fu(\000)22 b Fw(\004)1412 1607 y Fl(\003)1452 1648 y Fw(\))27 b(:)h Fv(D)1656 1607 y Fr(2)1653 1673 y Fs(x)1697 1648 y Fv(v)1748 1607 y Fs(\017)1744 1673 y(\016)1782 1648 y Fv(d\045)1883 1663 y Fs(\024)1950 1648 y Fw(+)22 b Fv(C)7 b(\024)17 b(:)0 1869 y Fw(That)33 b(is,)g(b)m(y)h(Assumption)g(6,)1035 2096 y Fv(o)1082 2111 y Fs(\016)1142 2096 y Fw(+)1240 1960 y Fp(Z)1367 2028 y Fv(m)p Fw(\()p Fv(!)t Fw(\))p 1366 2073 V 1366 2164 a Fv(T)i Fu(\000)23 b Fv(t)1621 2096 y(u)1677 2054 y Fs(\017)1725 2096 y Fv(d)p Fw(\002)875 2361 y Fu(\024)89 b Fw(\026)-55 b Fv(u)22 b Fw(+)1211 2225 y Fp(Z)1311 2361 y Fw(\()p Fv(u)1405 2320 y Fs(\017)1459 2361 y Fu(\000)1578 2335 y Fw(\026)1558 2361 y Fv( )t Fw(\))17 b Fv(d)6 b Fw(\026)-55 b Fv(\027)1779 2376 y Fs(T)1856 2361 y Fw(+)1954 2225 y Fp(Z)11 b(Z)2148 2293 y Fv(m)p Fw(\()p Fv(!)t Fw(\))p 2147 2338 V 2147 2429 a Fv(T)36 b Fu(\000)22 b Fv(t)2401 2361 y(u)2457 2320 y Fs(\017)2489 2361 y Fv(d\027)2594 2320 y Fs(\017)2588 2386 y(t)2644 2361 y Fv(d)p Fw(\002)1133 2620 y Fu(\000)1227 2484 y Fp(Z)1343 2620 y Fv(u)1399 2579 y Fs(\017)1431 2620 y Fv(d)6 b Fw(\026)-55 b Fv(\027)1530 2635 y Fs(t)1582 2620 y Fw(+)1680 2484 y Fp(Z)1780 2620 y Fw(\(\004)1883 2579 y Fs(\017)1937 2620 y Fu(\000)23 b Fw(\004)2102 2579 y Fl(\003)2141 2620 y Fw(\))28 b(:)g Fv(D)2346 2579 y Fr(2)2343 2644 y Fs(x)2387 2620 y Fv(v)2438 2579 y Fs(\017)2434 2644 y(\016)2472 2620 y Fv(d\045)22 b Fw(+)g Fv(C)7 b(\024)17 b(:)0 2840 y Fw(W)-8 b(e)33 b(no)m(w)g(send)h Fv(\017)28 b Fu(!)f Fw(0)33 b(and)f(w)m(e)i(obtain)1438 3014 y Fv(o)1485 3029 y Fs(\016)1545 3014 y Fw(+)22 b Fv(u)1699 2973 y Fr(0)1766 3014 y Fu(\024)34 b Fw(\026)-55 b Fv(u)21 b Fw(+)h Fv(C)7 b(\024)17 b(;)0 3187 y Fw(thanks)34 b(to)e(Prop)s(osition)h(24)f(and)g(form)m(ulas)i(\(58\))o(,)f(\(88\))f (and)g(\(89\).)0 3304 y(W)-8 b(e)27 b(no)m(w)f(send)i Fv(\016)i Fw(and)c Fv(\024)g Fw(to)g(zero,)i(getting)e Fv(u)1634 3267 y Fr(0)1701 3304 y Fu(\024)34 b Fw(\026)-55 b Fv(u)p Fw(.)41 b(This)27 b(and)g(Prop)s(osition)f(25)g(yield)h(the)g (desired)0 3420 y(claim.)3330 b Fi(\003)1018 3665 y Fw(11.)48 b Fx(Impr)n(o)n(ved)38 b(Moment)f(Estima)-7 b(tes)0 3839 y Fw(In)30 b(this)h(section)g(w)m(e)g(impro)m(v)m(e)h(the)e(momen)m(t)i (estimates)f(of)f(section)h(5)f(in)g(order)g(to)g(sho)m(w)h(that)f(for) 0 3955 y(small)h(time)g(\\optimal)g(tra)5 b(jectories)32 b(don't)e(go)g(to)s(o)g(far".)42 b(These)32 b(estimates)h(will)e(b)s(e) g(essen)m(tial)h(in)0 4072 y(establishing)38 b(that)e(the)h (homogenized)h(limit)e(is)h(a)f(viscosit)m(y)j(solution)e(of)f(a)g (suitable)i(equation.)0 4188 y(Here,)33 b(w)m(e)h(will)f(tak)m(e)h(a)e (small)h(time)g(step)h Fv(h)28 b Fu(2)g Fw(\(0)p Fv(;)17 b(T)d Fw(\).)42 b(W)-8 b(e)33 b(start)g(with)g(an)g(auxiliary)g (result:)0 4372 y FA(Lemma)39 b(27.)j Fo(Consider)34 b(the)g(di\013usion)848 4603 y Fv(d)p FA(x)958 4562 y Fr(0)1025 4603 y Fw(=)28 b Fv(f)1188 4493 y Fp(\020)1247 4603 y FA(x)1306 4562 y Fr(0)1346 4603 y Fv(;)1399 4536 y FA(x)1458 4500 y Fr(0)p 1399 4580 99 4 v 1429 4671 a Fv(\017)1508 4603 y(;)17 b Fw(0)p Fv(;)g(!)1710 4493 y Fp(\021)1768 4603 y Fv(dt)22 b Fw(+)g Fv(\033)2033 4493 y Fp(\020)2093 4603 y FA(x)2152 4562 y Fr(0)2191 4603 y Fv(;)2245 4536 y FA(x)2304 4500 y Fr(0)p 2245 4580 V 2274 4671 a Fv(\017)2353 4603 y(;)17 b Fw(0)p Fv(;)g(!)2555 4493 y Fp(\021)2613 4603 y Fv(dW)2756 4618 y Fs(t)2786 4603 y Fv(;)0 4816 y Fo(with)35 b(initial)f(c)-5 b(ondition)34 b FA(x)983 4780 y Fr(0)1023 4816 y Fw(\()p Fv(T)h Fu(\000)23 b Fv(h)p Fw(\))28 b(=)f Fv(x)p Fo(.)46 b(Then,)33 b(we)i(have)1195 4990 y Fu(j)p Fv(E)6 b Fw([)p FA(x)1387 4949 y Fr(0)1427 4990 y Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(h)p Fw(\))g Fu(\000)h FA(x)1932 4949 y Fr(0)1972 4990 y Fw(\()p Fv(T)14 b Fw(\)])p Fu(j)27 b(\024)h Fv(C)7 b(h;)0 5164 y Fo(and)1203 5297 y Fv(E)f Fu(j)p FA(x)1368 5256 y Fr(0)1407 5297 y Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(h)p Fw(\))h Fu(\000)f FA(x)1912 5256 y Fr(0)1952 5297 y Fw(\()p Fv(T)14 b Fw(\))p Fu(j)2127 5256 y Fr(2)2193 5297 y Fu(\024)28 b Fv(C)7 b(h;)0 5450 y Fo(wher)-5 b(e)34 b(the)h(c)-5 b(onstant)35 b Fv(C)41 b Fo(c)-5 b(an)35 b(b)-5 b(e)34 b(chosen)g(indep)-5 b(endent)34 b(of)g Fv(\017)p Fo(,)h Fv(h)g Fo(and)g Fv(x)p Fo(.)p eop end %%Page: 40 40 TeXDict begin 40 39 bop 168 251 a Fq(40)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y Fo(Pr)k(o)g(of.)41 b Fw(Fix)33 b Fv(h)27 b(>)h Fw(0,)k(let)h Fv(T)j Fu(\000)23 b Fv(h)28 b Fu(\024)g Fv(t)g Fu(\024)g Fv(T)14 b Fw(.)43 b(By)33 b(Dynkin's)h(F)-8 b(orm)m(ula)33 b(w)m(e)g(ha)m(v)m(e)168 667 y(\(90\))1053 599 y Fv(d)p 1036 644 86 4 v 1036 735 a(dt)1132 667 y(E)6 b Fw([)p FA(x)1296 626 y Fr(0)1335 667 y Fw(\()p Fv(T)36 b Fu(\000)23 b Fv(h)p Fw(\))f Fu(\000)h FA(x)1841 626 y Fr(0)1880 667 y Fw(\()p Fv(t)p Fw(\)])28 b(=)g Fu(\000)p Fv(E)6 b(f)2364 556 y Fp(\020)2423 667 y FA(x)2482 626 y Fr(0)2522 667 y Fv(;)2576 599 y FA(x)2635 563 y Fr(0)p 2576 644 99 4 v 2605 735 a Fv(\017)2684 667 y(;)17 b Fw(0)p Fv(;)g(!)2886 556 y Fp(\021)2944 667 y Fv(;)168 855 y Fw(whic)m(h,)34 b(b)m(y)f(Assumption)i(14,)d(yields)i(the)f(\014rst)g(part)f(of)h(the)g (claim.)168 972 y(Similarly)-8 b(,)33 b(w)m(e)h(ha)m(v)m(e)620 1121 y Fv(d)p 603 1165 86 4 v 603 1257 a(dt)698 1188 y(E)6 b Fu(j)p FA(x)863 1147 y Fr(0)903 1188 y Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(h)p Fw(\))g Fu(\000)h FA(x)1408 1147 y Fr(0)1448 1188 y Fw(\()p Fv(t)p Fw(\))p Fu(j)1587 1147 y Fr(2)1653 1188 y Fw(=2)p Fv(E)1873 1048 y Fp(\024)1925 1188 y Fw(\()p FA(x)2022 1147 y Fr(0)2062 1188 y Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(h)p Fw(\))h Fu(\000)f FA(x)2567 1147 y Fr(0)2607 1188 y Fw(\()p Fv(t)p Fw(\)\))p Fv(f)11 b Fw(\()p FA(x)2912 1147 y Fr(0)2951 1188 y Fv(;)3005 1121 y FA(x)3064 1085 y Fr(0)p 3005 1165 99 4 v 3034 1257 a Fv(\017)3113 1188 y(;)17 b Fw(0)p Fv(;)g(!)t Fw(\))3353 1048 y Fp(\025)1751 1467 y Fw(+)23 b(tr)16 b Fv(E)6 b Fw([)p Fv(\033)t Fw(\()p FA(x)2203 1426 y Fr(0)2243 1467 y Fv(;)2296 1399 y FA(x)2355 1363 y Fr(0)p 2296 1444 V 2326 1535 a Fv(\017)2405 1467 y(;)17 b Fw(0)p Fv(;)g(!)t Fw(\))p Fv(\033)2704 1426 y Fs(T)2757 1467 y Fw(\()p FA(x)2854 1426 y Fr(0)2894 1467 y Fv(;)2948 1399 y FA(x)3007 1363 y Fr(0)p 2948 1444 V 2977 1535 a Fv(\017)3056 1467 y(;)g Fw(0)p Fv(;)g(!)t Fw(\)])p Fv(:)-3182 b Fw(\(91\))168 1665 y(Th)m(us,)34 b(if)e(w)m(e)i(set)f Fv(A)p Fw(\()p Fv(t)p Fw(\))28 b(=)g Fv(E)6 b Fu(j)p FA(x)1308 1629 y Fr(0)1347 1665 y Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(h)p Fw(\))h Fu(\000)f FA(x)1852 1629 y Fr(0)1892 1665 y Fw(\()p Fv(t)p Fw(\))p Fu(j)2031 1629 y Fr(2)2103 1665 y Fw(w)m(e)33 b(ha)m(v)m(e)1316 1810 y Fv(d)p 1298 1855 86 4 v 1298 1946 a(dt)1394 1878 y(A)28 b Fu(\024)g Fv(A)22 b Fw(+)g Fv(C)r(;)212 b(A)p Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(h)p Fw(\))28 b(=)g(0)p Fv(:)168 2072 y Fw(A)k(Gron)m(w)m(all)h(estimate)h (yields)g Fu(j)p Fv(A)p Fw(\()p Fv(t)p Fw(\))p Fu(j)28 b(\024)g Fv(C)7 b(h)p Fw(.)1863 b Fi(\003)168 2247 y FA(Lemma)54 b(28.)49 b Fo(Consider)d(an)g(optimal)h(di\013usion)f FA(x)p Fw(\()p Fv(t)p Fw(\))p Fo(,)k(for)d Fv(T)e Fu(\000)31 b Fv(h)51 b Fu(\024)g Fv(t)f Fu(\024)h Fv(T)14 b Fo(,)50 b(with)d(initial)168 2363 y(c)-5 b(ondition)34 b FA(x)p Fw(\()p Fv(T)i Fu(\000)22 b Fv(h)p Fw(\))28 b(=)g Fv(x)p Fo(.)45 b(L)-5 b(et)1362 2594 y Fv(M)38 b Fw(:=)28 b Fv(E)1720 2458 y Fp(Z)1819 2485 y Fs(T)1775 2684 y(T)10 b Fl(\000)p Fs(h)1942 2594 y Fv(L)2008 2483 y Fp(\020)2068 2594 y FA(x)p Fv(;)2181 2527 y FA(x)p 2181 2571 60 4 v 2191 2662 a Fv(\017)2250 2594 y(;)17 b FA(v)r Fv(;)g(!)2464 2483 y Fp(\021)2522 2594 y Fv(dt:)168 2811 y Fo(Then,)34 b(we)g(have)1114 2932 y Fu(j)p Fv(E)6 b Fw([)p FA(x)p Fw(\()p Fv(T)35 b Fu(\000)23 b Fv(h)p Fw(\))f Fu(\000)h FA(x)p Fw(\()p Fv(T)14 b Fw(\)])p Fu(j)27 b(\024)h Fv(M)2249 2891 y Fr(1)p Fs(=)p Fr(2)2360 2932 y Fv(C)7 b(h)2493 2891 y Fr(1)p Fs(=)p Fr(2)2625 2932 y Fw(+)22 b Fv(C)7 b(h;)168 3070 y Fo(and)1231 3189 y Fv(E)f Fu(j)p FA(x)p Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(h)p Fw(\))g Fu(\000)h FA(x)p Fw(\()p Fv(T)14 b Fw(\))p Fu(j)2076 3147 y Fr(2)2143 3189 y Fu(\024)28 b Fv(M)10 b(C)d(h)23 b Fw(+)f Fv(C)7 b(h;)168 3327 y Fo(wher)-5 b(e)34 b Fv(C)42 b Fo(c)-5 b(an)34 b(b)-5 b(e)35 b(chosen)e(indep)-5 b(endently)34 b(of)h Fv(\017)p Fo(,)g Fv(h)g Fo(and)f Fv(x)p Fo(.)168 3503 y(Pr)-5 b(o)g(of.)41 b Fw(Using)33 b(again)f(\(90\))g(and)h (Assumption)h(15)e(w)m(e)i(obtain)662 3733 y Fv(E)6 b Fw([)p FA(x)p Fw(\()p Fv(T)36 b Fu(\000)23 b Fv(h)p Fw(\))f Fu(\000)h FA(x)p Fw(\()p Fv(T)14 b Fw(\)])27 b(=)h Fv(E)1731 3598 y Fp(Z)1831 3624 y Fs(T)1787 3823 y(T)10 b Fl(\000)p Fs(h)1954 3733 y Fv(f)2013 3623 y Fp(\020)2072 3733 y FA(x)p Fv(;)2185 3666 y FA(x)p 2185 3710 V 2195 3802 a Fv(\017)2254 3733 y(;)17 b FA(v)r Fv(;)g(!)2468 3623 y Fp(\021)2526 3733 y Fv(dt)1533 4046 y Fu(\024)28 b Fv(h)1694 4005 y Fr(1)p Fs(=)p Fr(2)1821 3905 y Fp(\022)1894 4046 y Fv(E)1989 3910 y Fp(Z)2089 3936 y Fs(T)2044 4136 y(T)10 b Fl(\000)p Fs(h)2212 3931 y Fp(\014)2212 3991 y(\014)2212 4051 y(\014)2245 4046 y Fv(f)2304 3935 y Fp(\020)2363 4046 y FA(x)p Fv(;)2476 3978 y FA(x)p 2476 4023 V 2486 4114 a Fv(\017)2545 4046 y(;)17 b FA(v)r Fv(;)g(!)2759 3935 y Fp(\021)2817 3931 y(\014)2817 3991 y(\014)2817 4051 y(\014)2850 3957 y Fr(2)2890 4046 y Fv(dt)2976 3905 y Fp(\023)3049 3917 y Fr(1)p Fs(=)p Fr(2)1533 4358 y Fu(\024)28 b Fv(C)7 b(h)1771 4317 y Fr(1)p Fs(=)p Fr(2)1898 4218 y Fp(\022)1971 4358 y Fv(E)2066 4223 y Fp(Z)2166 4249 y Fs(T)2121 4448 y(T)j Fl(\000)p Fs(h)2289 4358 y Fv(L)2355 4248 y Fp(\020)2415 4358 y FA(x)p Fv(;)2527 4291 y FA(x)p 2527 4335 V 2537 4426 a Fv(\017)2596 4358 y(;)17 b FA(v)r Fv(;)g(!)2810 4248 y Fp(\021)2891 4358 y Fw(+)22 b Fv(C)7 b(dt)3152 4218 y Fp(\023)3225 4230 y Fr(1)p Fs(=)p Fr(2)1533 4583 y Fu(\024)28 b Fv(C)7 b(M)1819 4542 y Fr(1)p Fs(=)p Fr(2)1930 4583 y Fv(h)1986 4542 y Fr(1)p Fs(=)p Fr(2)2118 4583 y Fw(+)22 b Fv(C)7 b(h:)168 4748 y Fw(Similarly)-8 b(,)33 b(b)m(y)i(\(91\),)d(w)m(e)i(ha)m (v)m(e)394 4893 y Fv(d)p 377 4938 86 4 v 377 5029 a(dt)472 4961 y(E)6 b Fu(j)p FA(x)p Fw(\()p Fv(T)36 b Fu(\000)23 b Fv(h)p Fw(\))f Fu(\000)h FA(x)p Fw(\()p Fv(t)p Fw(\))p Fu(j)1282 4919 y Fr(2)1349 4961 y Fw(=2)p Fv(E)1552 4850 y Fp(h)1598 4961 y Fw(\()p FA(x)p Fw(\()p Fv(T)36 b Fu(\000)23 b Fv(h)p Fw(\))f