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%%Title: jets.dvi
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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 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 56 /eight put
dup 61 /equal put
dup 100 /d put
dup 105 /i put
dup 115 /s put
dup 116 /t put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
/UniqueXX 5000789 def
currentdict end
currentfile eexec
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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 11 /alpha put
dup 12 /beta put
dup 16 /zeta put
dup 18 /theta put
dup 20 /kappa put
dup 26 /rho put
dup 33 /omega put
dup 59 /comma put
dup 61 /slash put
dup 63 /star put
dup 64 /partialdiff put
dup 66 /B put
dup 68 /D put
dup 69 /E put
dup 77 /M put
dup 82 /R put
dup 84 /T put
dup 97 /a put
dup 100 /d put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 110 /n put
dup 114 /r put
dup 120 /x put
dup 121 /y put
readonly def
/FontBBox{11 -250 1241 750}readonly def
/UniqueXX 5087381 def
currentdict end
currentfile eexec
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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 12 /fi put
dup 13 /fl put
dup 14 /ffi put
dup 34 /quotedblright put
dup 39 /quoteright 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 54 /six put
dup 58 /colon put
dup 59 /semicolon 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 92 /quotedblleft 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
readonly def
/FontBBox{-163 -250 1146 969}readonly def
/UniqueXX 5000828 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMBX10
%!PS-AdobeFont-1.1: CMBX10 1.00B
%%CreationDate: 1992 Feb 19 19:54:06

% 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 (CMBX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX10 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 46 /period 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 65 /A put
dup 67 /C put
dup 68 /D put
dup 71 /G put
dup 76 /L put
dup 78 /N put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 84 /T put
dup 90 /Z put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e 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 121 /y put
readonly def
/FontBBox{-301 -250 1164 946}readonly def
/UniqueXX 5000768 def
currentdict end
currentfile eexec
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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 16 /zeta put
dup 18 /theta put
dup 20 /kappa put
dup 22 /mu put
dup 26 /rho put
dup 27 /sigma put
dup 33 /omega put
dup 34 /epsilon put
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 61 /slash put
dup 62 /greater put
dup 63 /star put
dup 64 /partialdiff put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 72 /H put
dup 75 /K put
dup 77 /M put
dup 78 /N put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 97 /a put
dup 100 /d 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 117 /u put
dup 118 /v put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueXX 5087383 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 12 /vextendsingle put
dup 16 /parenleftBig put
dup 17 /parenrightBig put
dup 18 /parenleftbigg put
dup 19 /parenrightbigg put
dup 56 /bracelefttp put
dup 58 /braceleftbt put
dup 60 /braceleftmid put
dup 80 /summationtext put
dup 82 /integraltext put
dup 83 /uniontext put
dup 88 /summationdisplay put
dup 90 /integraldisplay put
dup 91 /uniondisplay put
dup 104 /bracketleftBig put
dup 105 /bracketrightBig put
dup 110 /braceleftBig put
dup 111 /bracerightBig put
dup 112 /radicalbig put
dup 114 /radicalbigg put
dup 115 /radicalBigg put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueXX 5000774 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 11 /alpha put
dup 12 /beta put
dup 13 /gamma put
dup 14 /delta put
dup 16 /zeta put
dup 18 /theta put
dup 20 /kappa put
dup 21 /lambda put
dup 22 /mu put
dup 23 /nu put
dup 26 /rho put
dup 27 /sigma put
dup 30 /phi put
dup 31 /chi put
dup 32 /psi put
dup 33 /omega put
dup 34 /epsilon put
dup 36 /pi1 put
dup 37 /rho1 put
dup 39 /phi1 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 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 96 /lscript put
dup 97 /a 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 110 /n 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{-32 -250 1048 750}readonly def
/UniqueXX 5087385 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40

% 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 (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 2 /Theta 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 56 /eight put
dup 57 /nine put
dup 61 /equal put
dup 67 /C put
dup 76 /L put
dup 87 /W put
dup 91 /bracketleft put
dup 93 /bracketright put
dup 97 /a put
dup 99 /c put
dup 100 /d put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 115 /s put
dup 116 /t put
dup 120 /x put
dup 126 /tilde put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueXX 5000791 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 6 /plusminus put
dup 15 /bullet put
dup 18 /reflexsubset put
dup 19 /reflexsuperset 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 56 /universal put
dup 59 /emptyset put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 71 /G put
dup 72 /H put
dup 74 /J put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 82 /R put
dup 84 /T put
dup 86 /V put
dup 88 /X put
dup 89 /Y put
dup 91 /union put
dup 92 /intersection put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
dup 112 /radical put
dup 114 /nabla put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueXX 5000820 def
currentdict end
currentfile eexec
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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. All Rights Reserved) readonly def
/FullName (CMBX12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX12 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 13 /fl put
dup 45 /hyphen put
dup 46 /period 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 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 71 /G put
dup 73 /I put
dup 76 /L put
dup 78 /N put
dup 80 /P put
dup 82 /R put
dup 84 /T put
dup 88 /X 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 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 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
dup 124 /emdash put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
/UniqueXX 5000769 def
currentdict end
currentfile eexec
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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 1 /Delta put
dup 2 /Theta put
dup 3 /Lambda put
dup 8 /Phi 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 16 /dotlessi put
dup 18 /grave put
dup 19 /acute put
dup 21 /breve put
dup 22 /macron put
dup 33 /exclam put
dup 34 /quotedblright put
dup 35 /numbersign 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 81 /Q 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 90 /Z put
dup 91 /bracketleft put
dup 92 /quotedblleft 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 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{-251 -250 1009 969}readonly def
/UniqueXX 5000793 def
currentdict end
currentfile eexec
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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. All Rights Reserved) readonly def