Fu(\000)h FA(x)p Fw(\()p Fv(t)p Fw(\)\))p Fv(f)2409 4850 y Fp(\020)2468 4961 y FA(x)p Fv(;)2581 4893 y FA(x)p 2581 4938 60 4 v 2591 5029 a Fv(\017)2650 4961 y(;)17 b FA(v)r Fv(;)g(!)2864 4850 y Fp(\021)o(i)1447 5181 y Fw(+)22 b(tr)16 b Fv(E)1715 5070 y Fp(h)1762 5181 y Fv(\033)1821 5070 y Fp(\020)1881 5181 y FA(x)p Fv(;)1994 5113 y FA(x)p 1994 5158 V 2004 5249 a Fv(\017)2063 5181 y(;)h FA(v)r Fv(;)g(!)2277 5070 y Fp(\021)2335 5181 y Fv(\033)2394 5140 y Fs(T)2449 5070 y Fp(\020)2508 5181 y FA(x)p Fv(;)2621 5113 y FA(x)p 2621 5158 V 2631 5249 a Fv(\017)2690 5181 y(;)g FA(v)r Fv(;)g(!)2904 5070 y Fp(\021)o(i)1347 5401 y Fu(\024)q Fv(C)7 b(h)1558 5360 y Fl(\000)p Fr(1)1652 5401 y Fv(E)f Fu(j)p FA(x)p Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(h)p Fw(\))g Fu(\000)h FA(x)p Fw(\()p Fv(t)p Fw(\))p Fu(j)2461 5360 y Fr(2)2523 5401 y Fw(+)f Fv(C)7 b(hE)f(L)2898 5291 y Fp(\020)2958 5401 y FA(x)p Fv(;)3070 5334 y FA(x)p 3070 5378 V 3080 5469 a Fv(\017)3140 5401 y(;)17 b FA(v)r Fv(;)g(!)3354 5291 y Fp(\021)3434 5401 y Fw(+)22 b Fv(C)r(:)p eop end %%Page: 41 41 TeXDict begin 41 40 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(41)0 450 y Fw(Th)m(us,)26 b(if)c(w)m(e)i(set)f Fv(A)p Fw(\()p Fv(t)p Fw(\))28 b(=)f Fv(E)6 b Fu(j)p FA(x)p Fw(\()p Fv(T)15 b Fu(\000)q Fv(h)p Fw(\))q Fu(\000)q FA(x)p Fw(\()p Fv(t)p Fw(\))p Fu(j)1660 414 y Fr(2)1724 450 y Fw(w)m(e)24 b(ha)m(v)m(e,)i(b)m(y)d(a)f(Gron)m(w)m(all)h (estimate,)i(that)e Fu(j)p Fv(A)p Fw(\()p Fv(t)p Fw(\))p Fu(j)k(\024)0 566 y Fv(C)7 b(M)j(h)23 b Fw(+)f Fv(C)7 b(h)p Fw(.)3066 b Fi(\003)0 736 y Fw(No)m(w)29 b(w)m(e)h(use)g(the)f(t) m(w)m(o)h(previous)g(results)g(to)e(pro)m(v)m(e)i(that)f(the)g(in)m (tegral)g(of)g(the)g(Lagrangian)f(along)0 853 y(an)k(optimal)h(tra)5 b(jectory)33 b(is)g(not)g(to)s(o)f(big:)0 1029 y FA(Prop)s(osition)38 b(29.)j Fo(Consider)34 b(an)g(optimal)h(di\013usion)f FA(x)h Fo(for)f Fv(T)i Fu(\000)23 b Fv(h)28 b Fu(\024)g Fv(t)g Fu(\024)g Fv(T)14 b Fo(,)34 b(minimizing)1091 1255 y Fv(E)1186 1119 y Fp(Z)1286 1146 y Fs(T)1241 1345 y(T)10 b Fl(\000)p Fs(h)1409 1255 y Fv(L)1475 1144 y Fp(\020)1534 1255 y FA(x)p Fv(;)1647 1188 y FA(x)p 1647 1232 60 4 v 1657 1323 a Fv(\017)1716 1255 y(;)17 b FA(v)r Fv(;)g(!)1930 1144 y Fp(\021)1988 1255 y Fv(dt)22 b Fw(+)g Fv( )t Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\)\))p Fv(;)0 1479 y Fo(with)35 b(initial)f(c)-5 b(ondition)34 b FA(x)p Fw(\()p Fv(T)j Fu(\000)22 b Fv(h)p Fw(\))28 b(=)g Fv(x)p Fo(.)46 b(Then)34 b(ther)-5 b(e)35 b(exists)g(a)f(c)-5 b(onstant)35 b Fv(C)42 b Fo(indep)-5 b(endent)34 b(of)g Fv(h)p Fo(,)0 1595 y Fv(\017)h Fo(and)g Fv(x)g Fo(such)g(that)1193 1761 y Fv(E)1287 1626 y Fp(Z)1387 1652 y Fs(T)1343 1851 y(T)10 b Fl(\000)p Fs(h)1510 1761 y Fv(L)1576 1651 y Fp(\020)1636 1761 y FA(x)p Fv(;)1749 1694 y FA(x)p 1749 1738 V 1759 1830 a Fv(\017)1818 1761 y(;)17 b FA(v)r Fv(;)g(!)2032 1651 y Fp(\021)2090 1761 y Fv(dt)27 b Fu(\024)h Fv(C)7 b(h:)0 1996 y Fo(Pr)-5 b(o)g(of.)41 b Fw(Consider)34 b(the)f(di\013usion)848 2214 y Fv(d)p FA(x)958 2172 y Fr(0)1025 2214 y Fw(=)28 b Fv(f)1188 2103 y Fp(\020)1247 2214 y FA(x)1306 2172 y Fr(0)1346 2214 y Fv(;)1399 2146 y FA(x)1458 2110 y Fr(0)p 1399 2191 99 4 v 1429 2282 a Fv(\017)1508 2214 y(;)17 b Fw(0)p Fv(;)g(!)1710 2103 y Fp(\021)1768 2214 y Fv(dt)22 b Fw(+)g Fv(\033)2033 2103 y Fp(\020)2093 2214 y FA(x)2152 2172 y Fr(0)2191 2214 y Fv(;)2245 2146 y FA(x)2304 2110 y Fr(0)p 2245 2191 V 2274 2282 a Fv(\017)2353 2214 y(;)17 b Fw(0)p Fv(;)g(!)2555 2103 y Fp(\021)2613 2214 y Fv(dW)2756 2229 y Fs(t)2786 2214 y Fv(;)0 2409 y Fw(with)33 b FA(x)p Fw(\()p Fv(T)j Fu(\000)23 b Fv(h)p Fw(\))28 b(=)f Fv(x)p Fw(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e,)h(b)m(y)f(the)g(optimalit)m(y)h(of) e FA(x)p Fw(,)h(that)813 2641 y Fv(E)908 2500 y Fp(\024)961 2505 y(Z)1060 2531 y Fs(T)1016 2731 y(T)10 b Fl(\000)p Fs(h)1183 2641 y Fv(L)1249 2530 y Fp(\020)1309 2641 y FA(x)p Fv(;)1422 2573 y FA(x)p 1422 2618 60 4 v 1432 2709 a Fv(\017)1491 2641 y(;)17 b FA(v)r Fv(;)g(!)1705 2530 y Fp(\021)1763 2641 y Fv(dt)22 b Fw(+)g Fv( )t Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\)\))2318 2500 y Fp(\025)1036 2930 y Fu(\024)28 b Fv(E)1236 2789 y Fp(\024)1288 2794 y(Z)1388 2820 y Fs(T)1344 3020 y(T)10 b Fl(\000)p Fs(h)1511 2930 y Fv(L)1577 2819 y Fp(\020)1637 2930 y FA(x)1696 2889 y Fr(0)1735 2930 y Fv(;)1789 2862 y FA(x)1848 2826 y Fr(0)p 1789 2907 99 4 v 1819 2998 a Fv(\017)1898 2930 y(;)17 b Fw(0)p Fv(;)g(!)2100 2819 y Fp(\021)2158 2930 y Fv(dt)22 b Fw(+)g Fv( )t Fw(\()p FA(x)2528 2889 y Fr(0)2567 2930 y Fw(\()p Fv(T)14 b Fw(\)\))2752 2789 y Fp(\025)2821 2930 y Fv(:)0 3158 y Fw(Moreo)m(v)m(er,)30 b(the)e(fact)g(that)f FA(x)p Fw(\()p Fv(T)f Fu(\000)12 b Fv(h)p Fw(\))29 b(=)e FA(x)1562 3122 y Fr(0)1602 3158 y Fw(\()p Fv(T)e Fu(\000)12 b Fv(h)p Fw(\))29 b(=)e Fv(x)p Fw(,)i(via)f(a)g(second)h(order)f(T)-8 b(a)m(ylor)28 b(expansion)0 3274 y(of)k Fv( )t Fw(,)h(giv)m(es)h(that) 711 3434 y Fv( )t Fw(\()p FA(x)875 3393 y Fr(0)915 3434 y Fw(\()p Fv(T)14 b Fw(\)\))21 b Fu(\000)i Fv( )t Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\)\))551 3580 y Fu(\024)83 b Fv(D)s( )t Fw(\()p Fv(x)p Fw(\))22 b Fu(\001)g Fw(\()p FA(x)1162 3539 y Fr(0)1202 3580 y Fw(\()p Fv(T)14 b Fw(\))21 b Fu(\000)i FA(x)1529 3539 y Fr(0)1569 3580 y Fw(\()p Fv(T)35 b Fu(\000)23 b Fv(h)p Fw(\)\))f Fu(\000)h Fv(D)s( )t Fw(\()p Fv(x)p Fw(\))f Fu(\001)g Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\))22 b Fu(\000)g FA(x)p Fw(\()p Fv(T)36 b Fu(\000)23 b Fv(h)p Fw(\)\))711 3757 y(+)p Fv(C)864 3647 y Fp(\020)923 3757 y Fu(j)p FA(x)1010 3716 y Fr(0)1050 3757 y Fw(\()p Fv(T)14 b Fw(\))21 b Fu(\000)i FA(x)1377 3716 y Fr(0)1417 3757 y Fw(\()p Fv(T)35 b Fu(\000)23 b Fv(h)p Fw(\))p Fu(j)1769 3716 y Fr(2)1830 3757 y Fw(+)f Fu(j)p FA(x)p Fw(\()p Fv(T)14 b Fw(\))22 b Fu(\000)h FA(x)p Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(h)p Fw(\))p Fu(j)2695 3716 y Fr(2)2734 3647 y Fp(\021)2794 3757 y Fv(:)0 3943 y Fw(By)33 b(the)g(previous)h(estimates)g(and)f(Lemmata)g (27)f(and)h(28,)49 4170 y Fv(M)39 b Fw(=)27 b Fv(E)380 4029 y Fp(\024)433 4034 y(Z)532 4060 y Fs(T)488 4260 y(T)10 b Fl(\000)p Fs(h)655 4170 y Fv(L)721 4059 y Fp(\020)781 4170 y FA(x)p Fv(;)894 4102 y FA(x)p 894 4147 60 4 v 904 4238 a Fv(\017)963 4170 y(;)17 b FA(v)r Fv(;)g(!)1177 4059 y Fp(\021)1235 4170 y Fv(dt)1321 4029 y Fp(\025)1401 4170 y Fu(\024)q Fv(E)1573 4029 y Fp(\024)1626 4034 y(Z)1725 4060 y Fs(T)1681 4260 y(T)10 b Fl(\000)p Fs(h)1848 4170 y Fv(L)1914 4059 y Fp(\020)1974 4170 y FA(x)2033 4128 y Fr(0)2073 4170 y Fv(;)2126 4102 y FA(x)2185 4066 y Fr(0)p 2126 4147 99 4 v 2156 4238 a Fv(\017)2235 4170 y(;)17 b Fw(0)p Fv(;)g(!)2437 4059 y Fp(\021)2495 4170 y Fv(dt)22 b Fw(+)g Fv( )t Fw(\()p FA(x)2865 4128 y Fr(0)2905 4170 y Fw(\()p Fv(T)14 b Fw(\)\))21 b Fu(\000)i Fv( )t Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\)\))3560 4029 y Fp(\025)1506 4395 y Fu(\024)28 b Fv(C)7 b Fw(\(1)22 b(+)g Fv(M)10 b Fw(\))p Fv(h)23 b Fw(+)f Fv(C)7 b(M)2395 4354 y Fr(1)p Fs(=)p Fr(2)2506 4395 y Fv(h)2562 4354 y Fr(1)p Fs(=)p Fr(2)2672 4395 y Fv(;)0 4555 y Fw(whic)m(h)34 b(implies)g(the)f(desired)h(b)s(ound.)2180 b Fi(\003)0 4725 y Fw(W)-8 b(e)33 b(will)g(also)g(need)g(the)g(follo)m(wing)g (estimate:)0 4901 y FA(Prop)s(osition)38 b(30.)j Fo(Fix)34 b Fv(r)d(>)c Fw(0)p Fo(.)45 b(Consider)34 b(the)g(optimal)h (di\013usion)935 5088 y Fv(d)p FA(x)28 b Fw(=)f Fv(f)1235 4977 y Fp(\020)1294 5088 y FA(x)p Fv(;)1407 5020 y FA(x)p 1407 5065 60 4 v 1417 5156 a Fv(\017)1476 5088 y(;)17 b FA(v)r Fv(;)g(!)1690 4977 y Fp(\021)1748 5088 y Fv(dt)22 b Fw(+)g Fv(\033)2013 4977 y Fp(\020)2073 5088 y FA(x)p Fv(;)2186 5020 y FA(x)p 2186 5065 V 2196 5156 a Fv(\017)2255 5088 y(;)17 b FA(v)r Fv(;)g(!)2469 4977 y Fp(\021)2527 5088 y Fv(dW)2670 5103 y Fs(t)2699 5088 y Fv(;)0 5284 y Fo(with)35 b FA(x)p Fw(\()p Fv(T)h Fu(\000)22 b Fv(h)p Fw(\))28 b(=)g Fv(x)p Fo(.)45 b(Then,)34 b(as)g Fv(h)28 b Fu(!)f Fw(0)1525 5248 y Fr(+)1584 5284 y Fo(,)35 b(we)f(have)1278 5445 y Fv(P)14 b Fw(\()p Fu(j)p FA(x)p Fw(\()p Fv(T)g Fw(\))21 b Fu(\000)h Fv(x)p Fu(j)28 b Fv(>)g(r)s Fw(\))f(=)h Fv(o)p Fw(\()p Fv(h)p Fw(\))p Fv(:)p eop end %%Page: 42 42 TeXDict begin 42 41 bop 168 251 a Fq(42)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y Fo(Pr)k(o)g(of.)41 b Fw(W)-8 b(rite)1774 567 y FA(x)28 b Fw(=)g FA(y)c Fw(+)e FA(z)p Fv(;)168 703 y Fw(where)927 827 y Fv(d)p FA(y)29 b Fw(=)f Fv(f)1229 716 y Fp(\020)1288 827 y FA(x)p Fv(;)1401 759 y FA(x)p 1401 804 60 4 v 1411 895 a Fv(\017)1470 827 y(;)17 b FA(v)r Fv(;)g(!)1684 716 y Fp(\021)1742 827 y Fv(dt;)211 b(d)p FA(z)28 b Fw(=)f Fv(\033)2357 716 y Fp(\020)2417 827 y FA(x)p Fv(;)2529 759 y FA(x)p 2529 804 V 2539 895 a Fv(\017)2598 827 y(;)17 b FA(v)r Fv(;)g(!)2812 716 y Fp(\021)2870 827 y Fv(dW)3013 842 y Fs(t)3043 827 y Fv(;)168 998 y Fw(and)32 b FA(y)r Fw(\()p Fv(T)k Fu(\000)23 b Fv(h)p Fw(\))k(=)h Fv(x)p Fw(,)33 b FA(z)p Fw(\()p Fv(T)j Fu(\000)22 b Fv(h)p Fw(\))28 b(=)f(0.)168 1115 y(W)-8 b(e)33 b(ha)m(v)m(e)1116 1233 y Fv(d)p 1098 1278 86 4 v 1098 1369 a(dt)1194 1301 y(E)6 b Fu(j)p FA(y)24 b Fu(\000)e Fv(x)p Fu(j)1565 1259 y Fr(2)1633 1301 y Fw(=)27 b(2)p Fv(E)6 b Fw(\()p FA(y)24 b Fu(\000)e Fv(x)p Fw(\))p Fv(f)2235 1190 y Fp(\020)2295 1301 y FA(x)p Fv(;)2408 1233 y FA(x)p 2408 1278 60 4 v 2418 1369 a Fv(\017)2477 1301 y(;)17 b FA(v)r Fv(;)g(!)2691 1190 y Fp(\021)1633 1514 y Fu(\024)28 b Fw(2)1804 1434 y Fp(\000)1849 1514 y Fv(E)6 b Fu(j)p FA(y)23 b Fu(\000)g Fv(x)p Fu(j)2220 1473 y Fr(2)2260 1434 y Fp(\001)2305 1456 y Fr(1)p Fs(=)p Fr(2)2432 1434 y Fp(\000)2478 1514 y Fv(E)6 b Fu(j)p Fv(f)11 b Fu(j)2671 1473 y Fr(2)2709 1434 y Fp(\001)2755 1456 y Fr(1)p Fs(=)p Fr(2)2881 1514 y Fv(;)168 1672 y Fw(whic)m(h)34 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Fl(\000)p Fs(h)2244 2539 y Fv(E)c Fu(j)p Fv(f)11 b Fu(j)2437 2497 y Fr(2)2476 2539 y Fv(dt)2562 2398 y Fp(\023)2635 2410 y Fr(1)p Fs(=)p Fr(2)1583 2851 y Fu(\024)28 b Fv(C)7 b(h)1821 2810 y Fr(1)p Fs(=)p Fr(2)1948 2711 y Fp(\022)2021 2851 y Fv(E)2116 2715 y Fp(Z)2216 2742 y Fs(T)2171 2941 y(T)j Fl(\000)p Fs(h)2339 2851 y Fv(L)22 b Fw(+)g Fv(C)7 b(dt)2688 2711 y Fp(\023)2761 2723 y Fr(1)p Fs(=)p Fr(2)2899 2851 y Fu(\024)28 b Fv(C)7 b(h;)168 3066 y Fw(due)33 b(to)f(Prop)s(osition)h(29,)f(and)h(so)1571 3223 y Fv(E)6 b Fu(j)p FA(y)r Fw(\()p Fv(t)p Fw(\))22 b Fu(\000)h Fv(x)p Fu(j)2054 3182 y Fr(2)2121 3223 y Fu(\024)28 b Fv(C)7 b(h)2359 3182 y Fr(2)2398 3223 y Fv(:)168 3381 y Fw(Accordingly)-8 b(,)34 b(b)m(y)f(Cheb)m(yc)m(hev's)k (inequalit)m(y)-8 b(,)34 b(w)m(e)g(ha)m(v)m(e)168 3596 y(\(92\))1063 b Fv(P)1482 3485 y Fp(\020)1541 3596 y Fu(j)p FA(y)r Fw(\()p Fv(T)14 b Fw(\))21 b Fu(\000)h Fv(x)p Fu(j)28 b Fv(>)2123 3528 y(r)p 2122 3573 49 4 v 2122 3664 a Fw(2)2180 3485 y Fp(\021)2268 3596 y Fu(\024)g Fv(C)2460 3528 y(h)2516 3492 y Fr(2)p 2460 3573 96 4 v 2465 3664 a Fv(r)2512 3635 y Fr(2)2565 3596 y Fv(:)168 3788 y Fw(T)-8 b(o)34 b(handle)h(the)g(second)g(term,)h(w)m(e)f(use)g (It^)-49 b(o's)35 b(form)m(ula)f(\(see,)i(e.g.,)f(page)g(105)e(in)i ([Ev)-5 b(a06b]\))34 b(and)168 3904 y(w)m(e)f(get)776 4042 y Fv(d)p 758 4086 86 4 v 758 4178 a(dt)854 4109 y(E)6 b(e)1032 4032 y Fm(\025)p Fj(j)p Fc(z)p Fj(j)1138 4011 y Fk(2)p 987 4050 230 3 v 987 4092 a Fm(t)p Fk(+2)p Fm(h)p Fj(\000)p Fm(T)386 4413 y Fw(=)27 b Fu(\000)p Fv(E)689 4345 y(\025)p Fu(j)p Fv(z)t Fu(j)851 4309 y Fr(2)890 4345 y Fv(e)990 4268 y Fm(\025)p Fj(j)p Fc(z)p Fj(j)1096 4247 y Fk(2)p 945 4286 V 945 4328 a Fm(t)p Fk(+2)p Fm(h)p Fj(\000)p Fm(T)p 655 4390 569 4 v 655 4481 a Fw(\()p Fv(t)22 b Fw(+)g(2)p Fv(h)g Fu(\000)h Fv(T)14 b Fw(\))1184 4452 y Fr(2)1255 4413 y Fw(+)1363 4345 y(1)p 1363 4390 49 4 v 1363 4481 a(2)1422 4413 y Fv(E)6 b(\033)t(\033)1618 4371 y Fs(T)1701 4413 y Fw(:)1755 4272 y Fp(\024)1966 4345 y Fw(2)p Fv(\025I)p 1818 4390 453 4 v 1818 4481 a(t)22 b Fw(+)g(2)p Fv(h)g Fu(\000)h Fv(T)2281 4413 y(e)2381 4335 y Fm(\025)p Fj(j)p Fc(z)p Fj(j)2487 4314 y Fk(2)p 2336 4354 230 3 v 2336 4395 a Fm(t)p Fk(+2)p Fm(h)p Fj(\000)p Fm(T)2601 4413 y Fw(+)2835 4345 y Fv(\025)2892 4309 y Fr(2)2931 4345 y Fv(z)k Fu(\012)c Fv(z)p 2709 4390 569 4 v 2709 4481 a Fw(\()p Fv(t)g Fw(+)f(2)p Fv(h)g Fu(\000)g Fv(T)14 b Fw(\))3238 4452 y Fr(2)3287 4413 y Fv(e)3387 4335 y Fm(\025)p Fj(j)p Fc(z)p Fj(j)3493 4314 y Fk(2)p 3343 4354 230 3 v 3343 4395 a Fm(t)p Fk(+2)p Fm(h)p Fj(\000)p Fm(T)3586 4272 y Fp(\025)386 4689 y Fu(\024)28 b Fv(E)6 b(\033)t(\033)687 4648 y Fs(T)770 4689 y Fw(:)825 4549 y Fp(\024)1060 4622 y Fv(\025I)p 887 4666 453 4 v 887 4757 a(t)22 b Fw(+)g(2)p Fv(h)h Fu(\000)f Fv(T)1350 4689 y(e)1450 4612 y Fm(\025)p Fj(j)p Fc(z)p Fj(j)1556 4591 y Fk(2)p 1405 4630 230 3 v 1405 4671 a Fm(t)p Fk(+2)p Fm(h)p Fj(\000)p Fm(T)1649 4549 y Fp(\025)386 4957 y Fu(\024)501 4889 y Fv(C)p 501 4934 77 4 v 511 5025 a(h)588 4957 y(E)6 b(e)766 4879 y Fm(\025)p Fj(j)p Fc(z)p Fj(j)872 4858 y Fk(2)p 721 4898 230 3 v 721 4939 a Fm(t)p Fk(+2)p Fm(h)p Fj(\000)p Fm(T)965 4957 y Fv(;)168 5149 y Fw(if)32 b Fv(\025)h Fw(is)g(su\016cien)m(tly)i (small)e(and)g Fv(t)28 b Fu(2)g Fw([)p Fv(T)36 b Fu(\000)22 b Fv(h;)17 b(T)d Fw(].)168 5265 y(Th)m(us,)34 b(recalling)f(that)f FA(z)p Fw(\()p Fv(T)k Fu(\000)23 b Fv(h)p Fw(\))28 b(=)f(0,)32 b(w)m(e)i(obtain)f(b)m(y)g(a)f(Gron)m(w)m(all)h(estimate)h(that)1694 5469 y Fv(E)6 b(e)1872 5392 y Fm(\025)p Fj(j)p Fc(z)p Fj(j)1978 5371 y Fk(2)p 1828 5411 V 1828 5452 a Fm(t)p Fk(+2)p Fm(h)p Fj(\000)p Fm(T)2099 5469 y Fu(\024)28 b Fv(C)r(;)p eop end %%Page: 43 43 TeXDict begin 43 42 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(43)0 450 y Fw(for)32 b(all)h Fv(T)i Fu(\000)23 b Fv(h)28 b Fu(\024)g Fv(t)g Fu(\024)g Fv(T)14 b Fw(.)43 b(Again)33 b(b)m(y)g(Cheb)m(yc)m(hev's)k(inequalit)m(y)-8 b(,)0 648 y(\(93\))1101 b Fv(P)1352 538 y Fp(\020)1411 648 y Fu(j)p FA(z)p Fw(\()p Fv(T)14 b Fw(\))p Fu(j)26 b(\025)1806 581 y Fv(r)p 1805 625 49 4 v 1805 716 a Fw(2)1864 538 y Fp(\021)1951 648 y Fu(\024)i Fv(C)7 b(e)2178 607 y Fl(\000)2243 580 y Fm(\025r)2311 559 y Fk(2)p 2244 592 102 3 v 2261 633 a(2)p Fm(h)2359 648 y Fv(;)0 849 y Fw(whic)m(h)47 b(is)f(exp)s(onen)m(tially)i(small)e(as)g Fv(h)k Fu(!)g Fw(0.)82 b(The)47 b(desired)g(result)g(then)f(follo)m(ws)h(from)e (\(92\))0 965 y(and)33 b(\(93\))o(.)3194 b Fi(\003)0 1141 y Fw(W)-8 b(e)40 b(will)h(need)g(a)f(\014nal)g(momen)m(t)h (estimate)g(for)f(the)g(optimal)g(measures)i(of)e(the)g(homogenized)0 1257 y(problem.)k(Suc)m(h)31 b(momen)m(t)g(estimate)h(will)f(only)g(b)s (e)f(in)h(use)g(in)g(the)g(pro)s(of)e(of)h(Prop)s(osition)g(33)g(and)0 1374 y(it)j(requires)h(the)f(follo)m(wing)g(additional)f(condition:)0 1553 y FA(Assumption)39 b(16.)i Fw(W)-8 b(e)34 b(assume)g(that)f(for)f (an)m(y)h(\014xed)h(\()p Fv(x)2186 1568 y Fr(0)2226 1553 y Fv(;)17 b($)2351 1568 y Fr(0)2390 1553 y Fv(;)g(\033)2489 1568 y Fr(0)2528 1553 y Fw(\))28 b Fu(2)h Ft(R)2761 1517 y Fs(n)2830 1553 y Fu(\002)23 b Ft(R)3002 1517 y Fs(n)3071 1553 y Fu(\002)g Fw(Mat\()p Fv(n)f Fu(\002)h Fv(n)p Fw(\))0 1669 y(there)39 b(exists)h(a)e(map)h Fv(x)f Fu(7!)f(S)1130 1684 y Fs(x)1170 1693 y Fk(0)1205 1684 y Fs(;$)1284 1693 y Fk(0)1318 1684 y Fs(;\033)1378 1693 y Fk(0)1416 1669 y Fw(\()p Fv(x)p Fw(\))h(=)g(\()p Fu(S)1804 1633 y Fs($)1797 1694 y(x)1837 1703 y Fk(0)1871 1694 y Fs(;$)1950 1703 y Fk(0)1984 1694 y Fs(;\033)2044 1703 y Fk(0)2083 1669 y Fw(\()p Fv(x)p Fw(\))p Fv(;)17 b Fu(S)2325 1633 y Fs(\033)2318 1694 y(x)2358 1703 y Fk(0)2392 1694 y Fs(;$)2471 1703 y Fk(0)2505 1694 y Fs(;\033)2565 1703 y Fk(0)2604 1669 y Fw(\()p Fv(x)p Fw(\)\))38 b Fu(2)g Ft(R)2987 1633 y Fs(n)3060 1669 y Fu(\002)26 b Fw(Mat\()p Fv(n)h Fu(\002)f Fv(n)p Fw(\))0 1785 y(with)33 b(the)g(prop)s(erties)h(listed)f(b)s(elo) m(w.)0 1917 y(Let)g(\004)240 1932 y Fr(0)307 1917 y Fw(=)420 1877 y Fr(1)p 420 1894 36 4 v 420 1951 a(2)466 1917 y Fv(\033)521 1932 y Fr(0)561 1917 y Fv(\033)620 1880 y Fs(T)616 1941 y Fr(0)675 1917 y Fw(,)f(and)h Fu(S)991 1880 y Fr(\004)984 1942 y Fs(x)1024 1951 y Fk(0)1059 1942 y Fs(;$)1138 1951 y Fk(0)1171 1942 y Fs(;)p Fr(\004)1238 1951 y Fk(0)1277 1917 y Fw(\()p Fv(x)p Fw(\))28 b(=)1549 1877 y Fr(1)p 1549 1894 V 1549 1951 a(2)1594 1917 y Fu(S)1661 1880 y Fs(\033)1654 1941 y(x)1694 1950 y Fk(0)1729 1941 y Fs(;$)1808 1950 y Fk(0)1842 1941 y Fs(;\033)1902 1950 y Fk(0)1941 1917 y Fw(\()p Fv(x)p Fw(\))2089 1836 y Fp(\000)2134 1917 y Fu(S)2201 1880 y Fs(\033)2194 1941 y(x)2234 1950 y Fk(0)2269 1941 y Fs(;$)2348 1950 y Fk(0)2382 1941 y Fs(;\033)2442 1950 y Fk(0)2480 1917 y Fw(\()p Fv(x)p Fw(\))2611 1836 y Fp(\001)2657 1858 y Fs(T)2712 1917 y Fw(.)44 b(Then)148 2062 y(\(1\))e Fu(S)375 2077 y Fs(x)415 2086 y Fk(0)449 2077 y Fs(;$)528 2086 y Fk(0)562 2077 y Fs(;\033)622 2086 y Fk(0)661 2062 y Fw(\()p Fv(x)754 2077 y Fr(0)793 2062 y Fw(\))28 b(=)g(\()p Fv($)1082 2077 y Fr(0)1121 2062 y Fv(;)17 b(\033)1220 2077 y Fr(0)1259 2062 y Fw(\);)148 2179 y(\(2\))42 b(the)33 b(mapping)g Fv(x)28 b Fu(7!)f(S)1154 2194 y Fs(x)1194 2203 y Fk(0)1229 2194 y Fs(;$)1308 2203 y Fk(0)1341 2194 y Fs(;\033)1401 2203 y Fk(0)1440 2179 y Fw(\()p Fv(x)p Fw(\))33 b(is)g(globally)g(b)s (ounded)g(and)g(globally)g(Lipsc)m(hitz;)148 2303 y(\(3\))42 b(the)33 b(mapping)g Fv(x)28 b Fu(7!)p 1094 2223 67 4 v 27 w Fv(L)p Fw(\()p Fu(S)1265 2267 y Fs($)1258 2327 y(x)1298 2336 y Fk(0)1333 2327 y Fs(;$)1412 2336 y Fk(0)1446 2327 y Fs(;\033)1506 2336 y Fk(0)1544 2303 y Fw(\()p Fv(x)p Fw(\))p Fv(;)17 b Fu(S)1786 2267 y Fr(\004)1779 2328 y Fs(x)1819 2337 y Fk(0)1854 2328 y Fs(;$)1933 2337 y Fk(0)1967 2328 y Fs(;)p Fr(\004)2034 2337 y Fk(0)2072 2303 y Fw(\()p Fv(x)p Fw(\))p Fv(;)g(x)p Fw(\))33 b(is)g(globally)g (Lipsc)m(hitz.)3584 2443 y Fi(J)0 2623 y Fw(Assumption)43 b(16)d(is)i(ob)m(viously)h(ful\014lled,)h(b)m(y)e(c)m(ho)s(osing)f Fu(S)2229 2638 y Fs(x)2269 2647 y Fk(0)2304 2638 y Fs(;$)2383 2647 y Fk(0)2417 2638 y Fs(;\033)2477 2647 y Fk(0)2515 2623 y Fw(\()p Fv(x)p Fw(\))h(to)e(b)s(e)i(iden)m(tically)h(equal)0 2747 y(to)33 b(\()p Fv($)239 2762 y Fr(0)278 2747 y Fv(;)17 b(\033)377 2762 y Fr(0)417 2747 y Fw(\),)34 b(when)p 771 2667 V 34 w Fv(L)g Fw(is)g(globally)g(Lipsc)m(hitz)i(in)d Fv(x)h Fw(and,)g(in)g(particular,)g(if)p 2860 2667 V 34 w Fv(L)g Fw(do)s(es)g(not)f(dep)s(end)0 2863 y(on)j Fv(x)g Fw(at)g(all)f({)h(this)g(is)h(the)f(case,)i(for)d(instance,)j (if)e Fv(L)g Fw(do)s(es)g(not)g(dep)s(end)h(on)f Fv(x)g Fw(\(though)g(it)g(ma)m(y)0 2979 y(dep)s(end)e(on)e Fv(X)k Fw(=)27 b Fv(x=\017)p Fw(\).)0 3158 y FA(Prop)s(osition)f(31.)35 b Fo(Supp)-5 b(ose)25 b(that)h Fv(T)42 b Fu(2)28 b Fw(\(0)p Fv(;)17 b Fw(1])25 b Fo(is)h(c)-5 b(onveniently)25 b(smal)5 b(l.)40 b(L)-5 b(et)27 b Fv(\045)f Fo(and)f Fv(\027)3153 3173 y Fs(T)3234 3158 y Fo(b)-5 b(e)26 b(optimal)0 3275 y(me)-5 b(asur)g(es)33 b(for)44 b Fw(\(72\))33 b Fo(under)h(the)g(c)-5 b(onstr)g(aint)33 b(in)41 b Fw(\(68\))33 b Fo(in)g(c)-5 b(ases)33 b(A)h(and)f(B,)g(or)h(optimal)f(me)-5 b(asur)g(es)0 3391 y(for)45 b Fw(\(73\))35 b Fo(under)f(the)h(c)-5 b(onstr)g(aint)35 b(in)42 b Fw(\(69\))34 b Fo(in)h(c)-5 b(ase)34 b(C,)g(with)h Fv(t)28 b Fw(=)g(0)p Fo(.)0 3507 y(L)-5 b(et)35 b(also)f(Assumption)h(16)g(hold.)44 b(Then,)0 3728 y Fw(\(94\))1362 3584 y Fp(\014)1362 3644 y(\014)1362 3703 y(\014)1362 3763 y(\014)1395 3593 y(Z)1495 3728 y Fw(\()p Fv(x)1588 3687 y Fl(0)1634 3728 y Fu(\000)22 b Fv(x)p Fw(\))p Fv(d\027)1925 3743 y Fs(T)1981 3584 y Fp(\014)1981 3644 y(\014)1981 3703 y(\014)1981 3763 y(\014)2041 3728 y Fu(\024)29 b Fv(C)7 b(T)h(;)0 4065 y Fw(\(95\))1395 3929 y Fp(Z)1511 4065 y Fu(j)p Fv(x)1594 4024 y Fl(0)1640 4065 y Fu(\000)22 b Fv(x)p Fu(j)1822 4024 y Fr(2)1862 4065 y Fv(d\045)28 b Fu(\024)g Fv(C)7 b(T)2244 4024 y Fr(2)0 4277 y Fo(and)0 4470 y Fw(\(96\))1377 4335 y Fp(Z)1494 4470 y Fu(j)p Fv(x)1577 4429 y Fl(0)1622 4470 y Fu(\000)23 b Fv(x)p Fu(j)1805 4429 y Fr(2)1844 4470 y Fv(d\027)1943 4485 y Fs(T)2026 4470 y Fu(\024)28 b Fv(C)7 b(T)h(:)0 4683 y Fo(F)-7 b(urthermor)i(e,)0 4904 y Fw(\(97\))1513 4759 y Fp(\014)1513 4819 y(\014)1513 4879 y(\014)1513 4939 y(\014)1546 4768 y(Z)p 1662 4824 V 1662 4904 a Fv(L)q(d\045)1830 4759 y Fp(\014)1830 4819 y(\014)1830 4879 y(\014)1830 4939 y(\014)1891 4904 y Fu(\024)28 b Fv(C)7 b(T)h(:)0 5139 y Fo(Pr)-5 b(o)g(of.)41 b Fw(First)33 b(of)f(all,)g(w)m(e)i(observ)m(e)g(that,)0 5304 y(\(98\))843 b Fu(j)p Fv($)s Fu(j)1157 5263 y Fr(2)1218 5304 y Fw(+)22 b Fu(j)p Fw(\004)p Fu(j)f Fw(+)h Fu(j)p Fw(\004)p Fu(j)1677 5263 y Fs(p)1744 5304 y Fu(\024)28 b Fv(C)7 b Fw(\(1)22 b(+)p 2133 5224 V 22 w Fv(L)p Fw(\()p Fv(x)2292 5263 y Fl(0)2316 5304 y Fv(;)17 b($)s(;)g Fw(\004\)\))0 5469 y(thanks)34 b(to)e(Prop)s(osition)h(15.)p eop end %%Page: 44 44 TeXDict begin 44 43 bop 168 251 a