/FullName (CMR9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR9 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 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period 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 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 72 /H put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 77 /M put
dup 78 /N put
dup 80 /P 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 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
readonly def
/FontBBox{-39 -250 1036 750}readonly def
/UniqueXX 5000792 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 18 /reflexsubset put
dup 48 /prime put
dup 49 /infinity put
dup 50 /element put
dup 54 /negationslash put
dup 59 /emptyset put
dup 66 /B put
dup 67 /C put
dup 92 /intersection put
readonly def
/FontBBox{-4 -948 1329 786}readonly def
/UniqueXX 5000816 def
currentdict end
currentfile eexec
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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 0 /minus put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
/UniqueXX 5000817 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMR7
%!PS-AdobeFont-1.1: CMR7 1.0
%%CreationDate: 1991 Aug 20 16:39:21

% 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 (CMR7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR7 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 40 /parenleft put
dup 41 /parenright put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
readonly def
/FontBBox{-27 -250 1122 750}readonly def
/UniqueXX 5000790 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 11 /alpha put
dup 59 /comma put
dup 105 /i put
dup 106 /j put
dup 110 /n put
readonly def
/FontBBox{0 -250 1171 750}readonly def
/UniqueXX 5087382 def
currentdict end
currentfile eexec
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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 1 /periodcentered put
dup 3 /asteriskmath put
dup 6 /plusminus put
dup 20 /lessequal put
dup 33 /arrowright put
dup 48 /prime put
dup 49 /infinity put
dup 50 /element put
dup 66 /B put
dup 77 /M put
dup 92 /intersection put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
/UniqueXX 5000818 def
currentdict end
currentfile eexec
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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 69 /E put
dup 86 /V put
dup 97 /a put
dup 99 /c put
dup 100 /d put
dup 105 /i put
dup 108 /l put
dup 110 /n put
dup 111 /o put
dup 114 /r put
readonly def
/FontBBox{-34 -251 988 750}readonly def
/UniqueXX 5000794 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMR17
%!PS-AdobeFont-1.1: CMR17 1.0
%%CreationDate: 1991 Aug 20 16:38:24

% 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 (CMR17) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR17 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 13 /fl put
dup 45 /hyphen put
dup 58 /colon put
dup 71 /G put
dup 76 /L put
dup 80 /P 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 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 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 119 /w put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-33 -250 945 749}readonly def
/UniqueXX 5000795 def
currentdict end
currentfile eexec
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$F2psEnd
rs

%%EndDocument
 @endspecial 1121 3115 a Fe(A)d(jet)h(of)h(\015uid)e(\\c)l(onstr)l
(aine)l(d)h(b)l(etwe)l(en)g(two)f(wal)t(ls")150 3651
y Fn(2)135 b(Existence)65 b(and)f(regularit)l(y)i(of)f(minimizers)g
(constrained)g(in)g(a)352 3800 y(strip)150 4003 y Fo(In)30
b(this)h(section,)h(w)m(e)f(recall)h(some)f(kno)m(wn)f(prop)s(erties)h
(of)f(existence)j(and)d(regularit)m(y)i(of)f(a)g(minimizer)150
4116 y(for)37 b(the)h(functional)g Fk(J)54 b Fo(when)37
b(w)m(e)h(prescrib)s(e)e(b)s(oundary)g(v)-5 b(alues)38
b(for)f Fi(u)g Fo(in)h(a)g(strip.)61 b(The)37 b(results)h(of)150
4229 y(this)d(section)h(are)f(an)g(o\013spring)f(of)h([A)m(C])h(and)e
([GG].)i(In)e(the)h(follo)m(wing,)j(for)d(a)g(v)m(ector)h
Fi(!)g Fk(2)c Ff(R)3521 4196 y Fg(n)3604 4229 y Fo(and)150
4342 y Fi(M)j Fk(2)25 b Ff(R)p Fo(,)31 b(w)m(e)g(denote)g(the)g(strip)e
(of)i(size)g Fi(M)41 b Fo(b)m(y)1237 4546 y Fi(S)1298
4508 y Fg(!)1293 4568 y(M)1397 4546 y Fo(:=)25 b Fk(f)p
Fi(x)h Fk(2)e Ff(R)1804 4508 y Fg(n)1892 4546 y Fo(s)p
Fi(:)p Fo(t)p Fi(:)41 b(x)21 b Fk(\001)f Fi(!)28 b Fk(2)d
Fo([0)p Fi(;)15 b(M)10 b Fo(])p Fk(g)15 b Fi(:)150 4750
y Fo(When)33 b Fi(!)g Fk(2)c Ff(Q)677 4717 y Fg(n)723
4750 y Fo(,)34 b(w)m(e)g(can)f(reco)m(v)m(er)i(compactness)f(b)m(y)f
(an)g(iden)m(ti\014cation)i(of)e(the)h(space.)49 b(Namely)-8
b(,)35 b(w)m(e)150 4863 y(consider)30 b(the)h(follo)m(wing)h(relation)f
Fk(\030)f Fo(induced)g(b)m(y)g Fi(!)s Fo(:)942 5067 y
Fi(x)25 b Fk(\030)g Fi(y)185 b Fo(i\013)k Fi(x)20 b Fk(\000)g
Fi(y)28 b Fo(=)d Fi(k)k Fk(2)c Ff(Z)2170 5030 y Fg(n)2307
5067 y Fo(with)91 b Fi(!)23 b Fk(\001)e Fi(k)28 b Fo(=)d(0)15
b Fi(:)606 b Fo(\(2.1\))150 5272 y(W)-8 b(e)41 b(will)f(denote)h(b)m(y)
951 5249 y(~)936 5272 y Fi(S)997 5239 y Fg(!)992 5299
y(M)1112 5272 y Fo(=)g Fi(S)1285 5239 y Fg(!)1280 5299
y(M)1358 5272 y Fi(=)h Fk(\030)e Fo(the)g(quotien)m(t)h(space)f
(induced)f(b)m(y)h(suc)m(h)g(relation)h(\(whic)m(h)f(is)150
5384 y(top)s(ologically)28 b(equiv)-5 b(alen)m(t)27 b(to)f(the)g(pro)s
(duct)e(of)h(an)h(\()p Fi(n)10 b Fk(\000)g Fo(1\))p Fk(\000)p
Fo(dimensional)26 b(torus)f(and)f(a)i(real)g(in)m(terv)-5
b(al\).)1927 5656 y(4)p eop
%%Page: 5 5
5 4 bop 291 399 a Fo(W)-8 b(e)32 b(also)f(set)1152 511
y Fk(X)1230 474 y Fg(!)1217 534 y(M)1321 511 y Fo(:=)25
b Fk(f)p Fi(u)h Fk(2)f Fo(L)1708 474 y Fj(1)1708 535
y(lo)r(c)1800 511 y Fo(\()1850 488 y(~)1835 511 y Fi(S)1896
474 y Fg(!)1891 534 y(M)1970 511 y Fo(\))41 b(s)p Fi(:)p
Fo(t)p Fi(:)g Fk(r)p Fi(u)25 b Fk(2)g Fo(L)2504 474 y
Fj(2)2543 511 y Fo(\()2594 488 y(~)2578 511 y Fi(S)2639
474 y Fg(!)2634 534 y(M)2713 511 y Fo(\))1133 678 y(with)30
b Fi(u)p Fo(\()p Fi(x)p Fo(\))c(=)f(1)30 b(for)h(an)m(y)f
Fi(x)41 b Fo(s)p Fi(:)p Fo(t)p Fi(:)g(!)23 b Fk(\001)d
Fi(x)25 b Fo(=)g(0)31 b(and)1220 845 y Fi(u)p Fo(\()p
Fi(x)p Fo(\))26 b(=)f Fk(\000)p Fo(1)30 b(for)g(an)m(y)h
Fi(x)41 b Fo(s)p Fi(:)p Fo(t)p Fi(:)g(!)23 b Fk(\001)d
Fi(x)25 b Fo(=)g Fi(M)10 b Fk(g)15 b Fi(:)150 1012 y
Fo(W)-8 b(e)30 b(also)g(denote)g(b)m(y)e Fk(L)965 979
y Fg(n)1041 1012 y Fo(the)h Fi(n)p Fk(\000)p Fo(dimensional)f(Leb)s
(esgue)h(measure.)40 b(In)29 b(the)g(sequel,)g(w)m(e)h(will)f(assume)
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7 6 bop 291 399 a Fo(W)-8 b(e)34 b(w)m(ould)e(lik)m(e)i(no)m(w)f(to)g
(deduce)g(some)g(other)g(densit)m(y)g(estimates,)i(that)e(will)g(b)s(e)
f(needed)g(in)h(the)150 511 y(application)e(of)e(the)h(Geometric)i
(Lemma)d(in)g(Section)i(6.)41 b(T)-8 b(o)30 b(do)f(this,)h(w)m(e)g
(need)f(some)h(b)s(etter)f(con)m(trol)150 624 y(on)35
b(the)g(\\width")h(of)f Fk(fj)p Fi(u)p Fk(j)f Fi(<)f
Fo(1)p Fk(g)j Fo(than)f(the)g(one)h(obtained)f(in)g(Lemma)g(3.2:)52
b(indeed,)36 b(in)f(Lemma)g(3.6)150 737 y(w)m(e)29 b(will)f(sho)m(w)g
(that)h Fm(b)-5 b(oth)29 b Fo(the)g(v)-5 b(alues)28 b(1)h(and)e
Fk(\000)p Fo(1)i(are)f(attained)i(in)e(a)g(univ)m(ersal)g(neigh)m(b)s
(orho)s(o)s(d)f(of)i(an)m(y)150 850 y(p)s(oin)m(t)38
b(in)f Fk(fj)p Fi(u)p Fk(j)h Fi(<)f Fo(1)p Fk(g)p Fo(.)64