Fq(44)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y Fw(Also,)36 b(b)m(y)g(taking)f Fv(')p Fw(\()p Fv(x)1016 414 y Fl(0)1040 450 y Fv(;)17 b(s)p Fw(\))31 b(=)h Fv(x)1362 414 y Fl(0)1362 475 y Fs(j)1423 450 y Fu(\000)24 b Fv(x)1579 465 y Fs(j)1651 450 y Fw(in)36 b(the)f(holonom)m(y)h(constrain)m(t)g (\(68\))f(or)f(\(69\))h(\(according)168 571 y(to)d(the)h(case\),)g (with)h Fv(j)f Fw(=)28 b(1)p Fv(;)17 b(:)g(:)g(:)e(;)i(n)p Fw(,)33 b(w)m(e)h(obtain)e(the)h(iden)m(tit)m(y)168 798 y(\(99\))1514 662 y Fp(Z)1630 798 y Fv($)s(d\045)27 b Fw(=)1946 662 y Fp(Z)2062 798 y Fv(x)2117 756 y Fl(0)2163 798 y Fu(\000)c Fv(xd\027)2417 813 y Fs(T)2472 798 y Fv(:)168 1032 y Fw(By)33 b(taking)g Fv(')27 b Fw(=)827 992 y Fr(1)p 827 1009 36 4 v 827 1066 a(2)872 1032 y Fw(\()p Fv(T)36 b Fu(\000)22 b Fv(s)p Fw(\))p Fu(j)p Fv(x)1269 995 y Fl(0)1315 1032 y Fu(\000)g Fv(x)p Fu(j)1497 995 y Fr(2)1537 1032 y Fw(,)33 b(w)m(e)g(obtain)168 1268 y(\(100\))947 1132 y Fp(Z)1046 1268 y Fw(\()p Fv(T)j Fu(\000)23 b Fv(s)p Fw(\)\()p Fv($)h Fu(\001)e Fw(\()p Fv(x)1647 1227 y Fl(0)1692 1268 y Fu(\000)h Fv(x)p Fw(\))g(+)38 b(tr)17 b(\004\))22 b Fu(\000)2349 1201 y Fw(1)p 2349 1245 49 4 v 2349 1336 a(2)2408 1268 y Fu(j)p Fv(x)2491 1227 y Fl(0)2537 1268 y Fu(\000)g Fv(x)p Fu(j)2719 1227 y Fr(2)2759 1268 y Fv(d\045)28 b Fw(=)f(0)p Fv(:)168 1502 y Fw(Similarly)-8 b(,)33 b(tak)m(e)h Fv(')27 b Fw(=)1018 1463 y Fr(1)p 1018 1479 36 4 v 1018 1536 a(2)1063 1502 y Fu(j)p Fv(x)1146 1466 y Fl(0)1192 1502 y Fu(\000)22 b Fv(x)p Fu(j)1374 1466 y Fr(2)1446 1502 y Fw(to)32 b(get)168 1738 y(\(101\))1110 1603 y Fp(Z)1226 1738 y Fv($)24 b Fu(\001)e Fw(\()p Fv(x)1474 1697 y Fl(0)1520 1738 y Fu(\000)g Fv(x)p Fw(\))h(+)39 b(tr)16 b(\004)h Fv(d\045)27 b Fw(=)2256 1603 y Fp(Z)2372 1738 y Fu(j)p Fv(x)2455 1697 y Fl(0)2501 1738 y Fu(\000)22 b Fv(x)p Fu(j)2683 1697 y Fr(2)2723 1738 y Fv(d\027)2822 1753 y Fs(T)2877 1738 y Fv(:)168 1960 y Fw(By)37 b(taking)h Fv(')d Fw(=)g(1)i(and)g Fv(')e Fw(=)h Fv(T)j Fu(\000)25 b Fv(s)37 b Fw(w)m(e)i(also)e(see)h(that)f (the)g(total)g(masses)i(of)d Fv(\027)3239 1975 y Fs(T)3332 1960 y Fw(and)h Fv(\045)g Fw(are)g(1)168 2076 y(and)k Fv(T)14 b Fw(,)44 b(resp)s(ectiv)m(ely)-8 b(.)74 b(Th)m(us,)45 b(using)f(\(99\))o(,)g(the)e(H\177)-49 b(older)42 b(Inequalit)m(y)i (and)e(\(98\))o(,)i(w)m(e)f(conclude)168 2192 y(that)168 2426 y(\(102\))633 2282 y Fp(\014)633 2342 y(\014)633 2402 y(\014)633 2461 y(\014)666 2291 y(Z)766 2426 y Fw(\()p Fv(x)859 2385 y Fl(0)905 2426 y Fu(\000)22 b Fv(x)p Fw(\))p Fv(d\027)1196 2441 y Fs(T)1251 2282 y Fp(\014)1251 2342 y(\014)1251 2402 y(\014)1251 2461 y(\014)1312 2426 y Fu(\024)29 b Fv(C)7 b(T)1566 2385 y Fr(1)p Fs(=)p Fr(2)1692 2286 y Fp(\022)1765 2291 y(Z)1881 2426 y Fu(j)p Fv($)s Fu(j)2021 2385 y Fr(2)2059 2426 y Fv(d\045)2160 2286 y Fp(\023)2234 2308 y Fr(1)p Fs(=)p Fr(2)2371 2426 y Fu(\024)29 b Fv(C)7 b(T)35 b Fw(+)22 b Fv(C)7 b(T)2892 2385 y Fr(1)p Fs(=)p Fr(2)3019 2286 y Fp(\022)3092 2291 y(Z)p 3208 2346 67 4 v 3208 2426 a Fv(L)q(d\045)3376 2286 y Fp(\023)3449 2308 y Fr(1)p Fs(=)p Fr(2)3576 2426 y Fv(:)168 2662 y Fw(Also,)33 b(from)f(\(100\))g(and)h(a)f(scaled)i (H\177)-49 b(older)33 b(Inequalit)m(y)-8 b(,)259 2753 y Fp(Z)375 2889 y Fu(j)p Fv(x)458 2848 y Fl(0)504 2889 y Fu(\000)22 b Fv(x)p Fu(j)686 2848 y Fr(2)726 2889 y Fv(d\045)27 b Fu(\024)i Fv(C)7 b(T)1124 2753 y Fp(Z)1240 2889 y Fu(j)p Fv($)s Fu(jj)p Fv(x)1463 2848 y Fl(0)1507 2889 y Fu(\000)23 b Fv(x)p Fu(j)f Fw(+)g Fu(j)p Fw(\004)p Fu(j)p Fv(d\045)27 b Fu(\024)2174 2822 y Fw(1)p 2174 2866 49 4 v 2174 2957 a(2)2250 2753 y Fp(Z)2366 2889 y Fu(j)p Fv(x)2449 2848 y Fl(0)2495 2889 y Fu(\000)22 b Fv(x)p Fu(j)2677 2848 y Fr(2)2717 2889 y Fv(d\045)g Fw(+)g Fv(C)7 b(T)3102 2753 y Fp(Z)3218 2889 y Fu(j)p Fv($)s Fu(j)3358 2848 y Fr(2)3419 2889 y Fw(+)22 b Fu(j)p Fw(\004)p Fu(j)p Fv(d\045)168 3116 y Fw(and)32 b(so,)h(b)m(y)i(\(98\),) 168 3351 y(\(103\))702 3216 y Fp(Z)818 3351 y Fu(j)p Fv(x)901 3310 y Fl(0)947 3351 y Fu(\000)22 b Fv(x)p Fu(j)1129 3310 y Fr(2)1169 3351 y Fv(d\045)28 b Fu(\024)g Fv(C)7 b(T)1567 3211 y Fp(\022)1640 3216 y(Z)1757 3351 y Fu(j)p Fv($)s Fu(j)1897 3310 y Fr(2)1957 3351 y Fw(+)22 b Fu(j)p Fw(\004)p Fu(j)p Fv(d\045)2277 3211 y Fp(\023)2377 3351 y Fu(\024)28 b Fv(C)7 b(T)2647 3211 y Fp(\022)2720 3351 y Fv(T)36 b Fw(+)2911 3216 y Fp(Z)p 3027 3271 67 4 v 3027 3351 a Fv(Ld\045)3194 3211 y Fp(\023)3284 3351 y Fv(:)168 3587 y Fw(W)-8 b(e)23 b(no)m(w)g(consider)g(an)g(additional)f (small)h(parameter)g Fv(d)k(>)h Fw(0.)40 b(From)22 b(\(101\))o(,)i(com) m(bined)h(with)e(\(98\))o(,)438 3689 y Fp(Z)554 3825 y Fu(j)p Fv(x)637 3784 y Fl(0)683 3825 y Fu(\000)g Fv(x)p Fu(j)866 3784 y Fr(2)905 3825 y Fv(d\027)1004 3840 y Fs(T)1142 3825 y Fu(\024)84 b Fv(C)1396 3684 y Fp(\022)1470 3689 y(Z)1596 3757 y Fu(j)p Fv(x)1679 3721 y Fl(0)1724 3757 y Fu(\000)23 b Fv(x)p Fu(j)1907 3721 y Fr(2)p 1596 3802 351 4 v 1746 3893 a Fv(d)1979 3825 y Fw(+)f Fv(d)p Fu(j)p Fv($)s Fu(j)2268 3784 y Fr(2)2327 3825 y Fw(+)g Fv(d)p Fu(j)p Fw(\004)p Fu(j)2597 3784 y Fs(p)2658 3825 y Fw(+)g Fv(d)2807 3784 y Fl(\000)p Fr(1)p Fs(=)p Fr(\()p Fs(p)p Fl(\000)p Fr(1\))3117 3825 y Fv(d\045)3218 3684 y Fp(\023)1142 4101 y Fu(\024)84 b Fv(C)7 b(d)1431 4060 y Fl(\000)p Fr(1)p Fs(=)p Fr(\()p Fs(p)p Fl(\000)p Fr(1\))1740 4101 y Fv(T)36 b Fw(+)1941 4034 y Fv(C)p 1941 4078 77 4 v 1954 4170 a(d)2045 3966 y Fp(Z)2161 4101 y Fu(j)p Fv(x)2244 4060 y Fl(0)2290 4101 y Fu(\000)22 b Fv(x)p Fu(j)2472 4060 y Fr(2)2512 4101 y Fv(d\045)g Fw(+)g Fv(C)7 b(d)2878 3961 y Fp(\022)2950 4101 y Fv(T)36 b Fw(+)3141 3966 y Fp(Z)p 3258 4021 67 4 v 3258 4101 a Fv(Ld\045)3425 3961 y Fp(\023)3532 4101 y Fv(:)168 4337 y Fw(Hence,)e(exploiting)h (\(103\))o(,)168 4564 y(\(104\))1110 4428 y Fp(Z)1226 4564 y Fu(j)p Fv(x)1309 4523 y Fl(0)1355 4564 y Fu(\000)23 b Fv(x)p Fu(j)1538 4523 y Fr(2)1577 4564 y Fv(d\027)1676 4579 y Fs(T)1759 4564 y Fu(\024)28 b Fv(C)1934 4579 y Fs(d)1974 4564 y Fv(T)36 b Fw(+)22 b Fv(C)7 b Fw(\()p Fv(d)22 b Fw(+)g Fv(T)14 b Fw(\))2577 4428 y Fp(Z)p 2692 4484 V 2692 4564 a Fv(L)q(d\045)j(;)168 4785 y Fw(where)34 b Fv(C)520 4800 y Fs(d)587 4785 y Fv(>)28 b Fw(0)k(ma)m(y)i(dep)s(end)f (on)g Fv(d)p Fw(.)168 4901 y(T)-8 b(o)25 b(get)g(the)g(desired)h (estimates,)j(it)c(remains)h(to)e(estimate)2333 4821 y Fp(R)p 2416 4821 V 2416 4901 a Fv(L)q(d\045)p Fw(.)41 b(Fixed)25 b Fv(x)j Fu(2)g Ft(R)3165 4865 y Fs(n)3212 4901 y Fw(,)f(w)m(e)f(tak)m(e)g Fv($)3687 4916 y Fr(0)3753 4901 y Fw(=)168 5022 y Fv($)249 5037 y Fr(0)288 5022 y Fw(\()p Fv(x)p Fw(\),)33 b Fv(\033)534 5037 y Fr(0)601 5022 y Fw(=)28 b Fv(\033)760 5037 y Fr(0)800 5022 y Fw(\()p Fv(x)p Fw(\))k(and)h(\004)1218 5037 y Fr(0)1285 5022 y Fw(=)28 b(\004)1454 5037 y Fr(0)1493 5022 y Fw(\()p Fv(x)p Fw(\))g(=)g Fv(\033)1811 5037 y Fr(0)1851 5022 y Fv(\033)1910 4985 y Fs(T)1906 5046 y Fr(0)1997 5022 y Fw(in)33 b(suc)m(h)h(a)e(w)m(a)m(y)i(that)951 5201 y(sup)961 5282 y Fs($)r(;)p Fr(\004)1114 5201 y Fu(\000)p Fv(D)1272 5216 y Fs(x)1317 5201 y Fv( )t Fw(\()p Fv(x)p Fw(\))22 b Fu(\001)g Fv($)i Fu(\000)f Fw(\004)28 b(:)g Fv(D)2024 5160 y Fr(2)2021 5225 y Fs(x)2064 5201 y Fv( )t Fw(\()p Fv(x)p Fw(\))23 b Fu(\000)p 2384 5121 V 22 w Fv(L)q Fw(\()p Fv(x;)17 b($)s(;)g Fw(\004\))1060 5430 y(=)27 b Fu(\000)p Fv(D)1321 5445 y Fs(x)1365 5430 y Fv( )t Fw(\()p Fv(x)p Fw(\))c Fu(\001)f Fv($)1717 5445 y Fr(0)1778 5430 y Fu(\000)g Fw(\004)1942 5445 y Fr(0)2010 5430 y Fw(:)27 b Fv(D)2148 5389 y Fr(2)2145 5455 y Fs(x)2189 5430 y Fv( )t Fw(\()p Fv(x)p Fw(\))c Fu(\000)p 2509 5350 V 22 w Fv(L)q Fw(\()p Fv(x;)17 b($)2794 5445 y Fr(0)2833 5430 y Fv(;)g Fw(\004)2942 5445 y Fr(0)2981 5430 y Fw(\))g Fv(:)p eop end %%Page: 45 45 TeXDict begin 45 44 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(45)0 455 y Fw(Indeed,)35 b(note)f(that,)g(since)p 1043 375 67 4 v 35 w Fv(L)g Fw(is)g(co)s(erciv)m(e,)i(b)m(y)g(\(98\))o(,)e (and)g(lo)m(w)m(er)h(semicon)m(tin)m(uous,)i(b)m(y)e(Prop)s(osi-)0 571 y(tion)25 b(14,)i(w)m(e)f(ha)m(v)m(e)h(that)e(suc)m(h)i Fv($)1197 586 y Fr(0)1262 571 y Fw(and)e(\004)1509 586 y Fr(0)1574 571 y Fw(exist,)j(although)e(they)g(ma)m(y)g(not)g(b)s(e)f (unique.)43 b(Consider)0 688 y(no)m(w)33 b(the)g(di\013usion)0 902 y(\(105\))732 b Fv(d)p FA(x)1065 917 y Fr(0)1132 902 y Fw(=)28 b Fu(S)1303 861 y Fs($)1296 926 y(x)1336 935 y Fk(0)1371 926 y Fs(;$)1450 935 y Fk(0)1483 926 y Fs(;\033)1543 935 y Fk(0)1582 902 y Fw(\()p FA(x)1679 917 y Fr(0)1719 902 y Fw(\))p Fv(dt)22 b Fw(+)g Fu(S)2030 861 y Fr(\004)2023 926 y Fs(x)2063 935 y Fk(0)2097 926 y Fs(;$)2176 935 y Fk(0)2210 926 y Fs(;)p Fr(\004)2277 935 y Fk(0)2315 902 y Fw(\()p FA(x)2412 917 y Fr(0)2452 902 y Fw(\))p Fv(dW)2633 917 y Fs(t)2679 902 y Fv(;)0 1120 y Fw(with)33 b FA(x)281 1135 y Fr(0)321 1120 y Fw(\(0\))27 b(=)h Fv(x)33 b Fw(and)f(let)h(the)g(measures)i Fv(d\045)1683 1135 y Fr(0)1722 1120 y Fw(\()p Fv(x)1815 1084 y Fl(0)1839 1120 y Fv(;)17 b(!)t(;)g Fw(\004)p Fv(;)g(s)p Fw(\))31 b(b)s(e)h(de\014ned)i(b)m(y)676 1252 y Fp(Z)731 1477 y Fn(R)779 1458 y Fm(n)821 1477 y Fl(\002)p Fn(R)924 1458 y Fm(n)967 1477 y Fl(\002)p Fr(Mat\()p Fs(n)p Fl(\002)p Fs(n)p Fr(\))1365 1387 y Fv(\036)p Fw(\()p Fv(x)1516 1346 y Fl(0)1540 1387 y Fv(;)17 b($)s(;)g Fw(\004)p Fv(;)g(s)p Fw(\))p Fv(d\045)2006 1402 y Fr(0)687 1686 y Fw(:=)27 b Fv(E)912 1550 y Fp(Z)1012 1577 y Fs(T)967 1776 y Fr(0)1083 1686 y Fv(\036)p Fw(\()p FA(x)1238 1701 y Fr(0)1277 1686 y Fw(\()p Fv(s)p Fw(\))p Fv(;)17 b Fu(S)1510 1645 y Fs($)1503 1711 y(x)1543 1720 y Fk(0)1578 1711 y Fs(;$)1657 1720 y Fk(0)1690 1711 y Fs(;\033)1750 1720 y Fk(0)1789 1686 y Fw(\()p FA(x)1886 1701 y Fr(0)1926 1686 y Fw(\()p Fv(s)p Fw(\)\))p Fv(;)g Fu(S)2197 1645 y Fr(\004)2190 1711 y Fs(x)2230 1720 y Fk(0)2264 1711 y Fs(;$)2343 1720 y Fk(0)2377 1711 y Fs(;)p Fr(\004)2444 1720 y Fk(0)2482 1686 y Fw(\()p FA(x)2579 1701 y Fr(0)2618 1686 y Fw(\()p Fv(s)p Fw(\)\))p Fv(;)g(s)p Fw(\))p Fv(ds)0 1957 y Fw(and)33 b Fv(\027)238 1972 y Fr(0)p Fs(;T)348 1957 y Fw(\()p Fv(x)441 1921 y Fl(0)464 1957 y Fw(\))g(b)m(y)1132 2050 y Fp(Z)1188 2276 y Fn(R)1236 2257 y Fm(n)1299 2186 y Fv(')p Fw(\()p Fv(x)1456 2145 y Fl(0)1479 2186 y Fw(\))p Fv(d\027)1616 2201 y Fr(0)p Fs(;T)1726 2186 y Fw(\()p Fv(x)1819 2145 y Fl(0)1843 2186 y Fw(\))27 b(:=)h Fv(E)6 b(')p Fw(\()p FA(x)2278 2201 y Fr(0)2318 2186 y Fw(\()p Fv(T)14 b Fw(\)\))p Fv(:)0 2431 y Fw(Then,)34 b(b)m(y)f(Dynkin's)h(F)-8 b(orm)m(ula)33 b(and)f(the)h(minimalit)m(y)i(of)d Fv(\045)h Fw(and)f Fv(\027)2494 2446 y Fs(T)2549 2431 y Fw(,)0 2692 y(\(106\))971 2557 y Fp(Z)p 1087 2612 V 1087 2692 a Fv(L)q(d\045)22 b Fw(+)1375 2557 y Fp(Z)1491 2692 y Fv( )t(d\027)1657 2707 y Fs(T)1740 2692 y Fu(\024)1845 2557 y Fp(Z)p 1961 2612 V 1961 2692 a Fv(Ld\045)2128 2707 y Fr(0)2190 2692 y Fw(+)2288 2557 y Fp(Z)2404 2692 y Fv( )t(d\027)2570 2707 y Fr(0)p Fs(;T)2680 2692 y Fv(:)0 2954 y Fw(By)33 b(Lemma)g(27,)f(w)m(e)i(ha)m(v)m(e)696 3079 y Fp(\014)696 3139 y(\014)696 3199 y(\014)696 3259 y(\014)729 3088 y(Z)829 3224 y Fw(\()p Fv(x)922 3183 y Fl(0)968 3224 y Fu(\000)22 b Fv(x)p Fw(\))p Fv(d\027)1259 3239 y Fr(0)p Fs(;T)1370 3079 y Fp(\014)1370 3139 y(\014)1370 3199 y(\014)1370 3259 y(\014)1430 3224 y Fu(\024)29 b Fv(C)7 b(T)1895 3088 y Fp(Z)2011 3224 y Fu(j)p Fw(\()p Fv(x)2132 3183 y Fl(0)2178 3224 y Fu(\000)22 b Fv(x)p Fw(\))p Fu(j)2398 3172 y Fr(2)2454 3224 y Fv(d\027)2553 3239 y Fr(0)p Fs(;T)2691 3224 y Fu(\024)28 b Fv(C)7 b(T)h(:)0 3494 y Fw(Th)m(us)467 3600 y Fp(\014)467 3660 y(\014)467 3720 y(\014)467 3780 y(\014)500 3609 y(Z)616 3745 y Fv( )t Fw(\()p Fv(x)776 3704 y Fl(0)800 3745 y Fw(\))22 b Fu(\000)h Fv( )t Fw(\()p Fv(x)p Fw(\))17 b Fv(d\027)1274 3760 y Fr(0)p Fs(;T)1383 3745 y Fw(\()p Fv(x)1476 3704 y Fl(0)1500 3745 y Fw(\))1538 3600 y Fp(\014)1538 3660 y(\014)1538 3720 y(\014)1538 3780 y(\014)673 4021 y Fu(\024)45 b(j)p Fv(D)s( )t Fw(\()p Fv(x)p Fw(\))p Fu(j)1165 3877 y Fp(\014)1165 3937 y(\014)1165 3996 y(\014)1165 4056 y(\014)1198 3886 y(Z)1298 4021 y Fw(\()p Fv(x)1391 3980 y Fl(0)1437 4021 y Fu(\000)23 b Fv(x)p Fw(\))p Fv(d\027)1729 4036 y Fr(0)p Fs(;T)1839 3877 y Fp(\014)1839 3937 y(\014)1839 3996 y(\014)1839 4056 y(\014)1894 4021 y Fw(+)f Fv(C)2086 3886 y Fp(Z)2202 4021 y Fu(j)p Fw(\()p Fv(x)2323 3980 y Fl(0)2368 4021 y Fu(\000)h Fv(x)p Fw(\))p 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Fv(x)p Fw(\))p Fu(j)2635 4798 y Fr(2)2691 4849 y Fv(d\027)2790 4864 y Fs(T)790 5149 y Fu(\024)29 b Fv(C)7 b(T)35 b Fw(+)22 b Fv(C)7 b(T)1311 5108 y Fr(1)p Fs(=)p Fr(2)1438 5009 y Fp(\022)1511 5014 y(Z)p 1627 5069 V 1627 5149 a Fv(L)q(d\045)1795 5009 y Fp(\023)1868 5031 y Fr(1)p Fs(=)p Fr(2)2000 5149 y Fw(+)22 b Fv(C)2168 5164 y Fs(d)2208 5149 y Fv(T)36 b Fw(+)22 b Fv(C)7 b Fw(\()p Fv(d)22 b Fw(+)g Fv(T)14 b Fw(\))2811 5014 y Fp(Z)p 2926 5069 V 2926 5149 a Fv(L)q(d\045)790 5417 y Fu(\024)29 b Fv(C)966 5432 y Fs(d)1006 5417 y Fv(T)36 b Fw(+)22 b Fv(C)7 b Fw(\()p Fv(d)21 b Fw(+)h Fv(T)14 b Fw(\))1608 5281 y Fp(Z)p 1724 5337 V 1724 5417 a Fv(Ld\045)j(:)0 4990 y Fw(\(108\))p eop end %%Page: 46 46 TeXDict begin 46 45 bop 168 251 a Fq(46)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y Fw(Therefore,)33 b(putting)g(together)g(\(106\))o(,)g(\(107\))f(and)g (\(108\),)824 540 y Fp(Z)p 941 596 67 4 v 941 676 a Fv(Ld\045)c Fu(\024)1241 531 y Fp(\014)1241 591 y(\014)1241 651 y(\014)1241 711 y(\014)1274 540 y(Z)1390 676 y Fv( )t Fw(\()p Fv(x)1550 635 y Fl(0)1574 676 y Fw(\))22 b Fu(\000)h Fv( )t Fw(\()p Fv(x)p Fw(\))p Fv(d\027)2031 691 y Fr(0)p Fs(;T)2141 531 y Fp(\014)2141 591 y(\014)2141 651 y(\014)2141 711 y(\014)2196 676 y Fw(+)2294 531 y Fp(\014)2294 591 y(\014)2294 651 y(\014)2294 711 y(\014)2327 540 y(Z)2444 676 y Fv( )t Fw(\()p Fv(x)p Fw(\))f Fu(\000)h Fv( )t Fw(\()p Fv(x)2924 635 y Fl(0)2947 676 y Fw(\))p Fv(d\027)3084 691 y Fs(T)3139 531 y Fp(\014)3139 591 y(\014)3139 651 y(\014)3139 711 y(\014)1325 944 y Fw(+)1423 808 y Fp(Z)p 1540 863 V 1540 944 a Fv(Ld\045)1707 959 y Fr(0)1136 1202 y Fu(\024)28 b Fv(C)1311 1217 y Fs(d)1351 1202 y Fv(T)36 b Fw(+)22 b Fv(C)7 b Fw(\()p Fv(d)22 b Fw(+)g Fv(T)14 b Fw(\))1954 1067 y Fp(Z)p 2069 1122 V 2069 1202 a Fv(L)q(d\045:)168 1419 y Fw(This)25 b(yields)h(the)f(estimate)h(in)e(\(97\).)41 b(Then,)27 b(com)m(bining)e(together)g(\(97\))o(,)i(\(102\))o(,)f (\(103\))d(and)i(\(104\))o(,)168 1535 y(w)m(e)33 b(obtain)g(\(94\))o(,) g(\(95\))f(and)g(\(96\).)2308 b Fi(\003)1408 1754 y Fw(12.)48 b Fx(Viscosity)38 b(Solutions)168 1929 y Fw(This)33 b(last)f(section)h (is)f(dedicated)h(to)e(the)i(pro)s(of)e(of)g(the)h(main)g(result)h(of)e (the)h(pap)s(er,)g(that)g(is)g(that)168 2045 y(the)d(homogenized)g (limit)g(is)g(a)g(viscosit)m(y)i(solution)e(to)f(an)g(E\013ectiv)m(e)j (Hamilton-Jacobi)d(equation.)168 2224 y FA(Prop)s(osition)36 b(32.)41 b Fo(Ther)-5 b(e)33 b(exists)h(a)f(function)h(c)-5 b(ontinuous)33 b(function)p 2839 2144 89 4 v 34 w Fv(H)i Fw(:)28 b(Mat\()p Fv(n)20 b Fu(\002)g Fv(n)p Fw(\))g Fu(\002)h Ft(R)3685 2187 y Fs(n)3752 2224 y Fu(\002)168 2340 y Ft(R)240 2304 y Fs(n)314 2340 y Fu(!)28 b Ft(R)34 b Fo(such)h(that)41 b Fw(\026)-55 b Fv(u)35 b Fo(is)f(a)h(visc)-5 b(osity)35 b(solution)f(of)168 2504 y Fw(\(109\))1012 b Fu(\000)29 b Fw(\026)-55 b Fv(u)1559 2519 y Fs(t)1610 2504 y Fw(+)p 1708 2424 V 22 w Fv(H)1814 2423 y Fp(\000)1859 2504 y Fv(D)1943 2463 y Fr(2)1940 2529 y Fs(x)1990 2504 y Fw(\026)g Fv(u;)17 b(D)2165 2519 y Fs(x)2214 2504 y Fw(\026)-55 b Fv(u;)17 b(x)2363 2423 y Fp(\001)2436 2504 y Fw(=)28 b(0)p Fv(:)168 2684 y Fo(Pr)-5 b(o)g(of.)41 b Fw(T)-8 b(o)22 b(pro)m(v)m(e)h(that)k(\026)-55 b Fv(u)21 b Fw(is)i(a)e(viscosit)m(y)j(solution,)h(w)m(e)d(are)g(going)g(to)f (use)i(the)f(results)h(from)e([Bit01].)168 2801 y(T)-8 b(o)32 b(do)h(so,)g(w)m(e)g(\014rst)g(de\014ne)h(an)f(op)s(erator:)168 3012 y(\(110\))874 b Fu(T)1319 3027 y Fs(t)1344 3036 y Fk(1)1379 3027 y Fs(;t)1424 3036 y Fk(2)1463 3012 y Fv( )t Fw(\()p Fv(x)p Fw(\))28 b(:=)g(inf)1820 3072 y Fs(\045;\027)1911 3080 y Fm(t)1955 2877 y Fp(Z)p 2071 2932 67 4 v 2071 3012 a Fv(Ld\045)22 b Fw(+)2358 2877 y Fp(Z)2475 3012 y Fv( )t(d\027)2641 3027 y Fs(t)2666 3036 y Fk(2)2705 3012 y Fv(;)168 3229 y Fw(where)33 b(the)e(in\014m)m (um)j(is)e(tak)m(en)g(o)m(v)m(er)h(all)e(measures)j Fv(\045)d Fw(on)h Ft(R)2406 3193 y Fs(n)2473 3229 y Fu(\002)20 b Ft(R)2642 3193 y Fs(n)2709 3229 y Fu(\002)h Fw([)p Fv(t)2869 3244 y Fr(1)2909 3229 y Fv(;)c(t)2988 3244 y Fr(2)3027 3229 y Fw(])32 b(\(or)f(on)g Ft(R)3448 3193 y Fs(n)3515 3229 y Fu(\002)21 b Ft(R)3685 3193 y Fs(n)3752 3229 y Fu(\002)168 3345 y Fw(Mat\()p Fv(n)j Fu(\002)g Fv(n)p Fw(\))g Fu(\002)g Fw([)p Fv(t)848 3360 y Fr(1)887 3345 y Fv(;)17 b(t)966 3360 y Fr(2)1006 3345 y Fw(]\),)35 b(and)g Fv(\027)1373 3360 y Fs(t)1438 3345 y Fw(on)f Ft(R)1647 3309 y Fs(n)1729 3345 y Fw(that)g(satisfy)k(\(68\))c(\(or)g (\(69\))o(,)i(according)f(to)f(case)i(A,)f(B)168 3462 y(or)d(C\))h(with)g Fv(T)46 b Fw(replaced)34 b(b)m(y)g Fv(t)1309 3477 y Fr(2)1381 3462 y Fw(and)e Fv(t)h Fw(replaced)h(b)m(y)f Fv(t)2193 3477 y Fr(1)2233 3462 y Fw(,)g(and)f Fv(\027)2530 3477 y Fs(t)2555 3486 y Fk(1)2622 3462 y Fw(=)c Fv(\016)t Fw(\()p Fv(x)p Fw(\).)168 3578 y(W)-8 b(e)33 b(also)f(set)168 3742 y(\(111\))1378 b Fu(T)1823 3757 y Fs(t)1898 3742 y Fw(:=)44 b Fu(T)2099 3757 y Fr(0)p Fs(;t)2200 3742 y Fv(:)168 3907 y Fw(Let)32 b Fv(Y)54 b Fw(b)s(e)33 b(the)g(set)g(of)f (all)h Fv( )f Fu(2)c Fv(C)1419 3870 y Fr(2)1458 3907 y Fw(\()p Ft(R)1568 3870 y Fs(n)1615 3907 y Fw(\))k(suc)m(h)i(that)168 4102 y(\(112\))789 b(sup)1194 4182 y Fs(\017>)p Fr(0)1344 4102 y Fw(sup)1365 4182 y Fs(x;!)1507 3988 y Fp(\014)1507 4048 y(\014)1507 4107 y(\014)1541 4102 y Fv(H)1630 3992 y Fp(\020)1689 4102 y Fv(D)1773 4061 y Fr(2)1812 4102 y Fv( )t(;)17 b(D)s( )t(;)g(x;)2226 4035 y(x)p 2226 4080 56 4 v 2234 4171 a(\017)2291 4102 y(;)g(\017;)g(!)2483 3992 y Fp(\021)2542 3988 y(\014)2542 4048 y(\014)2542 4107 y(\014)2603 4102 y Fv(<)27 b Fu(1)p Fv(:)168 4336 y Fw(Then,)34 b(b)m(y)h(\(72\))d(and)h(Prop)s(ositions)h(20,)f(24)f (and)h(26,)g(w)m(e)h(ha)m(v)m(e)g(that)f Fu(T)2818 4351 y Fs(t)2880 4336 y Fw(maps)h Fv(Y)54 b Fw(in)m(to)33 b(Lipsc)m(hitz)168 4452 y(functions,)46 b(for)c(all)h Fv(t)i Fu(\025)g Fw(0.)74 b(De\014ne)43 b Fv(X)53 b Fw(:=)45 b Fu([)1945 4467 y Fs(t)p Fl(\025)p Fr(0)2065 4452 y Fu(T)2119 4467 y Fs(t)2149 4452 y Fv(Y)21 b Fw(.)74 b(If)42 b(w)m(e)i(pro)m(v)m(e)g(that)e Fu(T)3137 4467 y Fs(t)3162 4476 y Fk(1)3198 4467 y Fs(;t)3243 4476 y Fk(2)3324 4452 y Fw(satis\014es)j(the)168 4568 y(semigroup)e(prop)s(ert)m(y:)64 b Fu(T)1154 4583 y Fs(t)1179 4592 y Fk(1)1214 4583 y Fs(;t)1259 4592 y Fk(2)1327 4568 y Fu(\016)28 b(T)1459 4583 y Fs(t)1484 4592 y Fk(2)1520 4583 y Fs(;t)1565 4592 y Fk(3)1648 4568 y Fw(=)44 b Fu(T)1822 4583 y Fs(t)1847 4592 y Fk(1)1882 4583 y Fs(;t)1927 4592 y Fk(3)1966 4568 y Fw(,)h(for)d(all)g(0)i Fu(\024)h Fv(t)2592 4583 y Fr(1)2676 4568 y Fv(<)f(t)2831 4583 y Fr(2)2915 4568 y Fv(<)g(t)3070 4583 y Fr(3)3110 4568 y Fw(,)h(it)d(is)h(then)g(clear)168 4684 y(that)36 b Fv(X)45 b Fw(is)38 b(an)e(in)m(v)-5 b(arian)m(t)38 b(set)g(for)e Fu(T)1529 4699 y Fs(t)1559 4684 y Fw(.)56 b(Note)37 b(that)g Fv(Y)58 b Fw(con)m(tains)38 b Fv(C)2677 4648 y Fl(1)2670 4709 y Fs(c)2752 4684 y Fw(\()p Ft(R)2862 4648 y Fs(n)2909 4684 y Fw(\),)g(thanks)g(to)e (\(8\),)i(th)m(us)168 4801 y(\(H1\))32 b(in)h([Bit01])f(holds.)168 4917 y(T)-8 b(o)35 b(address)i(the)g(semigroup)f(prop)s(ert)m(y)-8 b(,)37 b(it)f(su\016ces)i(to)d(observ)m(e)j(that)d(for)g Fv(\017)f(>)e Fw(0)k(the)g(ev)m(olution)168 5033 y(asso)s(ciated)c(to)f (the)h(Hamilton-Jacobi)g(equation)g Fu(T)2127 4997 y Fs(\017)2102 5058 y(t)2127 5067 y Fk(1)2162 5058 y Fs(;t)2207 5067 y Fk(2)2277 5033 y Fw(is)g(a)g(semigroup,)g(whic)m(h)h(con)m(v)m (erges)h(uni-)168 5149 y(formly)-8 b(,)33 b(and)f(therefore)i(the)f (limit)g(also)f(satis\014es)j(the)e(semigroup)g(prop)s(ert)m(y)-8 b(.)45 b(In)33 b(further)g(detail:)168 5265 y(one)g(considers)1302 5417 y Fu(T)1382 5376 y Fs(\017)1356 5441 y(t)1381 5450 y Fk(1)1416 5441 y Fs(;t)1461 5450 y Fk(2)1500 5417 y Fv( )f Fw(:=)27 b(inf)1860 5281 y Fp(Z)1976 5417 y Fv(Ld\026)2152 5376 y Fs(\017)2207 5417 y Fw(+)2305 5281 y Fp(Z)2421 5417 y Fv( )t(d\027)2593 5376 y Fs(\017)2587 5441 y(t)2612 5450 y Fk(2)2668 5417 y Fv(;)p eop end %%Page: 47 47 TeXDict begin 47 46 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(47)0 450 y Fw(where)43 b(the)f(in\014m)m(um)h(is)f(tak)m(en)g(on)g (the)f(measures)j Fv(\032)d Fw(on)h Ft(R)2297 414 y Fs(n)2372 450 y Fu(\002)29 b Ft(R)2550 414 y Fs(n)2625 450 y Fu(\002)g Fv(U)38 b Fu(\002)29 b Fw([)p Fv(t)3003 465 y Fr(1)3043 450 y Fv(;)17 b(t)3122 465 y Fr(2)3161 450 y Fw(])42 b(with)g(total)0 566 y(mass)e Fv(t)281 581 y Fr(2)347 566 y Fu(\000)27 b