b(This)37 b(result)g(will)h(b)s(e)f(a)h(consequence)g(of)g(some)g(prop)
s(erties)f(of)h(\\rigidit)m(y")i(of)150 963 y(minimizers,)29
b(related)g(to)g(the)f(\\linear)h(gro)m(wth)f(from)g(the)g(free)g(b)s
(oundary")f(pro)m(v)m(ed)h(b)m(y)g(Alt,)i(Ca\013arelli)150
1076 y(and)g(F)-8 b(riedman.)41 b(The)30 b(follo)m(wing)h(results)g
(will)f(lead)h(us)f(to)h(this)f(target.)291 1189 y(In)24
b(order)f(to)i(compare)g(the)g(functional)f(in)g(\(1.1\))i(with)e(the)h
(one)f(dealt)h(with,)h(for)e(instance,)i(in)e([A)m(C],)150
1302 y(w)m(e)31 b(observ)m(e)g(the)f(follo)m(wing:)150
1489 y Ff(Lemma)35 b(3.3.)46 b Fm(L)-5 b(et)39 b Fi(u)e
Fk(2)f(M)1187 1456 y Fg(!)1187 1516 y(M)1305 1489 y Fm(and)k(let)f
Fo(\012)f Fm(b)-5 b(e)39 b(an)h(op)-5 b(en)40 b(c)-5
b(onne)g(cte)g(d)40 b(set)f(c)-5 b(ontaine)g(d)41 b(in)e
Fi(S)3358 1456 y Fg(!)3353 1516 y(M)3431 1489 y Fm(,)i(with)f(a)150
1602 y(Lipschitz)34 b(b)-5 b(oundary.)43 b(If)33 b Fi(u)f
Fm(do)-5 b(es)34 b(not)f(take)g(the)g(value)g Fo(1)g
Fm(in)f Fo(\012)p Fm(,)g(then)h Fi(u)g Fm(minimizes)1338
1720 y Fh(Z)1388 1926 y Fj(\012)1458 1843 y Fi(a)1506
1857 y Fg(i;j)1602 1843 y Fi(@)1650 1857 y Fg(i)1678
1843 y Fi(v)19 b(@)1789 1857 y Fg(j)1825 1843 y Fi(v)39
b Fo(+)c Fi(Q)p Fo(\()p Fi(x)p Fo(\))p Fi(\037)2265 1862
y Fz(f)p Fg(v)r(>)p Fz(\000)p Fj(1)p Fz(g)2537 1843 y
Fi(;)150 2101 y Fm(among)f(al)5 b(l)33 b Fi(v)28 b Fk(2)d
Fo(L)788 2068 y Fj(1)788 2129 y(lo)r(c)880 2101 y Fo(\(\012\))33
b Fm(with)h Fk(r)p Fi(v)28 b Fk(2)d Fo(L)1538 2068 y
Fj(2)1577 2101 y Fo(\(\012\))33 b Fm(and)h Fi(v)28 b
Fo(=)d Fi(u)33 b Fm(on)g Fi(@)5 b Fo(\012)p Fm(.)291
2214 y(A)n(nalo)-5 b(gously,)34 b(if)e Fi(u)h Fm(do)-5
b(es)34 b(not)f(take)g(the)g(value)f Fk(\000)p Fo(1)h
Fm(in)g Fo(\012)p Fm(,)f(then)h Fi(u)f Fm(minimizes)1365
2331 y Fh(Z)1415 2538 y Fj(\012)1486 2455 y Fi(a)1534
2469 y Fg(i;j)1629 2455 y Fi(@)1677 2469 y Fg(i)1706
2455 y Fi(v)18 b(@)1816 2469 y Fg(j)1853 2455 y Fi(v)38
b Fo(+)e Fi(Q)p Fo(\()p Fi(x)p Fo(\))p Fi(\037)2293 2474
y Fz(f)p Fg(v)r(<)p Fj(1)p Fz(g)2510 2455 y Fi(;)150
2713 y Fm(among)e(al)5 b(l)33 b Fi(v)28 b Fk(2)d Fo(L)788
2680 y Fj(1)788 2741 y(lo)r(c)880 2713 y Fo(\(\012\))33
b Fm(with)h Fk(r)p Fi(v)28 b Fk(2)d Fo(L)1538 2680 y
Fj(2)1577 2713 y Fo(\(\012\))33 b Fm(and)h Fi(v)28 b
Fo(=)d Fi(u)33 b Fm(on)g Fi(@)5 b Fo(\012)p Fm(.)291
2901 y Fo(The)30 b(pro)s(of)g(is)g(straigh)m(tforw)m(ard.)150
3088 y Ff(Lemma)35 b(3.4.)46 b Fm(L)-5 b(et)33 b Fo(\012)f
Fm(b)-5 b(e)32 b(a)h(Lipschitz)h(domain.)43 b(L)-5 b(et)33
b Fi(u)g Fm(b)-5 b(e)32 b(a)h(minimizer)h(of)1227 3306
y Fo(~)1196 3329 y Fk(J)1258 3343 y Fj(\012)1312 3329
y Fo(\()p Fi(u)p Fo(\))26 b(:=)1581 3206 y Fh(Z)1632
3412 y Fj(\012)1702 3329 y Fi(a)1750 3343 y Fg(i;j)1830
3329 y Fi(@)1878 3343 y Fg(i)1907 3329 y Fi(u@)2007 3343
y Fg(j)2044 3329 y Fi(u)20 b Fo(+)g Fi(Q)p Fo(\()p Fi(x)p
Fo(\))p Fi(\037)2458 3348 y Fz(f)p Fg(u>)p Fj(0)p Fz(g)2679
3329 y Fi(;)859 b Fo(\(3.1\))150 3587 y Fm(among)37 b(al)5
b(l)36 b(the)g(functions)g Fi(v)f Fk(2)30 b Fo(L)1358
3554 y Fj(1)1358 3615 y(lo)r(c)1451 3587 y Fo(\(\012\))36
b Fm(such)g(that)h Fk(r)p Fi(v)d Fk(2)c Fo(L)2324 3554
y Fj(2)2364 3587 y Fo(\(\012\))36 b Fm(so)g(that)h Fi(v)d
Fo(=)d Fi(u)36 b Fm(on)g Fi(@)5 b Fo(\012)p Fm(.)51 b(Assume)150
3700 y(that)367 3677 y Fo(~)336 3700 y Fk(J)398 3714
y Fj(\012)453 3700 y Fo(\()p Fi(u)p Fo(\))26 b Fi(<)f
Fk(1)32 b Fm(and)i(that)f Fi(a)1230 3714 y Fg(i;j)1343
3700 y Fm(and)h Fi(Q)e Fm(satisfy)i(the)f(hyp)-5 b(otheses)35
b(state)-5 b(d)34 b(at)f(p)-5 b(age)33 b(2.)291 3813
y(Then,)g(for)g(any)g Fi(x)26 b Fk(2)e(f)p Fi(u)i(>)f
Fo(0)p Fk(g)p Fm(,)1436 4017 y Fo(dist\()p Fi(x;)15 b
Fk(f)p Fi(u)27 b Fo(=)e(0)p Fk(g)p Fo(\))h Fk(\024)f
Fi(C)7 b(u)p Fo(\()p Fi(x)p Fo(\))15 b Fi(;)150 4221
y Fm(pr)-5 b(ovide)g(d)31 b(that)e Fi(B)755 4240 y Fj(dist\()p
Fg(x;)p Fz(f)p Fg(u)p Fj(=0)p Fz(g)p Fj(\))1189 4221
y Fo(\()p Fi(x)p Fo(\))d Fk(\022)f Fo(\012)p Fm(,)k(wher)-5
b(e)29 b Fi(C)35 b Fm(is)28 b(a)h(p)-5 b(ositive)29 b(c)-5
b(onstant,)31 b(dep)-5 b(ending)29 b(only)g(on)g Fi(n)f
Fm(and)150 4334 y(on)33 b(the)g(b)-5 b(ounds)34 b(on)f
Fi(a)909 4348 y Fg(i;j)1022 4334 y Fm(and)g Fi(Q)p Fm(.)150
4522 y(Pr)-5 b(o)g(of.)47 b Fo(By)34 b(scaling,)j(w)m(e)e(ma)m(y)g
(assume)f Fi(x)e Fo(=)f(0)k(and)49 b(dist)15 b(\()p Fi(x;)g
Fk(f)p Fi(u)33 b Fo(=)f(0)p Fk(g)p Fo(\))h(=)f(1.)53
b(Set)34 b Fi(")e Fo(:=)g Fi(u)p Fo(\(0\).)54 b(Since)150
4635 y Fi(B)219 4649 y Fj(1)284 4635 y Fk(\022)25 b(f)p
Fi(u)g(>)g Fo(0)p Fk(g)p Fo(,)32 b(the)e(\014rst)g(v)-5
b(ariation)31 b(of)g(the)g(functional)f(sho)m(ws)g(that)h
Fi(@)2663 4649 y Fg(i)2692 4635 y Fo(\()p Fi(a)2775 4649
y Fg(i;j)2855 4635 y Fi(@)2903 4649 y Fg(j)2940 4635
y Fi(u)p Fo(\))26 b(=)f(0)30 b(on)h Fi(B)3420 4649 y
Fj(1)3459 4635 y Fo(.)41 b(Th)m(us,)150 4748 y(b)m(y)30
b(Harnac)m(k)i(inequalit)m(y)-8 b(,)1707 4861 y(sup)1700
4939 y Fg(B)1753 4953 y Fc(1)p Fd(=)p Fc(2)1865 4861
y Fi(u)25 b Fk(\024)g Fi(c)2077 4823 y Fz(\003)2117 4861
y Fi(")15 b(;)150 5101 y Fo(for)30 b(a)h(univ)m(ersal)f(constan)m(t)i
Fi(c)1155 5068 y Fz(\003)1195 5101 y Fo(.)1927 5656 y(7)p
eop
%%Page: 8 8
8 7 bop 291 399 a Fo(Let)29 b Fi(')f Fo(b)s(e)f(the)h(radially)h
(symmetric)f(function)f(de\014ned)g(as)h(follo)m(ws:)41
b Fi(')26 b Fo(=)f(0)j(in)f Fi(B)3115 417 y Fj(1)p Fg(=)p
Fj(4)3225 399 y Fo(,)i Fi(')d Fo(=)f(2)p Fi(c)3544 366
y Fz(\003)3584 399 y Fi(")j Fo(on)150 511 y Fi(@)5 b(B)272
530 y Fj(1)p Fg(=)p Fj(2)416 511 y Fo(and)32 b(gro)m(wing)i(linearly)g
(on)g(the)f(ra)m(ys)h(of)f Fi(B)1928 530 y Fj(1)p Fg(=)p
Fj(2)2060 511 y Fk(\000)22 b Fi(B)2222 530 y Fj(1)p Fg(=)p
Fj(4)2332 511 y Fo(.)50 b(Set)33 b Fi(v)h Fo(:=)c(min)o
Fk(f)p Fi(u;)15 b(')p Fk(g)p Fo(.)52 b(Then)32 b Fi(v)37
b Fo(is)c(an)150 624 y(admissible)d(function,)h(so)855
739 y Fh(Z)906 945 y Fg(B)959 959 y Fc(1)p Fd(=)p Fc(2)1075
863 y Fi(a)1123 877 y Fg(i;j)1203 863 y Fi(@)1251 877
y Fg(i)1280 863 y Fi(u@)1380 877 y Fg(j)1417 863 y Fi(u)20
b Fo(+)g Fi(Q\037)1709 881 y Fz(f)p Fg(u>)p Fj(0)p Fz(g)1940
863 y Fk(\024)2035 739 y Fh(Z)2086 945 y Fg(B)2139 959
y Fc(1)p Fd(=)p Fc(2)2255 863 y Fi(a)2303 877 y Fg(i;j)2384
863 y Fi(@)2432 877 y Fg(i)2460 863 y Fi(v)s(@)2555 877
y Fg(j)2592 863 y Fi(v)k Fo(+)c Fi(Q\037)2880 881 y Fz(f)p
Fg(v)r(>)p Fj(0)p Fz(g)3106 863 y Fk(\024)855 1129 y(\024)951
1005 y Fh(Z)1002 1211 y Fg(B)1055 1225 y Fc(1)p Fd(=)p
Fc(2)1171 1129 y Fi(a)1219 1143 y Fg(i;j)1299 1129 y
Fi(@)1347 1143 y Fg(i)1376 1129 y Fi(u@)1476 1143 y Fg(j)1513
1129 y Fi(u)g Fo(+)1676 1005 y Fh(Z)1726 1211 y Fg(B)1779
1225 y Fc(1)p Fd(=)p Fc(2)1896 1129 y Fi(a)1944 1143
y Fg(i;j)2024 1129 y Fi(@)2072 1143 y Fg(i)2100 1129
y Fi('@)2207 1143 y Fg(j)2245 1129 y Fi(')g Fo(+)2415
1005 y Fh(Z)2466 1211 y Fg(B)2519 1225 y Fc(1)p Fd(=)p
Fc(2)2616 1211 y Fz(\000)p Fg(B)2724 1225 y Fc(1)p Fd(=)p
Fc(4)2840 1129 y Fi(Q\037)2969 1147 y Fz(f)p Fg(v)r(>)p
Fj(0)p Fz(g)3186 1129 y Fi(:)150 1403 y Fo(Since)30 b
Fk(jr)p Fi(')p Fk(j)c(\024)40 b Fo(const)16 b Fi(")31
b Fo(and)f Fi(u)25 b(>)g Fo(0)31 b(on)f Fi(B)1626 1422
y Fj(1)p Fg(=)p Fj(4)1736 1403 y Fo(,)g(the)h(previous)f(inequalit)m(y)
i(implies)846 1659 y Fi(")888 1621 y Fj(2)953 1659 y
Fk(\025)1049 1535 y Fh(Z)1099 1741 y Fg(B)1152 1755 y
Fc(1)p Fd(=)p Fc(2)1269 1659 y Fi(Q\037)1398 1677 y Fz(f)p
Fg(u>)p Fj(0)p Fz(g)1623 1659 y Fk(\000)1714 1535 y Fh(Z)1765