Fv(t)486 581 y Fr(1)564 566 y Fw(and)39 b Fv(\027)808 581 y Fs(t)877 566 y Fw(probabilit)m(y)h(measures)g(on)f Ft(R)2020 530 y Fs(n)2106 566 y Fw(satisfying)i(\(30\).)62 b(From)38 b(Prop)s(osition)i(9)0 683 y(and)33 b(\(5\))o(,)593 868 y Fu(T)673 827 y Fs(\017)647 892 y(t)672 901 y Fk(1)708 892 y Fs(;t)753 901 y Fk(2)791 868 y Fv( )f Fw(=)c(inf)23 b Fv(E)1219 732 y Fp(Z)1318 758 y Fs(t)1343 767 y Fk(2)1274 958 y Fs(t)1299 967 y Fk(1)1399 868 y Fv(L)1482 727 y Fp(\022)1556 868 y FA(x)p Fw(\()p Fv(s)p Fw(\))p Fv(;)1790 800 y FA(x)p Fw(\()p Fv(s)p Fw(\))p 1790 845 181 4 v 1861 936 a Fv(\017)1981 868 y(;)17 b FA(v)r Fw(\()p Fv(s)p Fw(\))p Fv(;)g(!)2317 727 y Fp(\023)2422 868 y Fv(ds)22 b Fw(+)g Fv( )e Fw(\()p FA(x)p Fw(\()p Fv(t)2892 883 y Fr(2)2932 868 y Fw(\)\))33 b Fv(;)0 1072 y Fw(where)k(the)g(in\014m)m (um)g(ab)s(o)m(v)m(e)g(is)g(tak)m(en)g(o)m(v)m(er)g(all)f(tra)5 b(jectories)37 b(of)e(the)i(con)m(trolled)g(dynamics)h(\(3\))0 1188 y(with)h FA(x)p Fw(\()p Fv(t)360 1203 y Fr(1)400 1188 y Fw(\))e(=)g Fv(x)p Fw(.)61 b(Then,)40 b Fu(T)1099 1152 y Fs(\017)1073 1213 y(t)1098 1222 y Fk(1)1133 1213 y Fs(;t)1178 1222 y Fk(2)1255 1188 y Fw(is)f(a)f(semigroup,)j(b)m(y)e (the)g(optimalit)m(y)g(of)f(the)h(latter)f(dynamic)0 1304 y(programming)33 b(principle,)h(thence)g(so)e(is)h Fu(T)1610 1319 y Fs(t)1635 1328 y Fk(1)1671 1319 y Fs(;t)1716 1328 y Fk(2)1755 1304 y Fw(,)f(due)i(to)e(Prop)s(ositions)h(24)f(and)h (26,)f(as)h(desired.)0 1421 y(Let)23 b(no)m(w)g(the)g(translation)g(op) s(erator)e Fv(\034)34 b Fw(b)s(e)23 b(de\014ned)h(b)m(y)f Fv(\034)2074 1436 y Fs(y)2116 1421 y Fv( )t Fw(\()p Fv(x)p Fw(\))28 b(=)g Fv( )t Fw(\()p Fv(x)r Fw(+)r Fv(y)t Fw(\),)23 b(for)f(an)m(y)h Fv( )32 b Fw(:)27 b Ft(R)3360 1385 y Fs(n)3435 1421 y Fu(!)g Ft(R)p Fw(.)0 1537 y(The)38 b(h)m(yp)s(othesis) h(\(H2\))d(in)h([Bit01])g(requires)h(that)f(for)f(all)h Fv(f)45 b Fu(2)35 b Fv(X)45 b Fw(and)37 b(all)f Fv(y)i Fu(2)d Ft(R)3156 1501 y Fs(n)3203 1537 y Fw(,)i Fv(\034)3309 1552 y Fs(y)3351 1537 y Fv(f)45 b Fu(2)36 b Fv(X)8 b Fw(.)0 1653 y(Ho)m(w)m(ev)m(er,)33 b(this)d(is)h(not)f(really)g (necessary)-8 b(,)33 b(and)d(w)m(e)h(ma)m(y)g(replace)g(it)f(b)m(y)g (requiring)h(for)f(all)g Fv(y)g Fu(2)e Ft(R)3614 1617 y Fs(n)0 1769 y Fw(and)33 b Fv( )e Fu(2)d Fv(Y)22 b Fw(,)32 b Fv(\034)558 1784 y Fs(y)600 1769 y Fv( )g Fu(2)c Fv(Y)21 b Fw(.)44 b(This)33 b(condition)g(is)h(then)f(v)m(eri\014ed,)h(thanks)g (to)e(\(8\).)0 1886 y(Hyp)s(othesis)i(\(H3\))f(of)f([Bit01)o(])h(is)g (ob)m(viously)i(ful\014lled)e(here)h(to)s(o.)0 2002 y(Moreo)m(v)m(er,) 55 b(clearly)-8 b(,)50 b Fu(T)877 2017 y Fs(t)902 2026 y Fk(1)937 2017 y Fs(;t)982 2026 y Fk(2)1070 2002 y Fw(is)f(monotone,) 54 b(that)49 b(is,)k(if)2181 1976 y(\026)2161 2002 y Fv( )2224 2017 y Fr(1)2320 2002 y Fu(\024)2473 1976 y Fw(\026)2453 2002 y Fv( )2516 2017 y Fr(2)2605 2002 y Fw(then)c Fu(T)2897 2017 y Fs(t)2922 2026 y Fk(1)2958 2017 y Fs(;t)3003 2026 y Fk(2)3062 1976 y Fw(\026)3042 2002 y Fv( )3105 2017 y Fr(1)3200 2002 y Fu(\024)56 b(T)3387 2017 y Fs(t)3412 2026 y Fk(1)3448 2017 y Fs(;t)3493 2026 y Fk(2)3551 1976 y Fw(\026)3531 2002 y Fv( )3594 2017 y Fr(2)3634 2002 y Fw(.)0 2118 y(Therefore,)50 b(to)c(establish)h(the)f (result)h(it)e(su\016ces)j(to)d(c)m(hec)m(k)j(that)e(the)g(con)m(tin)m (uit)m(y)i(and)e(regu-)0 2234 y(larit)m(y)32 b(h)m(yp)s(othesis)i(I-IV) d(that)g(w)m(e)i(sp)s(ell)f(next.)44 b(W)-8 b(e)32 b(will)g(need)h(the) e(follo)m(wing)h(notation:)43 b(for)31 b(an)m(y)0 2351 y(sequence)k Fv(d)28 b Fw(=)f(\()p Fv(d)675 2366 y Fs(k)717 2351 y Fw(\))33 b(of)f(p)s(ositiv)m(e)i(reals)f(w)m(e)h(de\014ne)894 2509 y Fv(Q)971 2524 y Fs(d)1039 2509 y Fw(=)28 b Fu(f)p Fv(\021)j Fu(2)d Fv(C)1443 2468 y Fl(1)1436 2534 y Fs(c)1518 2509 y Fw(\()p Ft(R)1628 2468 y Fs(n)1675 2509 y Fw(\))p Fv(;)17 b Fu(k)p Fv(D)1891 2468 y Fs(\013)1939 2509 y Fv(\021)t Fu(k)2041 2524 y Fl(1)2143 2509 y Fu(\024)28 b Fv(d)2299 2524 y Fs(k)2342 2509 y Fv(;)17 b Fu(j)p Fv(\013)q Fu(j)26 b(\024)i Fv(k)s Fu(g)p Fv(:)238 2668 y Fw(I)42 b(Con)m(tin)m(uit)m(y:)54 b(for)37 b(ev)m(ery)i Fv( )g Fu(2)d Fv(X)45 b Fw(the)38 b(function)f(\()p Fv(t;)17 b(x)p Fw(\))36 b Fu(7!)f(T)2589 2683 y Fs(t)2619 2668 y Fw([)p Fv( )t Fw(]\()p Fv(x)p Fw(\))j(is)f(con)m(tin)m(uous)i(and)315 2784 y(for)32 b(all)g Fv(b)c(>)g(a)g Fu(\025)g Fw(0)k(there)h(exists)i Fv(C)f Fw(=)28 b Fv(C)7 b Fw(\()p Fv(a;)17 b(b;)g( )t Fw(\))32 b(suc)m(h)i(that)1612 2943 y Fu(jT)1694 2958 y Fs(t)1723 2943 y Fv( )t Fu(j)28 b(\024)g Fv(C)r(;)315 3101 y Fw(for)k(an)m(y)h Fv(t)28 b Fu(2)g Fw([)p Fv(a;)17 b(b)p Fw(].)200 3218 y(I)s(I)42 b(Lo)s(calit)m(y:)f(for)28 b(ev)m(ery)j Fv( )1190 3233 y Fr(1)1230 3218 y Fv(;)17 b( )1337 3233 y Fr(2)1404 3218 y Fu(2)28 b Fv(C)1575 3182 y Fl(1)1650 3218 y Fw(\()p Ft(R)1760 3182 y Fs(n)1806 3218 y Fw(\))14 b Fu(\\)g Fv(X)36 b Fw(and)29 b(an)m(y)g(\014xed)h Fv(x)e Fu(2)g Ft(R)2902 3182 y Fs(n)2949 3218 y Fw(,)h(and)g Fv(r)h(>)e Fw(0,)h(suc)m(h)315 3334 y(that)j Fv( )589 3349 y Fr(1)656 3334 y Fw(=)c Fv( )823 3349 y Fr(2)895 3334 y Fw(in)33 b(the)g(ball)g Fv(B)5 b Fw(\()p Fv(x;)17 b(r)s Fw(\))32 b(then)1274 3493 y Fu(T)1328 3508 y Fs(t)p Fl(\000)p Fs(h;t)1498 3493 y Fv( )1561 3508 y Fr(1)1623 3493 y Fu(\000)23 b(T)1777 3508 y Fs(t)p Fl(\000)p Fs(h;t)1948 3493 y Fv( )2011 3508 y Fr(2)2078 3493 y Fw(=)28 b Fv(o)p Fw(\()p Fv(h)p Fw(\))p Fv(;)315 3651 y Fw(as)k Fv(h)c Fu(!)f Fw(0)694 3615 y Fr(+)753 3651 y Fw(.)162 3767 y(I)s(I)s(I)42 b(Regularit)m(y:)d(for)21 b(an)m(y)j(sequence)h(of)c(p)s (ositiv)m(e)j(n)m(um)m(b)s(ers)g Fv(d)k Fw(=)f(\()p Fv(d)2638 3782 y Fs(k)2681 3767 y Fw(\),)d(an)m(y)f(compact)g(set)g Fv(K)35 b Fu(\032)315 3884 y Ft(R)387 3848 y Fs(n)465 3884 y Fw(and)c(for)g(ev)m(ery)i Fv( )f Fu(2)c Fv(C)1323 3848 y Fl(1)1398 3884 y Fw(\()p Ft(R)1508 3848 y Fs(n)1555 3884 y Fw(\))19 b Fu(\\)h Fv(X)40 b Fw(there)32 b(exists)h(a)e (function)h Fv(m)2881 3899 y Fs(K)q(;f)s(;d)3058 3884 y Fw(\()p Fu(\001)p Fw(\))c(:)f Ft(R)3316 3848 y Fr(+)3403 3884 y Fu(!)g Ft(R)3602 3848 y Fr(+)315 4000 y Fw(suc)m(h)34 b(that)e Fv(m)831 4015 y Fs(K)q(;f)s(;d)1009 4000 y Fw(\(0)1096 3964 y Fr(+)1154 4000 y Fw(\))c(=)f(0,)947 4159 y Fu(jT)1029 4174 y Fs(t)1058 4159 y Fw([)p Fv( )f Fw(+)c Fv(\025\021)t Fw(])g Fu(\000)h(T)1584 4174 y Fs(t)1614 4159 y Fw([)p Fv( )t Fw(])f Fu(\000)h Fv(\025\021)t Fw(\()p Fv(x)p Fw(\))p Fu(j)k(\024)h Fv(m)2342 4174 y Fs(K)q(;f)s(;d)2520 4159 y Fw(\()p Fv(\025)p Fw(\))p Fv(t;)315 4317 y Fw(for)k(an)m(y)h(\() p Fv(x;)17 b(\021)t Fw(\))27 b Fu(2)h Fv(K)i Fu(\002)22 b Fv(Q)1285 4332 y Fs(d)1358 4317 y Fw(and)33 b(an)m(y)g Fv(\025;)17 b(t)28 b Fu(\025)g Fw(0.)165 4434 y(IV)42 b(T)-8 b(ranslation:)63 b(for)41 b(an)m(y)i(compact)g(subset)h Fv(K)51 b Fu(\032)45 b Ft(R)2279 4397 y Fs(n)2368 4434 y Fw(and)d(ev)m(ery)i Fv( )k Fu(2)c Fv(C)3132 4397 y Fl(1)3125 4458 y Fs(c)3207 4434 y Fw(\()p Fv(K)7 b Fw(\),)45 b(there)315 4550 y(exists)34 b(a)e(function)h Fv(n)1105 4565 y Fs(K)q(; )1266 4550 y Fw(:)27 b Ft(R)1392 4514 y Fr(+)1479 4550 y Fu(!)g Ft(R)1678 4514 y Fr(+)1737 4550 y Fw(,)33 b(with)g Fv(n)2077 4565 y Fs(K)q(; )2210 4550 y Fw(\(0)2297 4514 y Fr(+)2356 4550 y Fw(\))27 b(=)h(0)k(suc)m(h)i (that)1038 4708 y Fu(j)p Fv(\034)1108 4723 y Fs(y)1149 4708 y Fu(T)1203 4723 y Fs(t)1233 4708 y Fw([)p Fv( )t Fw(]\()p Fv(x)p Fw(\))23 b Fu(\000)f(T)1661 4723 y Fs(t)1691 4708 y Fw([)p Fv(\034)1760 4723 y Fs(y)1802 4708 y Fv( )t Fw(]\()p Fv(x)p Fw(\))p Fu(j)28 b(\024)g Fv(n)2246 4723 y Fs(K)q(; )2379 4708 y Fw(\()p Fu(j)p Fv(y)t Fu(j)p Fw(\))p Fv(t;)315 4867 y Fw(for)k(an)m(y)h Fv(x)28 b Fu(2)g Fv(K)40 b Fw(and)32 b Fv(t)c Fu(\025)g Fw(0.)0 5005 y(Prop)s(ert)m(y)47 b([I])g(holds)g(b)m(y)i(\(74\))o(.)85 b(Prop)s(erties)48 b([I)s(I])e(and)h([I)s(I)s(I])f(follo)m(w)h(from)f (impro)m(v)m(ed)i(momen)m(t)0 5121 y(estimates)36 b(of)d(the)i (previous)g(section,)h(as)e(w)m(e)i(explain)f(next.)49 b(Prop)s(ert)m(y)35 b([IV])f(follo)m(ws)h(from)f(\(72\))0 5237 y(and)f(\(83\))o(.)0 5353 y(Let)46 b(us)g(no)m(w)g(giv)m(e)g(the)g (details)g(ab)s(out)f(ho)m(w)h(to)f(c)m(hec)m(k)i(Prop)s(erties)g([I)s (I])f(and)f([I)s(I)s(I].)h(First,)i(w)m(e)0 5469 y(will)41 b(use)h(Prop)s(osition)e(30)g(to)g(pro)m(v)m(e)i(that)e([I)s(I])h (holds.)68 b(Assume)42 b(that)e Fv( )2792 5484 y Fr(1)2873 5469 y Fw(and)g Fv( )3133 5484 y Fr(2)3213 5469 y Fw(satisfy)i(the)p eop end %%Page: 48 48 TeXDict begin 48 47 bop 168 251 a Fq(48)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y Fw(conditions)41 b(of)e([I)s(I],)i(and)f(let)g FA(x)h Fw(b)s(e)f(an)g(optimal)g(tra)5 b(jectory)41 b(for)e Fv( )2730 465 y Fr(2)2770 450 y Fw(,)j(that)e(is)h FA(x)f Fw(optimizes)h(\(5\))168 566 y(for)32 b Fv( )f Fw(:=)d Fv( )605 581 y Fr(2)677 566 y Fw(and)33 b FA(x)p Fw(\()p Fv(t)23 b Fu(\000)f Fv(h)p Fw(\))28 b(=)g Fv(x)p Fw(.)43 b(Then,)34 b(b)m(y)h(\(72\))d(and)h(Prop)s(ositions)g(24)f(and)h(26,) 687 811 y Fu(T)741 826 y Fs(t)p Fl(\000)p Fs(h;t)912 811 y Fv( )975 826 y Fr(2)1042 811 y Fw(=)28 b(lim)1146 871 y Fs(\017)p Fl(!)p Fr(0)1298 811 y Fv(E)1393 671 y Fp(\024)1445 676 y(Z)1545 702 y Fs(T)1501 901 y(t)1616 811 y Fv(L)1699 671 y Fp(\022)1773 811 y FA(x)p Fw(\()p Fv(s)p Fw(\))p Fv(;)2007 744 y FA(x)p Fw(\()p Fv(s)p Fw(\))p 2007 788 181 4 v 2078 880 a Fv(\017)2198 811 y(;)17 b FA(v)r Fw(\()p Fv(s)p Fw(\))p Fv(;)g(!)2534 671 y Fp(\023)2639 811 y Fv(ds)22 b Fw(+)g Fv( )2919 826 y Fr(2)2975 811 y Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\)\))3257 671 y Fp(\025)168 1040 y Fw(while)656 1220 y Fu(T)710 1235 y Fs(t)p Fl(\000)p Fs(h;t)881 1220 y Fv( )944 1235 y Fr(1)1011 1220 y Fu(\024)29 b Fw(lim)1117 1280 y Fs(\017)p Fl(!)p Fr(0)1269 1220 y Fv(E)1363 1079 y Fp(\024)1416 1084 y(Z)1515 1111 y Fs(T)1471 1310 y(t)1587 1220 y Fv(L)1670 1079 y Fp(\022)1743 1220 y FA(x)p Fw(\()p Fv(s)p Fw(\))p Fv(;)1978 1152 y FA(x)p Fw(\()p Fv(s)p Fw(\))p 1978 1197 V 2049 1288 a Fv(\017)2169 1220 y(;)17 b FA(v)r Fw(\()p Fv(s)p Fw(\))p Fv(;)g(!)2505 1079 y Fp(\023)2610 1220 y Fv(ds)22 b Fw(+)g Fv( )2890 1235 y Fr(2)2946 1220 y Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\)\))3228 1079 y Fp(\025)3313 1220 y Fv(;)168 1429 y Fw(via)33 b(a)f(suitable)i(subsequence.)168 1545 y(Consequen)m(tly)-8 b(,)35 b(using)f(that)e Fv( )1321 1560 y Fr(1)1393 1545 y Fw(and)h Fv( )1646 1560 y Fr(2)1719 1545 y Fw(agree)f(on)h Fv(B)5 b Fw(\()p Fv(x;)17 b(r)s Fw(\),)32 