1741 y Fg(B)1818 1755 y Fc(1)p Fd(=)p Fc(2)1915 1741
y Fz(\000)p Fg(B)2023 1755 y Fc(1)p Fd(=)p Fc(4)2139
1659 y Fi(Q\037)2268 1677 y Fz(f)p Fg(v)r(>)p Fj(0)p
Fz(g)2495 1659 y Fk(\025)25 b Fi(Q)2663 1673 y Fj(min)2784
1659 y Fk(j)p Fi(B)2878 1677 y Fj(1)p Fg(=)p Fj(4)2989
1659 y Fk(j)15 b Fi(:)p 3684 1927 4 62 v 3688 1869 55
4 v 3688 1927 V 3742 1927 4 62 v 150 2114 a Ff(Lemma)35
b(3.5.)46 b Fm(L)-5 b(et)28 b Fo(\012)p Fm(,)h Fi(u)f
Fm(and)1287 2091 y Fo(~)1256 2114 y Fk(J)44 b Fm(b)-5
b(e)28 b(as)h(in)f(L)-5 b(emma)29 b(3.4.)41 b(L)-5 b(et)28
b Fi(x)d Fk(2)g Fi(@)5 b Fk(f)p Fi(u)26 b(>)f Fo(0)p
Fk(g)p Fm(.)41 b(Then,)30 b(for)e(any)h Fi(r)f Fk(\025)d
Fo(0)p Fm(,)1710 2315 y Fo(sup)1687 2398 y Fg(B)1740
2406 y Fd(r)1775 2398 y Fj(\()p Fg(x)p Fj(\))1884 2315
y Fi(u)h Fk(\025)f Fi(C)7 b(r)17 b(;)150 2589 y Fm(wher)-5
b(e)42 b Fi(C)47 b Fm(is)40 b(a)h(p)-5 b(ositive)42 b(c)-5
b(onstant)42 b(\(dep)-5 b(ending)42 b(only)g(on)f Fi(n)f
Fm(and)i(on)f(the)g(b)-5 b(ounds)42 b(on)f Fi(a)3308
2603 y Fg(i;j)3428 2589 y Fm(and)h Fi(Q)p Fm(\),)150
2701 y(pr)-5 b(ovide)g(d)35 b Fi(B)578 2715 y Fg(r)616
2701 y Fo(\()p Fi(x)p Fo(\))26 b Fk(\022)f Fo(\012)p
Fm(.)150 2887 y(Pr)-5 b(o)g(of.)47 b Fo(By)36 b(scaling,)j(w)m(e)d(ma)m
(y)g(assume)g Fi(x)e Fo(=)g(0)i(and)g Fi(r)g Fo(=)f(1.)57
b(Since)36 b(0)f Fk(2)f Fi(@)5 b Fk(f)p Fi(u)35 b(>)f
Fo(0)p Fk(g)p Fo(,)k(for)e(an)m(y)g Fi(")f(>)f Fo(0)150
3000 y(there)d(exists)g(a)g(p)s(oin)m(t)f Fi(x)999 3014
y Fj(1)1064 3000 y Fk(2)25 b Fi(B)1219 3014 y Fg(")1256
3000 y Fo(,)31 b(so)f(that)i Fi(u)p Fo(\()p Fi(x)1760
3014 y Fj(1)1799 3000 y Fo(\))26 b Fi(>)f Fo(0.)42 b(W)-8
b(e)32 b(no)m(w)e(construct)h(a)g(p)s(olygonal,)g(starting)h(as)150
3113 y(close)g(as)e(w)m(e)h(wish)f(to)h(zero,)g(along)h(whic)m(h)e
Fi(u)g Fo(gro)m(ws)h(up)e(at)i(least)h(linearly)-8 b(.)42
b(Set)1228 3314 y Fi(d)1275 3328 y Fj(1)1340 3314 y Fo(:=)f(dist)15
b(\()p Fi(x)1726 3328 y Fj(1)1766 3314 y Fi(;)g(@)5 b
Fk(f)p Fi(u)26 b(>)f Fo(0)p Fk(g)p Fo(\))i Fk(\024)e(j)p
Fi(x)2403 3328 y Fj(1)2442 3314 y Fk(j)h(\024)f Fi(")15
b(:)150 3516 y Fo(By)31 b(Lemma)f(3.4,)1679 3628 y Fi(u)p
Fo(\()p Fi(x)1818 3642 y Fj(1)1858 3628 y Fo(\))c Fk(\025)f
Fi(c)2054 3591 y Fz(\003)2093 3628 y Fi(d)2140 3642 y
Fj(1)2195 3628 y Fi(;)1343 b Fo(\(3.2\))150 3794 y(for)30
b(a)h(suitable)g(constan)m(t)g Fi(c)1109 3761 y Fz(\003)1149
3794 y Fo(.)41 b(Let)31 b Fi(z)1420 3808 y Fj(1)1485
3794 y Fk(2)25 b Fi(@)5 b(B)1693 3809 y Fg(d)1729 3818
y Fc(1)1768 3794 y Fo(\()p Fi(x)1855 3808 y Fj(1)1895
3794 y Fo(\))20 b Fk(\\)g(f)p Fi(u)26 b Fo(=)f(0)p Fk(g)p
Fo(.)41 b(De\014ne)945 3995 y Fi(\032)992 4009 y Fj(1)1057
3995 y Fo(:=)25 b(sup)o Fk(f)p Fi(\032)41 b Fo(s)p Fi(:)p
Fo(t)p Fi(:)g(B)1679 4009 y Fg(\032)1719 3995 y Fo(\()p
Fi(z)1796 4009 y Fj(1)1836 3995 y Fo(\))26 b Fk(\022)f
Fi(B)2062 4009 y Fj(1)2141 3995 y Fo(and)93 b(sup)2343
4078 y Fg(B)2396 4086 y Fd(\032)2433 4078 y Fj(\()p Fg(z)2493
4087 y Fc(1)2528 4078 y Fj(\))2570 3995 y Fi(u)25 b Fk(\025)g
Fi(c)2782 3958 y Fz(\003)2822 3995 y Fi(\032)p Fk(g)15
b Fi(:)150 4270 y Fo(By)31 b(\(3.2\),)h Fi(\032)582 4284
y Fj(1)647 4270 y Fk(\025)25 b Fi(d)790 4284 y Fj(1)830
4270 y Fo(.)40 b(Also,)31 b(if)g Fi(B)1278 4284 y Fg(\032)1314
4293 y Fc(1)1353 4270 y Fo(\()p Fi(z)1430 4284 y Fj(1)1470
4270 y Fo(\))f(touc)m(hes)i Fi(@)5 b(B)1984 4284 y Fj(1)2023
4270 y Fo(,)31 b(then)1146 4514 y Fi(\032)1193 4528 y
Fj(1)1258 4514 y Fo(=)25 b(1)c Fk(\000)f(j)p Fi(z)1578
4528 y Fj(1)1618 4514 y Fk(j)25 b(\025)g Fo(1)c Fk(\000)f
Fi(d)1968 4528 y Fj(1)2028 4514 y Fk(\000)g Fi(")25 b
Fk(\025)g Fo(1)c Fk(\000)f Fo(2)p Fi(")26 b Fk(\025)2658
4452 y Fo(1)p 2658 4493 46 4 v 2658 4576 a(2)2728 4514
y Fi(;)150 4741 y Fo(if)k Fi(")c Fk(\024)f Fo(1)p Fi(=)p
Fo(4,)32 b(so)1501 4897 y(sup)1525 4976 y Fg(B)1578 4985
y Fc(1)1653 4897 y Fi(u)25 b Fk(\025)79 b Fo(sup)1826
4980 y Fg(B)1879 4988 y Fd(\032)1911 5003 y Fc(1)1950
4980 y Fj(\()p Fg(z)2010 4989 y Fc(1)2044 4980 y Fj(\))2087
4897 y Fi(u)25 b Fk(\025)2270 4836 y Fi(c)2309 4803 y
Fz(\003)p 2270 4877 79 4 v 2287 4960 a Fo(2)2374 4897
y Fi(;)150 5136 y Fo(and)30 b(the)g(claim)i(w)m(ould)e(b)s(e)g(pro)m(v)
m(en.)41 b(Th)m(us,)29 b(w)m(e)i(ma)m(y)g(restrict)g(ourselv)m(es)g(to)
g(the)g(case)g(in)f(whic)m(h)1685 5338 y(sup)1630 5421
y Fg(B)1683 5429 y Fd(\032)1715 5444 y Fc(1)1754 5421
y Fj(\()p Fg(z)1814 5430 y Fc(1)1849 5421 y Fj(\))1891
5338 y Fi(u)25 b Fo(=)g Fi(c)2103 5300 y Fz(\003)2143
5338 y Fi(\032)2190 5352 y Fj(1)2245 5338 y Fi(:)1293
b Fo(\(3.3\))1927 5656 y(8)p eop
%%Page: 9 9
9 8 bop 150 399 a Fo(T)-8 b(ak)m(e)30 b(~)-51 b Fi(x)414
413 y Fj(1)479 399 y Fk(2)586 376 y Fo(\026)565 399 y
Fi(B)634 413 y Fg(\032)670 422 y Fc(1)708 399 y Fo(\()p
Fi(z)785 413 y Fj(1)825 399 y Fo(\))23 b(attaining)i(the)d(sup)g(in)g
(\(3.3\).)40 b(In)22 b(fact,)k(b)m(y)c(Maxim)m(um)h(Principle,)30
b(~)-50 b Fi(x)3225 413 y Fj(1)3289 399 y Fk(2)25 b Fi(@)5
b(B)3497 413 y Fg(\032)3533 422 y Fc(1)3572 399 y Fo(\()p
Fi(z)3649 413 y Fj(1)3689 399 y Fo(\).)150 511 y(W)-8
b(e)32 b(w)m(an)m(t)f(no)m(w)f(to)h(sho)m(w)g(that)g(there)f(exist)h
(some)g(univ)m(ersal)g Fi(\016)e(>)c Fo(0)30 b(suc)m(h)g(that)1517
710 y(sup)1447 792 y Fg(B)1500 801 y Fc(2)p Fd(\032)1562
816 y Fc(1)1601 792 y Fj(\()p Fg(z)1661 801 y Fc(1)1696
792 y Fj(\))1739 710 y Fi(u)25 b Fk(\025)g Fo(\(1)c(+)f
Fi(\016)s Fo(\))15 b Fi(u)p Fo(\()6 b(~)-51 b Fi(x)2336
724 y Fj(1)2377 710 y Fo(\))15 b Fi(;)1111 b Fo(\(3.4\))150
987 y(pro)m(vided)30 b Fi(B)593 1001 y Fj(2)p Fg(\032)664
1010 y Fc(1)703 987 y Fo(\()p Fi(z)780 1001 y Fj(1)820
987 y Fo(\))c Fk(\022)f Fi(B)1046 1001 y Fj(1)1085 987
y Fo(.)41 b(T)-8 b(o)31 b(pro)m(v)m(e)g(this,)f(assume)1537
1185 y(sup)1467 1268 y Fg(B)1520 1277 y Fc(2)p Fd(\032)1582
1292 y Fc(1)1622 1268 y Fj(\()p Fg(z)1682 1277 y Fc(1)1716
1268 y Fj(\))1759 1185 y Fi(u)25 b(<)g Fo(\(1)c(+)f Fi(\016)s
Fo(\))15 b Fi(u)p Fo(\()6 b(~)-51 b Fi(x)2356 1199 y
Fj(1)2397 1185 y Fo(\))150 1458 y(and)30 b(de\014ne)1212
1586 y Fi(v)s Fo(\()p Fi(x)p Fo(\))c(:=)1559 1525 y(1)p
1538 1565 87 4 v 1538 1649 a Fi(\032)1585 1663 y Fj(1)1635
1586 y Fi(u)p Fo(\()p Fi(z)1764 1600 y Fj(1)1824 1586
y Fo(+)20 b Fi(\032)1962 1600 y Fj(1)2001 1586 y Fi(x)p
Fo(\))15 b Fi(;)198 b Fk(8)p Fi(x)25 b Fk(2)f Fi(B)2608
1600 y Fj(2)2663 1586 y Fi(:)150 1789 y Fo(Then)1171
1927 y Fk(jj)p Fi(v)s Fk(jj)1318 1945 y Fj(L)1362 1926
y Fq(1)1429 1945 y Fj(\()p Fg(B)1509 1954 y Fc(2)1544
1945 y Fj(\))1601 1927 y Fk(\024)1707 1865 y Fo(\(1)d(+)f
Fi(\016)s Fo(\))15 b Fi(u)p Fo(\()6 b(~)-51 b Fi(x)2131
1879 y Fj(1)2172 1865 y Fo(\))p 1707 1906 501 4 v 1914
1989 a Fi(\032)1961 2003 y Fj(1)2242 1927 y Fo(=)25 b(\(1)c(+)f
Fi(\016)s Fo(\))p Fi(c)2647 1889 y Fz(\003)2703 1927
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(uniform)f(b)s(ound)e(on)j Fk(jj)p Fi(v)s Fk(jj)2980
2152 y Fj(L)3024 2134 y Fq(1)3090 2152 y Fj(\()p Fg(B)3170
2161 y Fc(2)3205 2152 y Fj(\))3237 2134 y Fo(.)65 b(So,)41
b(b)m(y)e(the)150 2247 y(Theorem)30 b(of)h(Giaquin)m(ta)h(and)d(Giusti)
i(\(see)g([GG])h(or)e([G2]\),)j Fi(v)28 b Fk(2)d Fo(C)2513
2214 y Fg(\013)2562 2247 y Fo(\()p Fi(B)2666 2265 y Fj(3)p
Fg(=)p Fj(2)2776 2247 y Fo(\))31 b(and)1623 2445 y Fk(jj)p
Fi(v)s Fk(jj)1770 2463 y Fj(C)1821 2445 y Fd(\013)1867
2463 y Fj(\()p Fg(B)1947 2477 y Fc(3)p Fd(=)p Fc(2)2044
2463 y Fj(\))2101 2445 y Fk(\024)c Fo(~)-47 b Fi(c)16
b(;)1286 b Fo(\(3.6\))150 2643 y(for)30 b(a)h(univ)m(ersal)h(~)-47
b Fi(c)q Fo(.)40 b(Set)1560 2771 y Fi(w)27 b Fo(:=)1912
2710 y(2)p Fi(v)p 1784 2751 350 4 v 1784 2834 a(c)1823
2808 y Fz(\003)1862 2834 y Fo(\(1)21 b(+)f Fi(\016)s
Fo(\))2164 2771 y Fk(\000)g Fo(1)15 b Fi(;)150 2984 y
Fo(so)28 b(that)h Fi(@)502 2998 y Fg(i)530 2984 y Fo(\()p
Fi(a)613 2998 y Fg(i;j)694 2984 y Fi(@)742 2998 y Fg(j)779
2984 y Fi(w)r Fo(\))d Fk(\025)f Fo(0)j(in)g Fi(B)1249
3002 y Fj(3)p Fg(=)p Fj(2)1387 2984 y Fo(and)f Fi(w)h
Fk(\024)d Fo(1)j(b)s(ecause)h(of)f(\(3.5\).)41 b(Also,)30
b(b)m(y)e(\(3.6\))i(and)d(the)h(fact)h(that)150 3097
y Fi(w)r Fo(\(0\))h(=)f Fk(\000)p Fo(1,)k(w)m(e)g(deduce)f(that)h
Fk(L)1339 3064 y Fg(n)1386 3097 y Fo(\()p Fk(f)p Fi(w)f
Fk(\024)c Fo(0)p Fk(g)23 b(\\)e Fi(B)1926 3115 y Fj(3)p
Fg(=)p Fj(2)2036 3097 y Fo(\))29 b Fk(\025)43 b Fo(const)16
b(.)47 b(Then,)32 b(b)m(y)g(De)i(Giorgi)f(Oscillation)150
3210 y(Lemma)21 b(\(see,)k(for)c(instance)h(the)f(App)s(endix)e(of)j
([C])f(or)g(Theorem)g(4.9)i(in)e([HL])g(and)g(references)g(therein\),)
150 3323 y(w)m(e)31 b(infer)f(sup)631 3345 y Fg(B)684
3354 y Fc(1)738 3323 y Fi(w)d Fk(\024)e Fi(\025)p Fo(,)31
b(for)f(a)h(univ)m(ersal)f Fi(\025)c Fk(2)f Fo(\(0)p
Fi(;)15 b Fo(1\).)42 b(In)30 b(particular,)h(since)36
b(~)-50 b Fi(x)2900 3337 y Fj(1)2964 3323 y Fk(2)3072
3300 y Fo(\026)3050 3323 y Fi(B)3119 3337 y Fg(\032)3155
3346 y Fc(1)3194 3323 y Fo(\()p Fi(z)3271 3337 y Fj(1)3311
3323 y Fo(\),)1333 3579 y Fi(\025)26 b Fk(\025)f Fi(w)1590
3451 y Fh(\022)1673 3518 y Fo(~)-51 b Fi(x)1719 3532
y Fj(1)1779 3518 y Fk(\000)20 b Fi(z)1912 3532 y Fj(1)p
1667 3558 285 4 v 1766 3641 a Fi(\032)1813 3655 y Fj(1)1961
3451 y Fh(\023)2053 3579 y Fo(=)2237 3518 y(2)p 2159
3558 201 4 v 2159 3641 a(1)h(+)f Fi(\016)2390 3579 y
Fk(\000)g Fo(1)15 b Fi(;)150 3827 y Fo(whic)m(h)32 b(is)h(a)g(con)m
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b(This)32 b(pro)m(v)m(es)h(\(3.4\).)49 b(By)33 b(construction,)h
Fi(u)p Fo(\()6 b(~)-51 b Fi(x)3575 3841 y Fj(1)3615 3827
y Fo(\))29 b Fk(\025)150 3940 y Fi(u)p Fo(\()p Fi(x)289
3954 y Fj(1)329 3940 y Fo(\).)44 b(Hence,)32 b(there)g(exists)g(a)g(p)s
(oin)m(t)f Fi(x)1582 3954 y Fj(2)1648 3940 y Fk(2)1758
3917 y Fo(\026)1736 3940 y Fi(B)1805 3954 y Fj(2)p Fg(\032)1876
3963 y Fc(1)1915 3940 y Fo(\()p Fi(z)1992 3954 y Fj(1)2032
3940 y Fo(\))h(\(and)f(in)g(fact)h Fi(x)2651 3954 y Fj(2)2718
3940 y Fk(2)26 b Fi(@)5 b(B)2927 3954 y Fj(2)p Fg(\032)2998
3963 y Fc(1)3038 3940 y Fo(\()p Fi(z)3115 3954 y Fj(1)3155
3940 y Fo(\))31 b(b)m(y)g(Maxim)m(um)150 4053 y(Principle\))g(suc)m(h)f
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4251 y Fo(\))25 b Fk(\025)g Fi(u)p Fo(\()6 b(~)-51 b
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4214 y Fz(\003)1860 4251 y Fi(\032)1907 4265 y Fj(1)1972
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4675 y Fi(x)202 4689 y Fg(j)264 4675 y Fk(2)25 b Fi(@)5
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Fj(1)1367 4675 y Fk(j)25 b(\025)g Fi(\032)1560 4689 y
Fg(j)1627 4675 y Fo(and)1263 4873 y Fi(u)p Fo(\()p Fi(x)1402
4887 y Fg(j)1438 4873 y Fo(\))84 b Fk(\025)f Fo(\(1)21
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y Fo(\))26 b Fk(\025)f Fi(\022)s(`)2133 5421 y Fg(j)1927
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%%Page: 10 10
10 9 bop 150 399 a Fo(where)34 b Fi(`)455 413 y Fg(j)527
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(suitable)g Fi(j)2997 754 y Fz(\003)3069 787 y Fo(suc)m(h)f(that)i
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1519 1157 a(20)1620 1126 y Fk(\000)20 b Fi(")p Fo(.)p
3684 1126 4 62 v 3688 1068 55 4 v 3688 1126 V 3742 1126
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1528 y(M)1284 1501 y Fm(.)72 b(Assume)42 b(that)i Fi(u)p
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b(the)f(values)g Fo(1)g Fm(and)h Fk(\000)p Fo(1)150 1727
y Fm(in)i Fi(B)331 1741 y Fg(r)363 1750 y Fc(0)401 1727
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b(ovide)g(d)35 b(that)f Fi(B)1198 1741 y Fg(r)1230 1750
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1914 y(Pr)-5 b(o)g(of.)47 b Fo(By)31 b(Lemma)g(3.2,)h
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b Fk(\000)p Fo(1)g(in)g Fi(B)2306 1928 y Fg(r)2338 1937
y Fc(0)2376 1914 y Fo(\()6 b(\026)-51 b Fi(x)p Fo(\).)42
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(tradiction.)p 3684 2622 V 3688 2564 55 4 v 3688 2622
V 3742 2622 4 62 v 291 2810 a(W)-8 b(e)32 b(are)e(no)m(w)h(in)f(p)s
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Fm(and)g Fi(Q)p Fm(,)g(such)f(that)1574 3315 y Fk(J)1636
3333 y Fg(B)1689 3341 y Fd(r)1723 3333 y Fj(\()p Fg(x)p
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eop
%%Page: 11 11
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3460 y Fd(!)2087 3506 y(M)2170 3422 y Fi(u)p Fo(\()p
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4071 y(follo)m(wing)32 b(Prop)s(osition)e(4.3.)42 b(F)-8
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%%Page: 12 12
12 11 bop 150 399 a Fn(5)135 b(Birkho\013)45 b(prop)t(ert)l(y)g(for)g
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1876 y Fg(n)1398 1909 y Fm(.)41 b(Assume)31 b(that)i
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b Fk(2)d Ff(Z)691 2770 y Fg(n)770 2803 y Fm(and)34 b
Fi(k)23 b Fk(\001)e Fi(!)28 b Fk(\024)d Fo(0)p Fm(,)33
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2803 y Fi(>)25 b(\022)s Fk(g)g(\023)g(fT)2156 2818 y
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3288 y Fk(T)200 3303 y Fg(k)242 3288 y Fi(S)303 3255
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3315 y(M)1385 3288 y Fo(is)f(the)h(minimal)g(minimizer)f(in)g
Fk(T)2567 3303 y Fg(k)2609 3288 y Fi(S)2670 3255 y Fg(!)2665
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3490 y Fk(fT)1317 3505 y Fg(k)1359 3490 y Fi(u)1411 3452
y Fg(!)1411 3512 y(M)1516 3490 y Fi(>)25 b(\022)s Fk(g)g(\022)g(f)p
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Fg(!)2346 3512 y(M)2451 3490 y Fi(>)g(\022)s Fk(g)15
b Fi(:)150 3691 y Fo(Analogously)-8 b(,)32 b(if)f Fi(k)23
b Fk(\001)e Fi(!)28 b Fk(\025)d Fo(0,)31 b Fi(v)i Fo(is)e(a)f
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3691 y Fo(,)g(so)1305 3892 y Fk(f)p Fi(u)1402 3855 y
Fg(!)1402 3915 y(M)1507 3892 y Fi(>)25 b(\022)s Fk(g)g(\022)g(f)p
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3892 y Fi(u)2430 3855 y Fg(!)2430 3915 y(M)2509 3892
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%%Page: 15 15
15 14 bop 291 399 a Fo(W)-8 b(e)29 b(no)m(w)f(pro)m(v)m(e)h(that,)h(if)
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%%Page: 16 16
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Fi(H)7 b Fo(.)p 3684 3085 4 62 v 3688 3027 55 4 v 3688
3085 V 3742 3085 4 62 v 291 3273 a(In)32 b(order)f(to)i(clearly)h
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y(w)m(e)j(de\014ne)1114 3499 y Fi(S)1175 3461 y Fg(!)1170
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y Fo(and)1125 3779 y Fk(X)1203 3741 y Fg(!)1190 3801
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Fo(\()1877 3756 y(~)1861 3779 y Fi(S)1922 3741 y Fg(!)1917
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3801 y(M)2740 3779 y Fo(\))1096 3946 y(with)30 b Fi(u)p
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eop
%%Page: 17 17