b(w)m(e)i(gather)e(that)836 1718 y Fu(T)890 1733 y Fs(t)p Fl(\000)p Fs(h;t)1060 1718 y Fv( )1123 1733 y Fr(1)1185 1718 y Fu(\000)23 b(T)1339 1733 y Fs(t)p Fl(\000)p Fs(h;t)1509 1718 y Fv( )1572 1733 y Fr(2)1640 1718 y Fu(\024)28 b Fw(\()p Fu(k)p Fv( )1896 1733 y Fr(1)1936 1718 y Fu(k)1986 1733 y Fl(1)2082 1718 y Fw(+)22 b Fu(k)p Fv( )2293 1733 y Fr(2)2333 1718 y Fu(k)2383 1733 y Fl(1)2458 1718 y Fw(\))16 b Fv(P)e Fw(\()p Fu(j)p FA(x)21 b Fu(\000)i Fv(x)p Fu(j)28 b Fv(>)f(r)s Fw(\))p Fv(;)168 1891 y Fw(whic)m(h)34 b(is)f Fv(o)p Fw(\()p Fv(h)p Fw(\))f(b)m(y)i(Prop)s(osition)f(30.)168 2008 y(T)-8 b(o)35 b(pro)m(v)m(e)i([I)s(I)s(I],)f(w)m(e)g(argue)f(in)h (an)f(analogous)h(w)m(a)m(y)-8 b(.)53 b(Namely)-8 b(,)37 b(w)m(e)f(tak)m(e)h(an)e(optimal)g(tra)5 b(jectory)168 2124 y(for)30 b Fv( )36 b Fw(and)31 b(w)m(e)h(tak)m(e)g(limit)g(for)e Fv(\017)f Fu(!)e Fw(0,)k(as)h(ab)s(o)m(v)m(e.)43 b(In)32 b(this)g(case,)g(w)m(e)h(also)e(exploit)h(Lemma)g(28)e(to)168 2240 y(conclude)k(that)723 2413 y Fu(T)777 2428 y Fs(T)10 b Fl(\000)p Fs(h;T)998 2413 y Fw(\()p Fv( )26 b Fw(+)c Fv(\025\021)t Fw(\))g Fu(\000)g(T)1545 2428 y Fs(T)10 b Fl(\000)p Fs(h;T)1767 2413 y Fv( )26 b Fu(\000)d Fv(\025\021)750 2564 y Fu(\024)29 b Fw(lim)856 2624 y Fs(\017)p Fl(!)p Fr(0)1008 2564 y Fv(E)6 b( )t Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\)\))21 b(+)h Fv(\025\021)t Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\)\))22 b Fu(\000)g Fv( )t Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\)\))22 b Fu(\000)h Fv(\025\021)t Fw(\()p FA(x)p Fw(\()p Fv(T)35 b Fu(\000)23 b Fv(h)p Fw(\)\))750 2746 y Fu(\024)29 b Fw(lim)856 2806 y Fs(\017)p Fl(!)p Fr(0)1008 2746 y Fv(\025E)6 b Fw(\()p Fv(\021)t Fw(\()p FA(x)p Fw(\()p Fv(T)14 b Fw(\)\))21 b Fu(\000)i Fv(\021)t Fw(\()p FA(x)p Fw(\()p Fv(T)35 b Fu(\000)23 b Fv(h)p Fw(\)\)\))28 b Fu(\024)g Fv(C)7 b Fu(k)p Fv(D)s(\021)t Fu(k)2631 2761 y Fl(1)2704 2746 y Fv(\025h)28 b Fu(\024)g Fv(C)7 b(d)3078 2761 y Fr(1)3117 2746 y Fv(\025h)17 b(;)168 2956 y Fw(as)32 b(required.)45 b(The)34 b(desired)g(result)g(then)f (follo)m(ws)g(from)f(Theorem)i(3.1)f(of)f([Bit01)o(].)476 b Fi(\003)168 3142 y Fw(W)-8 b(e)33 b(remark)g(that)f(Theorem)i(3.1)e (of)g([Bit01])h(explicitly)i(giv)m(es)f(that)168 3369 y(\(113\))719 b Fu(\000)p 1210 3289 89 4 v 23 w Fv(H)7 b Fw(\()p Fv(D)1420 3328 y Fr(2)1460 3369 y Fv( )t Fw(\()p Fv(x)p Fw(\))p Fv(;)17 b(D)s( )t Fw(\()p 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Fw(\004)28 b(:)g Fv(M)33 b Fu(\000)p 2893 4095 67 4 v 22 w Fv(L)q Fw(\()p Fv(x;)17 b($)s(;)g Fw(\004\))g Fv(:)168 4441 y Fo(Pr)-5 b(o)g(of.)41 b Fw(W)-8 b(e)29 b(will)g(b)s(e)g(using)g(form) m(ula)f(\(113\))g(to)g(establish)i(the)f(result.)43 b(W)-8 b(e)29 b(only)g(consider)h(case)g(C,)168 4558 y(since)k(cases)g(A)e (and)h(B)g(are)f(analogous,)h(with)g(just)g(less)h(v)-5 b(ariables)33 b(in)m(v)m(olv)m(ed.)168 4674 y(By)39 b(taking)g Fv(')p Fw(\()p Fv(x;)17 b(t)p Fw(\))39 b(:=)f Fv(t)i Fw(and)f Fv(')p Fw(\()p Fv(x;)17 b(t)p Fw(\))38 b(:=)h(1)f(in)h (\(69\),)h(w)m(e)g(see)g(that)f(if)g Fv(\045)g Fw(and)45 b(\026)-55 b Fv(\027)3222 4689 y Fs(T)3316 4674 y Fw(satisfy)42 b(\(69\))o(,)168 4790 y(then)32 b(they)g(ha)m(v)m(e)h(total)e(mass)i Fv(T)45 b Fw(and)32 b(1,)g(resp)s(ectiv)m(ely)-8 b(.)46 b(Also,)32 b(the)g(suprem)m(um)i(in)e(\(114\))e(is)i(\014nite)168 4906 y(and)42 b(attained,)k(due)d(to)f(Prop)s(ositions)i(14)e(and)g (15.)73 b(Th)m(us,)47 b(tak)m(e)c Fv($)2815 4921 y Fr(0)2899 4906 y Fw(=)h Fv($)3100 4921 y Fr(0)3139 4906 y Fw(\()p Fv(x)p Fw(\),)i Fv(\033)3398 4921 y Fr(0)3482 4906 y Fw(=)f Fv(\033)3658 4921 y Fr(0)3698 4906 y Fw(\()p Fv(x)p Fw(\))168 5023 y(and)32 b(\004)422 5038 y Fr(0)490 5023 y Fw(=)27 b(\004)658 5038 y Fr(0)698 5023 y Fw(\()p Fv(x)p Fw(\))h(=)f Fv(\033)1015 5038 y Fr(0)1055 5023 y Fv(\033)1114 4986 y Fs(T)1110 5047 y Fr(0)1202 5023 y Fw(that)32 b(satisfy)951 5201 y(sup)961 5282 y Fs($)r(;)p Fr(\004)1114 5201 y Fu(\000)p Fv(D)1272 5216 y Fs(x)1317 5201 y Fv( )t Fw(\()p Fv(x)p Fw(\))22 b Fu(\001)g Fv($)i Fu(\000)f Fw(\004)28 b(:)g Fv(D)2024 5160 y Fr(2)2021 5225 y Fs(x)2064 5201 y Fv( )t Fw(\()p Fv(x)p Fw(\))23 b Fu(\000)p 2384 5121 V 22 w Fv(L)q Fw(\()p Fv(x;)17 b($)s(;)g Fw(\004\))1060 5430 y(=)27 b Fu(\000)p Fv(D)1321 5445 y Fs(x)1365 5430 y Fv( )t Fw(\()p Fv(x)p Fw(\))c Fu(\001)f Fv($)1717 5445 y Fr(0)1778 5430 y Fu(\000)g Fw(\004)1942 5445 y Fr(0)2010 5430 y Fw(:)27 b Fv(D)2148 5389 y Fr(2)2145 5455 y Fs(x)2189 5430 y Fv( )t Fw(\()p Fv(x)p Fw(\))c Fu(\000)p 2509 5350 V 22 w Fv(L)q Fw(\()p Fv(x;)17 b($)2794 5445 y Fr(0)2833 5430 y Fv(;)g Fw(\004)2942 5445 y Fr(0)2981 5430 y Fw(\))g Fv(:)p eop end %%Page: 49 49 TeXDict begin 49 48 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(49)0 450 y Fw(W)-8 b(e)33 b(also)g(consider)h(the)f(dynamics)h(of)e (the)h(asso)s(ciated)g(con)m(trol)g(problem)955 611 y Fv(d)p FA(x)1065 626 y Fr(0)1132 611 y Fw(=)28 b Fu(S)1303 570 y Fs($)1296 636 y(x)1336 645 y Fk(0)1371 636 y Fs(;$)1450 645 y Fk(0)1483 636 y Fs(;\033)1543 645 y Fk(0)1582 611 y Fw(\()p FA(x)1679 626 y Fr(0)1719 611 y Fw(\))p Fv(dt)22 b Fw(+)g Fu(S)2030 570 y Fr(\004)2023 636 y Fs(x)2063 645 y Fk(0)2097 636 y Fs(;$)2176 645 y Fk(0)2210 636 y Fs(;)p Fr(\004)2277 645 y Fk(0)2315 611 y Fw(\()p FA(x)2412 626 y Fr(0)2452 611 y Fw(\))p Fv(dW)2633 626 y Fs(t)2679 611 y Fv(;)0 777 y Fw(and)33 b(corresp)s(onding)g(measures)i Fv(\045)1285 792 y Fr(0)1324 777 y Fw(\()p Fv(x)1417 741 y Fl(0)1441 777 y Fv(;)17 b(!)t(;)g Fw(\004)p Fv(;)g(s)p Fw(\))31 b(and)h Fv(\027)2055 792 y Fr(0)p Fs(;T)2166 777 y Fw(\()p Fv(x)2259 741 y Fl(0)2282 777 y Fw(\),)h(giv)m(en)h(b)m (y)676 858 y Fp(Z)731 1083 y Fn(R)779 1064 y Fm(n)821 1083 y Fl(\002)p Fn(R)924 1064 y Fm(n)967 1083 y Fl(\002)p Fr(Mat\()p Fs(n)p Fl(\002)p Fs(n)p Fr(\))1365 993 y Fv(\036)p Fw(\()p Fv(x)1516 952 y Fl(0)1540 993 y Fv(;)17 b($)s(;)g Fw(\004)p Fv(;)g(s)p Fw(\))p Fv(d\045)2006 1008 y Fr(0)687 1292 y Fw(:=)27 b Fv(E)912 1156 y Fp(Z)1012 1183 y Fs(T)967 1382 y Fr(0)1083 1292 y Fv(\036)p Fw(\()p FA(x)1238 1307 y Fr(0)1277 1292 y Fw(\()p Fv(s)p Fw(\))p Fv(;)17 b Fu(S)1510 1251 y Fs($)1503 1317 y(x)1543 1326 y Fk(0)1578 1317 y Fs(;$)1657 1326 y Fk(0)1690 1317 y Fs(;\033)1750 1326 y Fk(0)1789 1292 y Fw(\()p FA(x)1886 1307 y Fr(0)1926 1292 y Fw(\()p Fv(s)p Fw(\)\))p Fv(;)g Fu(S)2197 1251 y Fr(\004)2190 1317 y Fs(x)2230 1326 y Fk(0)2264 1317 y Fs(;$)2343 1326 y Fk(0)2377 1317 y Fs(;)p Fr(\004)2444 1326 y Fk(0)2482 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2347 y Fs(x)1843 2323 y Fv( )e Fu(\000)p 2031 2243 V 22 w Fv(L)q(d\045)2199 2338 y Fr(0)748 2581 y Fw(=)908 2446 y Fp(Z)1024 2581 y Fu(\000)p Fv(D)1182 2596 y Fs(x)1226 2581 y Fv( )t Fw(\()p Fv(x)p Fw(\))d Fu(\001)e Fv($)1577 2596 y Fr(0)1639 2581 y Fu(\000)h Fw(\004)1803 2596 y Fr(0)1870 2581 y Fw(:)28 b Fv(D)2009 2540 y Fr(2)2006 2606 y Fs(x)2050 2581 y Fv( )t Fw(\()p Fv(x)p Fw(\))22 b Fu(\000)p 2370 2501 V 23 w Fv(L)p Fw(\()p Fv(x;)17 b($)2654 2596 y Fr(0)2694 2581 y Fv(;)g Fw(\004)2803 2596 y Fr(0)2842 2581 y Fw(\))p Fv(d\045)2981 2596 y Fr(0)1005 2840 y Fw(+)1098 2704 y Fp(Z)1214 2726 y(\014)1214 2785 y(\014)1214 2845 y(\014)1247 2840 y Fv(D)1328 2855 y Fs(x)1372 2840 y Fv( )26 b Fu(\001)c Fv($)i Fw(+)e(\004)28 b(:)g Fv(D)1946 2799 y Fr(2)1943 2865 y Fs(x)1987 2840 y Fv( )1103 3078 y Fw(+)p 1179 2998 V Fv(L)22 b Fu(\000)h Fv(D)1448 3093 y Fs(x)1492 3078 y Fv( )t Fw(\()p Fv(x)p Fw(\))f Fu(\001)g Fv($)1843 3093 y Fr(0)1904 3078 y Fu(\000)h Fw(\004)2069 3093 y Fr(0)2136 3078 y Fw(:)28 b Fv(D)2275 3037 y Fr(2)2272 3102 y Fs(x)2315 3078 y Fv( )t Fw(\()p Fv(x)p Fw(\))23 b Fu(\000)p 2635 2998 V 22 w Fv(L)q Fw(\()p Fv(x;)17 b($)2920 3093 y Fr(0)2959 3078 y Fv(;)g Fw(\004)3068 3093 y Fr(0)3107 3078 y Fw(\))3145 2963 y Fp(\014)3145 3023 y(\014)3145 3083 y(\014)3195 3078 y Fv(d\045)3296 3093 y Fr(0)3335 3078 y Fv(:)747 3268 y Fu(\025)84 b Fv(T)30 b Fw(sup)1005 3349 y Fs($)r(;)p Fr(\004)1158 3268 y Fu(\000)p Fv(D)1316 3283 y Fs(x)1361 3268 y Fv( )t Fw(\()p Fv(x)p Fw(\))22 b Fu(\001)g Fv($)i Fu(\000)f Fw(\004)28 b(:)g Fv(D)2068 3227 y Fr(2)2065 3293 y Fs(x)2108 3268 y Fv( )t Fw(\()p Fv(x)p Fw(\))23 b Fu(\000)p 2428 3188 V 22 w Fv(L)q Fw(\()p Fv(x;)17 b($)s(;)g Fw(\004\))908 3564 y Fu(\000)p Fv(C)1079 3429 y Fp(Z)1178 3455 y Fs(T)1134 3654 y Fr(0)1250 3564 y Fv(E)6 b Fu(j)p FA(x)1415 3579 y Fr(0)1454 3564 y Fw(\()p Fv(s)p Fw(\))22 b Fu(\000)h Fv(x)p Fu(j)17 b Fv(ds;)0 3777 y Fw(thanks)34 b(to)e(Assumption)i(16.)0 3893 y(That)f(is,)648 4035 y Fv( )26 b Fu(\000)d(T)891 4050 y Fs(T)946 4035 y Fv( )p 648 4079 366 4 v 795 4170 a(T)1051 4102 y Fu(\025)28 b Fw(sup)1166 4183 y Fs($)r(;)p Fr(\004)1320 4102 y Fu(\000)p Fv(D)1478 4117 y Fs(x)1522 4102 y Fv( )t Fw(\()p Fv(x)p Fw(\))22 b Fu(\001)g Fv($)j Fu(\000)d Fw(\004)28 b(:)g Fv(D)2229 4061 y Fr(2)2226 4127 y Fs(x)2270 4102 y Fv( )t Fw(\()p Fv(x)p Fw(\))22 b Fu(\000)p 2590 4022 67 4 v 23 w Fv(L)p Fw(\()p Fv(x;)17 b($)s(;)g Fw(\004\))856 4398 y Fu(\000)965 4331 y Fv(C)p 965 4375 77 4 v 968 4467 a(T)1085 4263 y Fp(Z)1185 4289 y Fs(T)1141 4488 y Fr(0)1257 4398 y Fv(E)6 b Fu(j)p FA(x)1422 4413 y Fr(0)1461 4398 y Fw(\()p Fv(s)p Fw(\))22 b Fu(\000)g Fv(x)p Fu(j)17 b Fv(ds)g(:)0 4251 y Fw(\(115\))0 4611 y(W)-8 b(e)33 b(no)m(w)g(observ)m(e)i(that,)d(b)m(y)i(Lemma)f(27,)636 4702 y Fp(Z)735 4729 y Fs(T)691 4928 y Fr(0)807 4838 y Fv(E)6 b Fu(j)p FA(x)972 4853 y Fr(0)1011 4838 y Fw(\()p Fv(s)p Fw(\))22 b Fu(\000)h Fv(x)p Fu(j)17 b Fv(ds)27 b Fu(\024)1584 4702 y Fp(Z)1684 4729 y Fs(T)1639 4928 y Fr(0)1739 4838 y Fw(\()p Fv(E)6 b Fu(j)p FA(x)1942 4853 y Fr(0)1981 4838 y Fw(\()p Fv(s)p Fw(\))22 b Fu(\000)g Fv(x)p Fu(j)2307 4797 y Fr(2)2347 4838 y Fw(\))2385 4797 y Fr(1)p Fs(=)p Fr(2)2512 4838 y Fv(ds)27 b Fu(\024)h Fv(C)7 b(T)2889 4797 y Fr(3)p Fs(=)p Fr(2)2999 4838 y Fv(;)0 5050 y Fw(and)33 b(so)0 5258 y(\(116\))1275 5191 y(1)p 1264 5235 71 4 v 1264 5327 a Fv(T)1361 5123 y Fp(Z)1461 5149 y Fs(T)1416 5348 y Fr(0)1532 5258 y Fv(E)6 b Fu(j)p FA(x)1697 5273 y Fr(0)1737 5258 y Fw(\()p Fv(s)p Fw(\))22 b Fu(\000)g Fv(x)p Fu(j)17 b Fv(ds)27 b Fu(!)h Fw(0)p Fv(;)0 5469 y Fw(as)33 b Fv(T)41 b Fu(!)27 b Fw(0.)p eop end %%Page: 50 50 TeXDict begin 50 49 bop 168 251 a Fq(50)970 b(DIOGO)25 b(A.)g(GOMES)h(&)f(ENRICO)g(V)-9 b(ALDINOCI)168 450 y Fw(Consequen)m(tly)h(,)35 b(making)f(use)f(of)39 b(\(113\))o(,)33 