17 16 bop 150 399 a Ff(Remark)35 b(7.3.)46 b Fo(F)-8
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%%Page: 18 18
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%%Page: 21 21
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%%Page: 22 22
22 21 bop 150 399 a Fm(Pr)-5 b(o)g(of.)47 b Fo(De\014ne)1249
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%%Page: 23 23
23 22 bop 150 399 a Ff(Prop)s(osition)36 b(10.5.)47 b
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%%Page: 24 24
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25 24 bop 291 399 a Fo(The)32 b(reader)g(ma)m(y)h(easily)g(con)m(vince)
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Fk(\\)20 b(f)p Fi(x)1529 4066 y Fg(n)1602 4052 y Fi(>)25
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Fk(\024)g Fi(v)s Fo(\()p Fi(x)p Fo(\).)291 4165 y(Hence,)51
b(b)m(y)c(Maxim)m(um)f(Principle,)51 b Fi(v)k Fk(\025)c
Fi(\020)i Fo(in)46 b Fi(B)2132 4179 y Fj(1)2171 4165
y Fo(.)89 b(In)45 b(particular,)51 b(b)m(y)46 b(\(12.4\))j(and)c
(\(12.5\),)150 4278 y Fi(v)s Fo(\(0)p Fi(;)15 b(:)g(:)g(:)j(;)d
Fo(0)p Fi(;)g Fk(\000)p Fo(1)21 b(+)f Fi(\022)s Fo(\))25
b Fk(\025)g Fi(\026)p Fo(\()p Fi(\014)h Fk(\000)20 b
Fi(c\022)1338 4245 y Fg(\013)1407 4278 y Fk(\000)g Fi(\022)s
Fo(\))k Fk(\025)h Fi(\026\014)5 b(=)p Fo(2,)32 b(if)e
Fi(\022)j Fo(is)d(small)h(enough)f(with)g(resp)s(ect)h(to)g
Fi(\014)5 b Fo(.)p 3684 4278 4 62 v 3688 4220 55 4 v
3688 4278 V 3742 4278 4 62 v 291 4464 a(W)-8 b(e)33 b(w)m(ould)f(lik)m
(e)h(to)g(remark)e(that)i(an)e(alternativ)m(e)k(pro)s(of)c(of)h(the)g
(Lemma)g(ab)s(o)m(v)m(e)h(can)f(b)s(e)f(accom-)150 4576
y(plished)f(b)m(y)g(using)g(the)g(b)s(oundary)f(Harnac)m(k)i(inequalit)
m(y)-8 b(.)150 4753 y Ff(Lemma)35 b(12.2.)46 b Fm(L)-5
b(et)33 b Fi(u)25 b Fk(2)g(M)1210 4720 y Fg(!)1210 4780
y(M)1289 4753 y Fm(.)42 b(Supp)-5 b(ose)34 b(that)g Fi(x)1943
4767 y Fj(0)2008 4753 y Fk(2)25 b Fi(S)2155 4720 y Fg(!)2150
4780 y(M)2261 4753 y Fm(veri\014es)1208 4943 y Fo(dist)15
b(\()p Fi(x)1457 4957 y Fj(0)1497 4943 y Fi(;)g(@)5 b(S)1651
4906 y Fg(!)1646 4966 y(M)1725 4943 y Fo(\))26 b Fi(>)f
Fo(2)15 b(dist)g(\()p Fi(x)2191 4957 y Fj(0)2231 4943
y Fi(;)g Fk(fj)p Fi(u)p Fk(j)27 b Fo(=)e(1)p Fk(g)p Fo(\))15
b Fi(:)150 5134 y Fm(Then,)944 5247 y Fo(min)o Fk(f)p
Fo(1)21 b Fk(\000)f Fi(u)p Fo(\()p Fi(x)1436 5261 y Fj(0)1476
5247 y Fo(\))p Fi(;)31 b Fo(1)21 b(+)f Fi(u)p Fo(\()p
Fi(x)1863 5261 y Fj(0)1902 5247 y Fo(\))p Fk(g)26 b(\024)f
Fi(C)d Fo(dist)15 b(\()p Fi(x)2440 5261 y Fj(0)2480 5247
y Fi(;)g Fk(fj)p Fi(u)p Fk(j)27 b Fo(=)e(1)p Fk(g)p Fo(\))15
b Fi(;)150 5407 y Fm(wher)-5 b(e)34 b Fi(C)39 b Fm(is)32
b(a)h(suitable)g(c)-5 b(onstant,)34 b(dep)-5 b(ending)34
b(only)g(on)f Fi(n)p Fm(,)f(the)h(b)-5 b(ounds)34 b(on)f
Fi(a)2902 5421 y Fg(i;j)3014 5407 y Fm(and)h Fi(F)13
b Fm(.)1905 5656 y Fo(25)p eop
%%Page: 26 26
26 25 bop 150 399 a Fm(Pr)-5 b(o)g(of.)47 b Fo(If)30
b Fi(u)p Fo(\()p Fi(x)665 413 y Fj(0)705 399 y Fo(\))c(=)f
Fk(\006)p Fo(1,)31 b(there)g(is)g(nothing)f(to)i(pro)m(v)m(e.)42
b(Set)31 b Fi(U)2304 413 y Fz(\006)2389 399 y Fo(=)25
b(1)c Fk(\006)f Fi(u)p Fo(,)31 b(and)f Fi(d)2974 413
y Fj(0)3040 399 y Fo(:=)40 b(dist)15 b(\()p Fi(x)3425
413 y Fj(0)3465 399 y Fi(;)g Fk(fj)p Fi(u)p Fk(j)27 b
Fo(=)150 511 y(1)p Fk(g)p Fo(\).)291 624 y(W)-8 b(e)46
b(assume)e(that)h(min)p Fk(f)p Fi(U)1260 638 y Fj(+)1319
624 y Fo(\()p Fi(x)1406 638 y Fj(0)1446 624 y Fo(\))p
Fi(;)31 b(U)1599 638 y Fz(\000)1658 624 y Fo(\()p Fi(x)1745
638 y Fj(0)1785 624 y Fo(\))p Fk(g)49 b(\025)f Fi(M)10
b(d)2178 638 y Fj(0)2218 624 y Fo(,)48 b(and)c(deriv)m(e)h(an)g(upp)s
(er)d(b)s(ound)g(for)j Fi(M)10 b Fo(.)150 737 y(Notice)32
b(that)f Fi(B)701 756 y Fj(9)p Fg(d)772 765 y Fc(0)807
756 y Fg(=)p Fj(10)917 737 y Fo(\()p Fi(x)1004 751 y
Fj(0)1044 737 y Fo(\))26 b Fk(\032)f(fj)p Fi(u)p Fk(j)h
Fi(<)f Fo(1)p Fk(g)p Fo(,)31 b(so)g(b)m(y)f(Lemma)h(2.4)g(and)f(Harnac)
m(k)h(inequalit)m(y)1087 936 y(inf)975 997 y Fg(B)1028
1011 y Fc(3)p Fd(d)1090 1026 y Fc(0)1125 1011 y Fd(=)p
Fc(4\()p Fd(x)1246 1026 y Fc(0)1281 1011 y(\))1324 936
y Fi(U)1386 950 y Fz(\006)1471 936 y Fk(\025)40 b Fo(const)c
Fk(\001)119 b Fo(sup)1870 1014 y Fg(B)1923 1028 y Fc(3)p
Fd(d)1985 1043 y Fc(0)2020 1028 y Fd(=)p Fc(4\()p Fd(x)2141
1043 y Fc(0)2176 1028 y(\))2219 936 y Fi(U)2281 950 y
Fz(\006)2366 936 y Fk(\025)40 b Fo(const)16 b Fi(M)10
b(d)2845 950 y Fj(0)2900 936 y Fi(:)593 b Fo(\(12.6\))150
1215 y(Let)36 b Fi(y)g Fk(2)c Fi(B)561 1230 y Fg(d)597
1239 y Fc(0)636 1215 y Fo(\()p Fi(x)723 1229 y Fj(0)763
1215 y Fo(\))24 b Fk(\\)f(fj)p Fi(u)p Fk(j)34 b Fo(=)e(1)p
Fk(g)p Fo(.)56 b(Without)36 b(loss)f(of)h(generalit)m(y)-8
b(,)39 b(w)m(e)c(ma)m(y)h(assume)f Fi(y)h Fo(=)c(0.)56
b(W)-8 b(e)36 b(no)m(w)150 1328 y(denote,)31 b(for)f(short,)h
Fi(B)e Fo(:=)d Fi(B)1155 1343 y Fg(d)1191 1352 y Fc(0)1260
1328 y Fo(and)k Fi(U)35 b Fo(:=)25 b Fi(U)1717 1342 y
Fz(\006)1776 1328 y Fo(.)291 1441 y(Let)31 b Fi(v)e Fo(:=)c
Fi(v)692 1455 y Fz(\006)781 1441 y Fo(b)s(e)30 b(the)h(minim)m(um)e(of)
1696 1556 y Fh(Z)1746 1762 y Fg(B)1822 1679 y Fi(a)1870
1693 y Fg(i;j)1950 1679 y Fi(@)1998 1693 y Fg(i)2027
1679 y Fi(\020)7 b(@)2122 1693 y Fg(j)2158 1679 y Fi(\020)150
1932 y Fo(among)31 b(the)f(functions)g Fi(\020)i Fk(2)25
b Fo(H)1217 1899 y Fj(1)1257 1932 y Fo(\()p Fi(B)5 b
Fo(\))30 b(with)g Fi(\020)i Fo(=)25 b Fi(U)1868 1946
y Fz(\006)1957 1932 y Fo(on)31 b Fi(@)5 b(B)g Fo(.)40
b(Hence,)1030 2172 y(const)1268 2049 y Fh(Z)1319 2255
y Fg(B)1394 2172 y Fk(jr)p Fo(\()p Fi(U)31 b Fk(\000)20
b Fi(v)s Fo(\))p Fk(j)1821 2135 y Fj(2)1887 2172 y Fk(\024)1982
2049 y Fh(Z)2033 2255 y Fg(B)2109 2172 y Fi(a)2157 2186
y Fg(i;j)2237 2172 y Fi(@)2285 2186 y Fg(i)2313 2172
y Fo(\()p Fi(U)31 b Fk(\000)20 b Fi(v)s Fo(\))p Fi(@)2662
2186 y Fg(j)2699 2172 y Fo(\()p Fi(U)31 b Fk(\000)20
b Fi(v)s Fo(\))26 b(=)861 2412 y(=)1015 2288 y Fh(Z)1066
2494 y Fg(B)1141 2412 y Fi(a)1189 2426 y Fg(i;j)1270
2412 y Fi(@)1318 2426 y Fg(i)1346 2412 y Fi(u@)1446 2426
y Fg(j)1483 2412 y Fi(u)20 b Fk(\000)1646 2288 y Fh(Z)1697
2494 y Fg(B)1772 2412 y Fi(a)1820 2426 y Fg(i;j)1901
2412 y Fi(@)1949 2426 y Fg(i)1977 2412 y Fi(v)s(@)2072
2426 y Fg(j)2109 2412 y Fi(v)f(:)1321 b Fo(\(12.7\))150
2657 y(By)42 b(construction,)k Fi(V)64 b Fo(=)44 b Fi(V)1150
2671 y Fz(\006)1254 2657 y Fo(:=)g Fk(\006)p Fo(\()p
Fi(v)31 b Fk(\000)d Fo(1\))45 b(=)f Fi(u)e Fo(on)f Fi(@)5
b(B)g Fo(:)64 b(hence)42 b(b)m(y)f(\(12.7\))j(and)e(the)g(fact)g(that)
150 2770 y Fi(u)25 b Fk(2)g(M)422 2737 y Fg(!)422 2797
y(M)501 2770 y Fo(,)897 2805 y Fh(Z)948 3011 y Fg(B)1023
2929 y Fk(jr)p Fo(\()p Fi(U)31 b Fk(\000)20 b Fi(v)s
Fo(\))1425 2891 y Fj(2)1465 2929 y Fk(j)26 b(\024)40
b Fo(const)1865 2805 y Fh(Z)1915 3011 y Fg(B)1991 2929
y Fi(Q\037)2120 2947 y Fj(\()p Fz(\000)p Fj(1)p Fg(;)p
Fj(1\))2324 2929 y Fo(\()p Fi(V)20 b Fo(\))h Fk(\000)f
Fi(\037)2636 2947 y Fj(\()p Fz(\000)p Fj(1)p Fg(;)p Fj(1\))2840
2929 y Fo(\()p Fi(u)p Fo(\))15 b Fi(:)516 b Fo(\(12.8\))150
3134 y(By)31 b(Maxim)m(um)g(Principle)g(and)f(the)h(fact)g(that)h(0)26
b Fk(\024)f Fi(U)36 b Fk(\024)25 b Fo(2,)32 b(w)m(e)f(deduce)f(that)i
(either)f(0)26 b Fi(<)f(v)k(<)d Fo(2)31 b(or)g Fi(v)150
3247 y Fo(is)f(constan)m(t)i(on)e Fi(B)5 b Fo(.)291 3360
y(In)24 b(the)h(latter)h(case,)h(since)e Fi(u)g Fk(2)g(M)1494
3327 y Fg(!)1494 3386 y(M)1573 3360 y Fo(,)h Fi(u)f Fo(m)m(ust)g(b)s(e)
f(constan)m(t)h(in)g Fi(B)5 b Fo(:)37 b(in)25 b(particular)f
Fi(u)p Fo(\()p Fi(x)3289 3374 y Fj(0)3329 3360 y Fo(\))i(=)f
Fi(u)p Fo(\(0\))h(=)150 3472 y Fk(\006)p Fo(1,)31 b(and)e(the)i(claim)g
(is)g(pro)m(v)m(en.)291 3585 y(Th)m(us,)f(w)m(e)h(can)f(no)m(w)h
(restrict)g(ourselv)m(es)g(to)g(the)f(case)i(0)26 b Fi(<)f(v)j(<)d
Fo(2)31 b(in)f Fi(B)5 b Fo(,)30 b(i.e.)41 b Fk(\000)p
Fo(1)26 b Fi(<)f(V)45 b(<)25 b Fo(1.)41 b(F)-8 b(rom)150
3698 y(\(12.8\),)1096 3729 y Fh(Z)1147 3935 y Fg(B)1222
3853 y Fk(jr)p Fo(\()p Fi(U)31 b Fk(\000)20 b Fi(v)s
Fo(\))p Fk(j)1649 3815 y Fj(2)1715 3853 y Fk(\024)40
b Fo(const)15 b Fk(L)2111 3815 y Fg(n)2158 3853 y Fo(\()p
Fk(fj)p Fi(u)p Fk(j)27 b Fo(=)e(1)p Fk(g)c(\\)f Fi(B)5
b Fo(\))15 b Fi(:)714 b Fo(\(12.9\))150 4063 y(F)-8 b(rom)35
b(\(12.6\))j(and)c(Lemma)h(12.1,)j(there)d(is)g(a)g(suitable)h(constan)
m(t)g Fi(\024)d Fk(2)g Fo(\(0)p Fi(;)15 b Fo(1\))37 b(and)d(a)h
(suitable)h(p)s(oin)m(t)150 4176 y Fi(x)202 4143 y Fg(?)267
4176 y Fk(2)25 b Fi(B)422 4191 y Fg(\024d)499 4200 y
Fc(0)567 4176 y Fo(so)31 b(that)g Fi(v)s Fo(\()p Fi(x)1010
4143 y Fg(?)1050 4176 y Fo(\))26 b Fk(\025)40 b Fo(const)16
b Fi(M)10 b(d)1590 4190 y Fj(0)1629 4176 y Fo(.)41 b(Harnac)m(k)31
b(inequalit)m(y)h(then)e(implies)1571 4375 y(inf)1547
4435 y Fg(B)1600 4447 y Fd(\024d)1668 4462 y Fc(0)1722
4375 y Fi(v)e Fk(\025)40 b Fo(const)16 b Fi(M)10 b(d)2273
4389 y Fj(0)2328 4375 y Fi(:)1119 b Fo(\(12.10\))150
4644 y(Set)31 b(no)m(w)f Fi(v)543 4611 y Fz(\003)583
4644 y Fo(\()p Fi(x)p Fo(\))c(:=)880 4608 y Fj(1)p 862
4623 71 4 v 862 4675 a Fg(d)898 4684 y Fc(0)943 4644
y Fi(v)s Fo(\()p Fi(d)1072 4658 y Fj(0)1112 4644 y Fi(x)p
Fo(\),)31 b(for)f Fi(x)25 b Fk(2)g Fi(B)1626 4658 y Fj(1)1666
4644 y Fo(.)40 b(Let)31 b(us)f(no)m(w)g(in)m(tro)s(duce)g(the)h(follo)m
(wing)h(barrier:)1414 4872 y Fi(w)r Fo(\()p Fi(x)p Fo(\))26
b(=)h(\026)-47 b Fi(c)q(M)10 b Fo(\()p Fi(e)1940 4834
y Fz(\000)p Fg(\026)p Fz(j)p Fg(x)p Fz(j)2117 4811 y
Fc(2)2176 4872 y Fk(\000)20 b Fi(e)2309 4834 y Fz(\000)p
Fg(\026)2410 4872 y Fo(\))15 b Fi(;)150 5071 y Fo(with)32
b(\026)-47 b Fi(c)31 b Fo(and)e Fi(\026)i Fo(suitable)f(constan)m(ts,)i
(prop)s(erly)d(c)m(hosen)i(to)g(satisfy)1020 5269 y Fi(@)1068
5283 y Fg(i)1097 5269 y Fo(\()p Fi(a)1180 5283 y Fg(i;j)1260
5269 y Fi(@)1308 5283 y Fg(j)1345 5269 y Fi(w)r Fo(\))26
b Fi(>)f Fo(0)182 b(in)41 b Fi(B)1982 5283 y Fj(1)2041
5269 y Fk(\000)20 b Fi(B)2201 5283 y Fg(\024)2261 5269
y Fi(;)1020 5407 y(w)28 b Fo(=)d(0)182 b(on)41 b Fi(@)5
b(B)1695 5421 y Fj(1)1916 5407 y Fo(and)182 b Fi(w)27
b Fk(\024)e Fi(v)2480 5370 y Fz(\003)2702 5407 y Fo(on)40
b Fi(@)5 b(B)2960 5421 y Fg(\024)3020 5407 y Fi(;)1905
5656 y Fo(26)p eop
%%Page: 27 27
27 26 bop 150 399 a Fo(where)30 b(the)g(last)i(relation)f(can)g(b)s(e)f
(ful\014lled)f(in)h(ligh)m(t)i(of)f(\(12.10\).)43 b(So,)31
b(b)m(y)f(Maxim)m(um)h(Principle,)960 603 y Fi(v)1007
565 y Fz(\003)1047 603 y Fo(\()p Fi(x)p Fo(\))26 b Fk(\025)f
Fi(w)r Fo(\()p Fi(x)p Fo(\))h Fk(\025)40 b Fo(const)16
b Fi(M)10 b Fo(\(1)21 b Fk(\000)f(j)p Fi(x)p Fk(j)q Fo(\))182
b(in)40 b Fi(B)2635 617 y Fj(1)2695 603 y Fk(\000)20
b Fi(B)2855 617 y Fg(\024)2914 603 y Fi(:)150 807 y Fo(Recalling)32
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y Fz(\003)1315 1011 y Fo(\()p Fi(x)p Fo(\))c Fk(\025)40
b Fo(const)16 b Fi(M)10 b Fo(\(1)21 b Fk(\000)f(j)p Fi(x)p
Fk(j)p Fo(\))183 b(in)40 b Fi(B)2592 1025 y Fj(1)2646
1011 y Fi(:)150 1215 y Fo(Rescaling)32 b(bac)m(k:)1177
1371 y Fi(v)s Fo(\()p Fi(x)p Fo(\))26 b Fk(\025)40 b
Fo(const)16 b Fi(M)10 b(d)1851 1385 y Fj(0)1891 1371
y Fo(\(1)21 b Fk(\000)2093 1310 y(j)p Fi(x)p Fk(j)p 2093
1350 103 4 v 2101 1433 a Fi(d)2148 1447 y Fj(0)2205 1371
y Fo(\))182 b(in)41 b Fi(B)2608 1386 y Fg(d)2644 1395
y Fc(0)2698 1371 y Fi(:)749 b Fo(\(12.11\))150 1578 y(W)-8
b(e)31 b(can)g(no)m(w)f(follo)m(w)h([A)m(C],)g(p.)40
b(112:)i(Fix)31 b(an)m(y)f Fi(y)e Fk(2)d Fi(B)2041 1596
y Fg(d)2077 1605 y Fc(0)2112 1596 y Fg(=)p Fj(2)2217
1578 y Fo(and)k(consider)h(the)g(follo)m(wing)i(transforma-)150
1690 y(tion)f(of)f Fi(B)509 1705 y Fg(d)545 1714 y Fc(0)615
1690 y Fo(suc)m(h)g(that)h Fi(y)i Fo(b)s(ecomes)d(the)h(new)f(origin:)
1482 1944 y Fi(T)1535 1958 y Fg(y)1577 1944 y Fo(\()p
Fi(x)p Fo(\))c(:=)1856 1882 y Fi(d)1903 1896 y Fj(0)1963
1882 y Fk(\000)20 b(j)p Fi(x)p Fk(j)p 1856 1923 301 4
v 1963 2006 a Fi(d)2010 2020 y Fj(0)2166 1944 y Fi(y)j
Fo(+)d Fi(x)15 b(;)150 2188 y Fo(and)30 b(set,)h(for)f(an)m(y)h
(function)f Fi(f)10 b Fo(,)30 b Fi(f)1316 2202 y Fg(y)1357
2188 y Fo(\()p Fi(x)p Fo(\))c(:=)f Fi(f)10 b Fo(\()p
Fi(T)1769 2202 y Fg(y)1810 2188 y Fo(\()p Fi(x)p Fo(\)\).)291
2301 y(F)-8 b(or)31 b(an)m(y)g Fi(\020)h Fk(2)25 b Fi(@)5
b(B)906 2315 y Fj(1)945 2301 y Fo(,)31 b(de\014ne)1160
2505 y Fi(r)1201 2520 y Fg(\020)1265 2505 y Fo(:=)26
b(inf)6 b Fk(f)p Fi(r)29 b Fk(2)24 b Fo([)p Fi(d)1769
2519 y Fj(0)1809 2505 y Fi(=)p Fo(8)p Fi(;)15 b(d)1986
2519 y Fj(0)2027 2505 y Fo(])41 b(s)p Fi(:)p Fo(t)p Fi(:)g(U)10
b Fo(\()p Fi(r)s(\020)d Fo(\))25 b(=)g(0)p Fk(g)15 b
Fi(;)150 2709 y Fo(if)30 b(this)h(set)g(is)f(non-empt)m(y)-8
b(.)41 b(F)-8 b(or)31 b(almost)h(all)f(suc)m(h)f Fi(\020)37
b Fo(w)m(e)31 b(ha)m(v)m(e)255 2995 y Fi(v)299 3009 y
Fg(y)341 2995 y Fo(\()p Fi(r)417 3010 y Fg(\020)457 2995
y Fi(\020)7 b Fo(\))25 b(=)660 2871 y Fh(Z)751 2897 y
Fg(d)787 2906 y Fc(0)710 3077 y Fg(r)742 3089 y Fd(\020)841
2995 y Fk(r)p Fo(\()p Fi(U)1014 3009 y Fg(y)1076 2995
y Fk(\000)19 b Fi(v)1210 3009 y Fg(y)1252 2995 y Fo(\)\()p
Fi(\032\020)7 b Fo(\))21 b Fk(\001)f Fi(\020)i(d\032)j
Fk(\024)1794 2834 y Fh(s)p 1885 2834 992 4 v 37 x(Z)1976
2897 y Fg(d)2012 2906 y Fc(0)1936 3077 y Fg(r)1968 3089
y Fd(\020)2066 2995 y Fk(jr)p Fo(\()p Fi(U)2264 3009
y Fg(y)2326 2995 y Fk(\000)20 b Fi(v)2461 3009 y Fg(y)2503
2995 y Fo(\)\()p Fi(\032\020)7 b Fo(\))p Fk(j)2727 2968
y Fj(2)2782 2995 y Fi(d\032)36 b Fk(\001)2957 2918 y
Fh(p)p 3048 2918 279 4 v 77 x Fi(d)3095 3009 y Fj(0)3155
2995 y Fk(\000)20 b Fi(r)3287 3010 y Fg(\020)3342 2995
y Fi(:)105 b Fo(\(12.12\))150 3270 y(By)31 b(\(12.11\),)i(for)d(an)m(y)
h Fi(r)d Fk(2)d Fo([0)p Fi(;)15 b(d)1249 3284 y Fj(0)1289
3270 y Fo(],)743 3517 y Fi(v)787 3531 y Fg(y)829 3517
y Fo(\()p Fi(r)s(\020)7 b Fo(\))82 b Fk(\025)98 b Fo(const)16
b Fi(M)1562 3417 y Fh(\020)1616 3517 y Fi(d)1663 3531
y Fj(0)1723 3517 y Fk(\000)1814 3413 y Fh(\014)1814 3467
y(\014)1814 3522 y(\014)1854 3456 y Fi(d)1901 3470 y
Fj(0)1961 3456 y Fk(\000)k Fi(r)p 1854 3497 242 4 v 1932
3580 a(d)1979 3594 y Fj(0)2106 3517 y Fi(y)j Fo(+)d Fi(r)s(\020)2356
3413 y Fh(\014)2356 3467 y(\014)2356 3522 y(\014)2385
3417 y(\021)2465 3517 y Fk(\025)1072 3751 y(\025)98 b
Fo(const)16 b Fi(M)1562 3650 y Fh(\020)1616 3751 y Fi(d)1663
3765 y Fj(0)1723 3751 y Fk(\000)1824 3690 y Fi(d)1871
3704 y Fj(0)1931 3690 y Fk(\000)k Fi(r)p 1824 3730 V
1901 3814 a(d)1948 3828 y Fj(0)2076 3751 y Fk(j)p Fi(y)s
Fk(j)g(\000)g Fi(r)2329 3650 y Fh(\021)2408 3751 y Fk(\025)40
b Fo(const)16 b Fi(M)2865 3690 y(d)2912 3704 y Fj(0)2972
3690 y Fk(\000)k Fi(r)p 2865 3730 V 2963 3814 a Fo(2)3132
3751 y Fi(;)150 3995 y Fo(since)31 b Fk(j)p Fi(y)s Fk(j)25
b(\024)g Fi(d)639 4009 y Fj(0)679 3995 y Fi(=)p Fo(2.)42
b(F)-8 b(rom)30 b(this)h(and)f(\(12.12\))1004 4261 y
Fi(M)1102 4223 y Fj(2)1141 4261 y Fo(\()p Fi(d)1223 4275
y Fj(0)1283 4261 y Fk(\000)20 b Fi(r)1415 4276 y Fg(\020)1455