b(\(115\))f(and)g(\(116\),)g(w)m(e)i(th)m(us)f(deduce)i(that)168 632 y(\(117\))p 641 552 89 4 v 250 w Fv(H)7 b Fw(\()p Fv(D)851 591 y Fr(2)890 632 y Fv( )t Fw(\()p Fv(x)p Fw(\))p Fv(;)17 b(D)s( )t Fw(\()p Fv(x)p Fw(\))p Fv(;)g(x)p Fw(\))28 b Fu(\025)g Fw(sup)1694 714 y Fs($)r(;)p Fr(\004)1848 632 y Fu(\000)p Fv(D)2006 647 y Fs(x)2050 632 y Fv( )t Fw(\()p Fv(x)p Fw(\))23 b Fu(\001)e Fv($)k Fu(\000)d Fw(\004)28 b(:)g Fv(D)2757 591 y Fr(2)2754 657 y Fs(x)2798 632 y Fv( )t Fw(\()p Fv(x)p Fw(\))22 b Fu(\000)p 3118 552 67 4 v 23 w Fv(L)p Fw(\()p Fv(x;)17 b($)s(;)g Fw(\004\))p Fv(:)168 880 y Fw(No)m(w)45 b(w)m(e)h(will)g(establish)g(the)g(rev)m (erse)h(inequalit)m(y)g(and)e(complete)h(the)g(pro)s(of)e(of)51 b(\(114\))o(.)81 b(F)-8 b(or)168 997 y(that)40 b(matter,)j(let)e Fv(\045)p Fw(\()p Fv(x)1036 961 y Fl(0)1060 997 y Fv(;)17 b($)s(;)g Fw(\004)p Fv(;)g(s)p Fw(\))38 b(and)j Fv(\027)1709 1012 y Fs(T)1764 997 y Fw(\()p Fv(x)1857 961 y Fl(0)1881 997 y Fw(\))f(b)s(e)h(minimizing)h(measures)g(for)e(\(73\))g(under)i (the)168 1113 y(constrain)m(t)33 b(in)g(\(69\))o(,)g(with)g Fv(t)28 b Fw(=)g(0.)43 b(Then,)417 1295 y Fv(T)30 b Fw(sup)514 1376 y Fs($)r(;)p Fr(\004)668 1295 y Fu(\000)p Fv(D)826 1310 y Fs(x)870 1295 y Fv(\036)p Fw(\()p Fv(x)p Fw(\))22 b Fu(\001)g Fv($)i Fu(\000)f Fw(\004)28 b(:)f Fv(D)1567 1254 y Fr(2)1564 1320 y Fs(x)1608 1295 y Fv(\036)p Fw(\()p Fv(x)p Fw(\))22 b Fu(\000)p 1919 1215 V 23 w Fv(L)p Fw(\()p Fv(x;)17 b($)s(;)g Fw(\004\))339 1572 y Fu(\025)433 1437 y Fp(Z)550 1572 y Fu(\000)p Fv(D)708 1587 y Fs(x)752 1572 y Fv(\036)p Fw(\()p Fv(x)p Fw(\))22 b Fu(\001)g(S)1080 1531 y Fs($)1073 1597 y(x)1113 1578 y Fj(0)1135 1597 y Fs(;$)r(;)p Fr(\004)1286 1572 y Fw(\()p Fv(x)p Fw(\))h Fu(\000)f(S)1606 1531 y Fr(\004)1599 1597 y Fs(x)1639 1578 y Fj(0)1662 1597 y Fs(;$)r(;)p Fr(\004)1813 1572 y Fw(\()p Fv(x)p Fw(\))28 b(:)f Fv(D)2110 1531 y Fr(2)2107 1597 y Fs(x)2151 1572 y Fv(\036)p Fw(\()p Fv(x)p Fw(\))c Fu(\000)p 2462 1492 V 22 w Fv(L)q Fw(\()p Fv(x;)17 b Fu(S)2733 1531 y Fs($)2726 1597 y(x)2766 1578 y Fj(0)2788 1597 y Fs(;$)r(;)p Fr(\004)2939 1572 y Fw(\()p Fv(x)p Fw(\))p Fv(;)g Fu(S)3181 1531 y Fr(\004)3174 1597 y Fs(x)3214 1578 y Fj(0)3236 1597 y Fs(;$)r(;)p Fr(\004)3387 1572 y Fw(\()p Fv(x)p Fw(\)\))p Fv(d\045)339 1831 y Fu(\025)433 1695 y Fp(Z)550 1831 y Fu(\000)p Fv(D)708 1846 y Fs(x)752 1831 y Fv(\036)p Fw(\()p Fv(x)p Fw(\))22 b Fu(\001)g Fv($)i Fu(\000)f Fw(\004)28 b(:)f Fv(D)1449 1790 y Fr(2)1446 1856 y Fs(x)1490 1831 y Fv(\036)p Fw(\()p Fv(x)p Fw(\))c Fu(\000)p 1801 1751 V 22 w Fv(L)q Fw(\()p Fv(x;)17 b($)s(;)g Fw(\004\))g Fv(d\045)j Fu(\000)j Fv(C)2567 1695 y Fp(Z)2684 1831 y Fu(j)p Fv(x)2767 1790 y Fl(0)2812 1831 y Fu(\000)g Fv(x)p Fu(j)p Fv(d\045)17 b(;)168 2055 y Fw(thanks)33 b(to)g(Assumption)h(16.)168 2171 y(The)f(ab)s(o)m(v)m(e)h(estimate,)f (the)g(holonom)m(y)h(constrain)m(t)f(and)g(Prop)s(osition)g(31)f(then)h (giv)m(e)1038 2349 y(sup)1048 2430 y Fs($)r(;)p Fr(\004)1201 2349 y Fu(\000)p Fv(D)1359 2364 y Fs(x)1404 2349 y Fv(\036)p Fw(\()p Fv(x)p Fw(\))22 b Fu(\001)g Fv($)i Fu(\000)f Fw(\004)k(:)h Fv(D)2101 2307 y Fr(2)2098 2373 y Fs(x)2142 2349 y Fv(\036)p Fw(\()p Fv(x)p Fw(\))22 b Fu(\000)p 2453 2269 V 23 w Fv(L)p Fw(\()p 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3432 y Fv( )t Fw(\()p Fv(x)p Fw(\))23 b Fu(\001)e Fv($)k Fu(\000)e Fw(\004)k(:)h Fv(D)1517 3391 y Fr(2)1514 3456 y Fs(x)1558 3432 y Fv( )t Fw(\()p Fv(x)p Fw(\))22 b Fu(\000)p 1878 3352 67 4 v 23 w Fv(L)p Fw(\()p Fv(x;)17 b($)s(;)g Fw(\004\))27 b Fu(\025)p 2444 3352 89 4 v 28 w Fv(H)8 b Fw(\()p Fv(D)2655 3391 y Fr(2)2694 3432 y Fv( )t Fw(\()p Fv(x)p Fw(\))p Fv(;)17 b(D)s( )t Fw(\()p Fv(x)p Fw(\))p Fv(;)g(x)p Fw(\))22 b Fu(\000)h Fw(0)p Fv(:)168 3680 y Fw(This)33 b(and)g(\(117\))f(complete)i(the)f (pro)s(of)e(of)39 b(\(114\).)1715 b Fi(\003)1625 3935 y Fw(13.)48 b Fx(Conclusion)168 4110 y Fo(Pr)-5 b(o)g(of)34 b(of)h(The)-5 b(or)g(em)34 b(1.)41 b Fw(Prop)s(ositions)33 b(4)f(and)h(9)f(imply)i(the)f(\014rst)g(claim)g(of)f(Theorem)i(1.)168 4226 y(The)f(second)h(claim)f(follo)m(ws)h(from)e(Prop)s(ositions)h(18) f(and)h(20)f(and)h(Remark)g(19.)168 4342 y(By)42 b(collecting)h(the)g (results)g(in)g(Prop)s(ositions)f(24,)i(26)e(and)g(32,)i(w)m(e)f (obtain)f(the)h(last)f(claim)h(of)168 4459 y(Theorem)34 b(1.)3096 b Fi(\003)1708 4714 y Fx(References)168 4872 y Fz([AB02])132 b(Olivier)17 b(Alv)-5 b(arez)19 b(and)f(Martino)g (Bardi.)g(Viscosit)n(y)g(solutions)g(metho)r(ds)h(for)f(singular)g(p)r (erturbations)g(in)551 4971 y(deterministic)i(and)g(sto)r(c)n(hastic)g (con)n(trol.)f Fb(SIAM)k(J.)g(Contr)l(ol)h(Optim.)p Fz(,)e (40\(4\):1159{1188)15 b(\(electronic\),)551 5071 y(2001/02.)168 5171 y([Alv99])124 b(O.)21 b(Alv)-5 b(arez.)22 b(Homogenization)e(of)i (Hamilton-Jacobi)f(equations)g(in)h(p)r(erforated)f(sets.)h Fb(J.)j(Di\013er)l(ential)551 5270 y(Equations)p Fz(,)j (159\(2\):543{577,)23 b(1999.)168 5370 y([A)n(tt84])129 b(H.)29 b(A)n(ttouc)n(h.)g Fb(V)-6 b(ariational)33 b(c)l(onver)l(genc)l (e)e(for)h(functions)f(and)h(op)l(er)l(ators)p Fz(.)e(Applicable)g (Mathematics)551 5469 y(Series.)d(Pitman)g(\(Adv)-5 b(anced)28 b(Publishing)f(Program\),)e(Boston,)i(MA,)h(1984.)p eop end %%Page: 51 51 TeXDict begin 51 50 bop 653 251 a Fq(GENERALIZED)32 b(MA)-6 b(THER)33 b(PR)n(OBLEM)h(AND)e(HOMOGENIZA)-6 b(TION)574 b(51)0 450 y Fz([Bil95])148 b(P)n(atric)n(k)24 b(Billingsley)-7 b(.)25 b Fb(Pr)l(ob)l(ability)30 b(and)f(me)l(asur)l(e)p Fz(.)d(Wiley)g(Series)f(in)h(Probabilit)n(y)e(and)i(Mathematical)383 550 y(Statistics.)34 b(John)g(Wiley)h(&)f(Sons)g(Inc.,)j(New)d(Y)-7 b(ork,)36 b(third)f(edition,)h(1995.)d(A)i(Wiley-In)n(terscience)383 649 y(Publication.)0 749 y([Bit01])139 b(Sam)n(uel)46 b(Biton.)h(Nonlinear)e(monotone)h(semigroups)f(and)i(viscosit)n(y)e (solutions.)h Fb(A)n(nn.)g(Inst.)h(H.)383 849 y(Poinc)l(ar)n(\023)-40 b(e)31 b(A)n(nal.)f(Non)g(Lin)n(\023)-40 b(eair)l(e)p Fz(,)29 b(18\(3\):383{402,)23 b(2001.)0 948 y([BLP76])85 b(Alain)27 b(Bensoussan,)f(Jacques-Louis)e(Lions,)j(and)g(Georges)e(P)n (apanicolaou.)f(Homog)n(\023)-39 b(en)n(\023)g(eisation,)24 b(cor-)383 1048 y(recteurs)k(et)h(probl)n(\022)-39 b(emes)27 b(non-lin)n(\023)-39 b(eaires.)27 b Fb(C.)32 b(R.)f(A)l(c)l(ad.)g(Sci.) h(Paris)g(S)n(\023)-40 b(er.)32 b(A-B)p Fz(,)d(282\(22\):Aii,)f(A1277{) 383 1147 y(A1282,)e(1976.)0 1247 y([CC95])133 b(Luis)23 b(A.)h(Ca\013arelli)e(and)h(Xa)n(vier)f(Cabr)n(\023)-39 b(e.)22 b Fb(F)-6 b(ul)t(ly)26 b(nonline)l(ar)g(el)t(liptic)i(e)l (quations)p Fz(,)c(v)n(olume)f(43)f(of)i Fb(A)n(meri-)383 1347 y(c)l(an)29 b(Mathematic)l(al)h(So)l(ciety)f(Col)t(lo)l(quium)h (Public)l(ations)p Fz(.)e(American)e(Mathematical)g(So)r(ciet)n(y)-7 b(,)27 b(Pro)n(v-)383 1446 y(idence,)h(RI,)f(1995.)0 1546 y([CDI01])100 b(I.)19 b(Capuzzo-Dolcetta)e(and)i(H.)g(Ishii.)g(On) g(the)g(rate)f(of)h(con)n(v)n(ergence)e(in)i(homogenization)e(of)i (Hamilton-)383 1646 y(Jacobi)26 b(equations.)h Fb(Indiana)k(Univ.)f (Math.)h(J.)p Fz(,)d(50\(3\):1113{1129,)22 b(2001.)0 1745 y([Con96])105 b(Marie)20 b(C.)g(Concordel.)g(P)n(erio)r(dic)f (homogenization)g(of)h(Hamilton-Jacobi)f(equations:)33 b(additiv)n(e)20 b(eigen-)383 1845 y(v)-5 b(alues)27 b(and)g(v)-5 b(ariational)27 b(form)n(ula.)f Fb(Indiana)31 b(Univ.)f(Math.)h(J.)p Fz(,)d(45\(4\):1095{1117,)23 b(1996.)0 1944 y([CS02])147 b(Mo)r(on)25 b(Jung)g(Cho)g(and)g(Ric)n(hard)g(H.)g (Sto)r(c)n(kbridge.)g(Linear)f(programming)f(form)n(ulation)i(for)g (optimal)383 2044 y(stopping)i(problems.)g Fb(SIAM)i(J.)h(Contr)l(ol)g (Optim.)p Fz(,)f(40\(6\):1965{1982)22 b(\(electronic\),)28 b(2002.)0 2144 y([DiB95])108 b(Emman)n(uele)36 b(DiBenedetto.)h Fb(Partial)i(di\013er)l(ential)g(e)l(quations)p Fz(.)e(Birkh\177)-42 b(auser)35 b(Boston)g(Inc.,)k(Boston,)383 2243 y(MA,)28 b(1995.)0 2343 y([EG01])131 b(L.)19 b(C.)g(Ev)-5 b(ans)19 b(and)g(D.)h(Gomes.)e(E\013ectiv)n(e)h(Hamiltonians)g(and)g(a)n(v)n (eraging)d(for)j(Hamiltonian)g(dynamics.)383 2443 y(I.)28 b Fb(A)n(r)l(ch.)i(R)l(ation.)g(Me)l(ch.)h(A)n(nal.)p Fz(,)d(157\(1\):1{33,)c(2001.)0 2542 y([Ev)-5 b(a82])115 b(La)n(wrence)19 b(C.)h(Ev)-5 b(ans.)20 b(Classical)f(solutions)h(of)g (fully)h(nonlinear,)g(con)n(v)n(ex,)f(second-order)e(elliptic)j(equa-) 383 2642 y(tions.)27 b Fb(Comm.)k(Pur)l(e)f(Appl.)h(Math.)p 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Fb(Visc)l(osity)j(solutions)g(and)g(applic)l(ations) h(\(Monte)l(c)l(atini)f(T)-6 b(erme,)29 b(1995\))p Fz(,)e(v)n(olume)d (1660)383 3439 y(of)j Fb(L)l(e)l(ctur)l(e)i(Notes)h(in)g(Math.)p Fz(,)f(pages)d(98{133.)f(Springer,)i(Berlin,)g(1997.)0 3539 y([Ev)-5 b(a98])115 b(La)n(wrence)25 b(C.)h(Ev)-5 b(ans.)26 b Fb(Partial)k(di\013er)l(ential)g(e)l(quations)p Fz(,)d(v)n(olume)f(19)g(of)g Fb(Gr)l(aduate)j(Studies)g(in)g(Math-)383 3638 y(ematics)p Fz(.)f(American)f(Mathematical)h(So)r(ciet)n(y)-7 b(,)27 b(Pro)n(vidence,)f(RI,)i(1998.)0 3738 y([Ev)-5 b(a06a])73 b(La)n(wrence)46 b(C.)j(Ev)-5 b(ans.)47 b Fb(A)n(n)h(Intr)l(o)l(duction)g(to)h(Mathematic)l(al)h(Optimal)f(Contr) l(ol)h(The)l(ory)p Fz(.)f(2006.)383 3837 y(Preprin)n(t)26 b(V)-7 b(ersion)27 b(0.1.)g(Av)-5 b(ailable)27 b(on-line)g(at)h Fa(http://math.berk)o(el)o(ey.)o(ed)o(u/)p Ff(\030)13 b Fa(ev)o(an)o(s/)o Fz(.)0 3937 y([Ev)-5 b(a06b])69 b(La)n(wrence)33 b(C.)h(Ev)-5 b(ans.)34 b Fb(A)n(n)i(Intr)l(o)l(duction)f(to)h(Sto)l (chastic)h(Di\013er)l(ential)g(Equations)p Fz(.)e(2006.)e(Preprin)n(t) 383 4037 y(V)-7 b(ersion)27 b(1.2.)g(Av)-5 b(ailable)27 b(on-line)g(at)h Fa(http://math.ber)o(ke)o(ley)o(.e)o(du/)o Ff(\030)14 b Fa(e)o(va)o(ns)o(/)p Fz(.)0 4136 y([FR75])138 b(W)-7 b(endell)24 b(H.)f(Fleming)g(and)g(Ra)n(ymond)f(W.)i(Rishel.)f Fb(Deterministic)j(and)g(sto)l(chastic)g(optimal)h(c)l(ontr)l(ol)p Fz(.)383 4236 y(Springer-V)-7 b(erlag,)25 b(Berlin,)i(1975.)f (Applications)h(of)h(Mathematics,)f(No.)h(1.)0 4336 y([FS06])153 b(W)-7 b(endell)19 b(H.)h(Fleming)e(and)h(H.)g(Mete)g(Soner.)g Fb(Contr)l(ol)t(le)l(d)k(Markov)g(pr)l(o)l(c)l(esses)f(and)g(visc)l (osity)h(solutions)p Fz(,)383 4435 y(v)n(olume)37 b(25)g(of)h Fb(Sto)l(chastic)i(Mo)l(del)t(ling)h(and)f(Applie)l(d)h(Pr)l(ob)l (ability)p Fz(.)f(Springer,)f(New)f(Y)-7 b(ork,)40 b(second)383 4535 y(edition,)28 b(2006.)0 4634 y([Gom03])77 b(Diogo)25 b(Aguiar)h(Gomes.)g(P)n(erturbation)f(theory)h(for)g(viscosit)n(y)f 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