4261 y Fo(\))26 b Fk(\024)40 b Fo(const)1865 4137 y Fh(Z)1956
4164 y Fg(d)1992 4173 y Fc(0)1915 4343 y Fg(r)1947 4355
y Fd(\020)2046 4261 y Fk(jr)p Fo(\()p Fi(U)2244 4275
y Fg(y)2306 4261 y Fk(\000)20 b Fi(v)2441 4275 y Fg(y)2482
4261 y Fo(\)\()p Fi(\032\020)7 b Fo(\))p Fk(j)2706 4223
y Fj(2)2762 4261 y Fi(d\032)15 b(:)576 b Fo(\(12.13\))150
4528 y(Also,)31 b(using)f(p)s(olar)g(co)s(ordinates)h(and)f(the)h
(de\014nition)f(of)g Fi(r)2218 4543 y Fg(\020)2258 4528
y Fo(,)526 4655 y Fh(Z)577 4861 y Fg(B)630 4873 y Fd(d)662
4888 y Fc(0)701 4861 y Fz(\000)p Fg(B)809 4875 y Fd(d)841
4890 y Fc(0)875 4875 y Fd(=)p Fc(4)942 4861 y Fj(\()p
Fg(y)r Fj(\))1053 4779 y Fi(\037)1110 4797 y Fz(f)p Fg(U)7
b Fj(=0)p Fz(g)1355 4779 y Fk(\024)40 b Fo(const)1704
4655 y Fh(Z)1755 4861 y Fg(B)1808 4873 y Fd(d)1840 4888
y Fc(0)1879 4861 y Fz(\000)p Fg(B)1987 4875 y Fd(d)2019
4890 y Fc(0)2053 4875 y Fd(=)p Fc(8)2139 4779 y Fi(\037)2196
4797 y Fz(f)p Fg(U)2279 4805 y Fd(y)2316 4797 y Fj(=0)p
Fz(g)2471 4779 y Fk(\024)526 5067 y(\024)g Fo(const)875
4944 y Fh(Z)926 5150 y Fg(@)t(B)1020 5159 y Fc(1)1074
4944 y Fh(Z)1165 4970 y Fg(d)1201 4979 y Fc(0)1124 5150
y Fg(d)1160 5159 y Fc(0)1195 5150 y Fg(=)p Fj(8)1285
5067 y Fi(\037)1342 5086 y Fz(f)p Fg(U)1425 5094 y Fd(y)1463
5086 y Fj(=0)p Fz(g)1592 5067 y Fo(\()p Fi(\032\020)7
b Fo(\))15 b Fi(d\032)g(d\020)33 b Fo(=)40 b(const)2349
4944 y Fh(Z)2400 5150 y Fg(@)t(B)2494 5159 y Fc(1)2548
4944 y Fh(Z)2639 4970 y Fg(d)2675 4979 y Fc(0)2598 5150
y Fg(r)2630 5162 y Fd(\020)2729 5067 y Fi(\037)2786 5086
y Fz(f)p Fg(U)2869 5094 y Fd(y)2906 5086 y Fj(=0)p Fz(g)3036
5067 y Fo(\()p Fi(\032\020)7 b Fo(\))15 b Fi(d\032)g(d\020)33
b Fk(\024)526 5328 y(\024)40 b Fo(const)875 5204 y Fh(Z)926
5411 y Fg(@)t(B)1020 5420 y Fc(1)1059 5328 y Fo(\()p
Fi(d)1141 5342 y Fj(0)1201 5328 y Fk(\000)20 b Fi(r)1333
5343 y Fg(\020)1373 5328 y Fo(\))15 b Fi(d\020)22 b(:)1905
5656 y Fo(27)p eop
%%Page: 28 28
28 27 bop 150 399 a Fo(F)-8 b(rom)31 b(this,)f(in)m(tegrating)j(b)s
(oth)c(sides)i(of)f(\(12.13\),)j(it)e(follo)m(ws)h(that)701
664 y(const)15 b Fi(M)1021 627 y Fj(2)1076 541 y Fh(Z)1127
747 y Fg(B)1180 759 y Fd(d)1212 774 y Fc(0)1251 747 y
Fz(\000)p Fg(B)1359 761 y Fd(d)1391 776 y Fc(0)1425 761
y Fd(=)p Fc(4)1492 747 y Fj(\()p Fg(y)r Fj(\))1603 664
y Fi(\037)1660 683 y Fz(f)p Fg(U)7 b Fj(=0)p Fz(g)1905
664 y Fk(\024)2001 541 y Fh(Z)2051 747 y Fg(@)t(B)2145
756 y Fc(1)2200 541 y Fh(Z)2291 567 y Fg(d)2327 576 y
Fc(0)2250 747 y Fg(d)2286 756 y Fc(0)2321 747 y Fg(=)p
Fj(8)2411 664 y Fk(jr)p Fo(\()p Fi(U)2609 678 y Fg(y)2671
664 y Fk(\000)20 b Fi(v)2806 678 y Fg(y)2847 664 y Fo(\)\()p
Fi(\032\020)7 b Fo(\))p Fk(j)3071 627 y Fj(2)3127 664
y Fi(d\032)15 b(d\020)32 b Fk(\024)532 933 y(\024)98
b Fo(const)939 810 y Fh(Z)989 1016 y Fg(B)1042 1028 y
Fd(d)1074 1043 y Fc(0)1132 933 y Fk(jr)p Fo(\()p Fi(U)31
b Fk(\000)20 b Fi(v)s Fo(\))p Fk(j)1559 896 y Fj(2)1614
933 y Fi(:)150 1210 y Fo(Applying)35 b(this)h(result)f(to)h(t)m(w)m(o)h
(p)s(oin)m(ts)f Fi(y)1607 1224 y Fj(1)1681 1210 y Fo(and)f
Fi(y)1908 1224 y Fj(2)1982 1210 y Fo(in)h Fi(B)2163 1229
y Fg(d)2199 1238 y Fc(0)2233 1229 y Fg(=)p Fj(2)2344
1210 y Fo(so)f(that)i Fi(B)2732 1229 y Fg(d)2768 1238
y Fc(0)2802 1229 y Fg(=)p Fj(4)2877 1210 y Fo(\()p Fi(y)2957
1224 y Fj(1)2997 1210 y Fo(\))e(and)g Fi(B)3318 1229
y Fg(d)3354 1238 y Fc(0)3389 1229 y Fg(=)p Fj(4)3464
1210 y Fo(\()p Fi(y)3544 1224 y Fj(2)3583 1210 y Fo(\))h(do)150
1323 y(not)31 b(o)m(v)m(erlap,)h(and)e(summing)f(up)g(the)i(t)m(w)m(o)h
(estimates,)g(w)m(e)f(obtain)1101 1564 y Fi(M)1199 1527
y Fj(2)1254 1440 y Fh(Z)1304 1647 y Fg(B)1357 1659 y
Fd(d)1389 1674 y Fc(0)1447 1564 y Fi(\037)1504 1583 y
Fz(f)p Fg(U)7 b Fj(=0)p Fz(g)1749 1564 y Fk(\024)40 b
Fo(const)2098 1440 y Fh(Z)2149 1647 y Fg(B)2202 1659
y Fd(d)2234 1674 y Fc(0)2292 1564 y Fk(jr)p Fo(\()p Fi(U)30
b Fk(\000)20 b Fi(v)s Fo(\))p Fk(j)2718 1527 y Fj(2)2774
1564 y Fi(:)150 1841 y Fo(Th)m(us,)30 b(recalling)h(\(12.9\),)i(and)d
(writing)g Fi(U)36 b Fo(=)25 b Fi(U)1803 1855 y Fz(\006)1892
1841 y Fo(explicitly:)887 2045 y Fi(M)985 2008 y Fj(2)1024
2045 y Fk(L)1087 2008 y Fg(n)1134 2045 y Fo(\()p Fk(f)p
Fi(U)1276 2059 y Fz(\006)1361 2045 y Fo(=)g(0)p Fk(g)c(\\)f
Fi(B)1718 2060 y Fg(d)1754 2069 y Fc(0)1793 2045 y Fo(\))26
b Fk(\024)40 b Fo(const)15 b Fk(L)2250 2008 y Fg(n)2297
2045 y Fo(\()p Fk(fj)p Fi(u)p Fk(j)27 b Fo(=)e(1)p Fk(g)c(\\)f
Fi(B)2863 2060 y Fg(d)2899 2069 y Fc(0)2938 2045 y Fo(\))15
b Fi(:)150 2249 y Fo(Since)30 b Fk(fj)p Fi(u)p Fk(j)d
Fo(=)e(1)p Fk(g)h(\022)f(f)p Fi(U)976 2263 y Fj(+)1061
2249 y Fo(=)f(0)p Fk(g)e([)d(f)p Fi(U)1455 2263 y Fz(\000)1540
2249 y Fo(=)25 b(0)p Fk(g)p Fo(,)32 b(w)m(e)e(deduce)h(that)873
2454 y Fi(M)971 2416 y Fj(2)1011 2454 y Fk(L)1074 2416
y Fg(n)1120 2454 y Fo(\()p Fk(fj)p Fi(u)p Fk(j)c Fo(=)e(1)p
Fk(g)c(\\)f Fi(B)1686 2469 y Fg(d)1722 2478 y Fc(0)1761
2454 y Fo(\))26 b Fk(\024)f Fo(2)15 b(const)h Fk(L)2264
2416 y Fg(n)2311 2454 y Fo(\()p Fk(fj)p Fi(u)p Fk(j)26
b Fo(=)f(1)p Fk(g)c(\\)f Fi(B)2876 2469 y Fg(d)2912 2478
y Fc(0)2951 2454 y Fo(\))15 b Fi(:)150 2658 y Fo(Accordingly)-8
b(,)44 b(recalling)e(also)f(\(12.9\),)k(if)40 b Fi(M)50
b Fo(w)m(ere)41 b(to)s(o)g(big,)h(w)m(e)f(get)g Fk(j)p
Fi(u)p Fk(j)h Fi(<)g Fo(1)e(in)g Fi(B)3191 2673 y Fg(d)3227
2682 y Fc(0)3266 2658 y Fo(,)j(whic)m(h)c(is)i(a)150
2771 y(con)m(tradiction.)p 3684 2771 4 62 v 3688 2713
55 4 v 3688 2771 V 3742 2771 4 62 v 150 2958 a Ff(Theorem)35
b(12.3.)46 b Fm(If)37 b Fi(u)32 b Fk(2)h(M)1241 2925
y Fg(!)1241 2985 y(M)1320 2958 y Fm(,)k(then)h Fi(u)32
b Fk(2)g Fo(C)1835 2925 y Fj(0)p Fg(;)p Fj(1)1929 2958
y Fo(\()p Fi(S)2025 2925 y Fg(!)2020 2985 y(M)2099 2958
y Fo(\))p Fm(.)55 b(F)-7 b(urthermor)i(e,)40 b(for)d(any)g(c)-5
b(omp)g(act)39 b(domain)150 3071 y Fi(D)c Fm(wel)5 b(l)33
b(c)-5 b(ontaine)g(d)35 b(in)d Fi(S)1025 3038 y Fg(!)1020
3098 y(M)1132 3071 y Fm(with)1577 3184 y Fi(D)23 b Fk(\\)d(fj)p
Fi(u)p Fk(j)26 b Fo(=)f(1)p Fk(g)h(6)p Fo(=)f Fk(;)15
b Fi(;)1150 b Fo(\(12.14\))150 3351 y Fm(ther)-5 b(e)40
b(exists)g(a)f(suitable)h(p)-5 b(ositive)40 b(c)-5 b(onstant)41
b Fi(C)1829 3365 y Fg(D)1893 3351 y Fm(,)g(dep)-5 b(ending)40
b(only)g(on)55 b Fo(dist)15 b(\()p Fi(D)s(;)g(@)5 b(S)3176
3318 y Fg(!)3171 3378 y(M)3251 3351 y Fo(\))p Fm(,)41
b Fi(n)d Fm(and)j(the)150 3464 y(b)-5 b(ounds)34 b(on)f
Fi(a)628 3478 y Fg(i;j)741 3464 y Fm(and)g Fi(F)46 b
Fm(such)32 b(that)1659 3577 y Fk(jj)p Fi(u)p Fk(jj)1811
3597 y Fj(C)1862 3578 y Fc(0)p Fd(;)p Fc(1)1976 3577
y Fk(\024)25 b Fi(C)2137 3591 y Fg(D)2216 3577 y Fi(:)1231
b Fo(\(12.15\))150 3765 y Fm(Pr)-5 b(o)g(of.)47 b Fo(First)21
b(of)f(all,)k(w)m(e)d(sho)m(w)f(that)h Fk(r)p Fi(u)f
Fo(is)h(b)s(ounded)d(in)i Fi(D)s Fk(\\N)13 b Fo(,)23
b(where)d Fk(N)33 b Fo(is)21 b(a)f(suitable)h(neigh)m(b)s(orho)s(o)s(d)
150 3878 y(of)31 b Fi(@)5 b Fk(fj)p Fi(u)p Fk(j)26 b
Fo(=)f(1)p Fk(g)p Fo(.)291 3990 y(Indeed:)39 b(if)44
b(dist)15 b(\()p Fi(x;)g Fk(fj)p Fi(u)p Fk(j)27 b Fo(=)e(1)p
Fk(g)p Fo(\))i Fk(\024)1552 3955 y Fj(1)p 1535 3970 71
4 v 1535 4022 a(10)1630 3990 y Fo(dist)16 b(\()p Fi(x;)f(@)5
b(S)2034 3957 y Fg(!)2029 4017 y(M)2108 3990 y Fo(\),)30
b(w)m(e)f(are)h(under)d(the)i(h)m(yp)s(otheses)g(of)g(Lemma)150
4103 y(12.2,)j(so)f(that)g(\(setting)h Fi(B)d Fo(:=)d
Fi(B)1311 4122 y Fj(dist)11 b(\()p Fg(x;)p Fz(fj)p Fg(u)p
Fz(j)p Fj(=1)p Fz(g)p Fj(\))p Fg(=)p Fj(10)1901 4103
y Fo(\()p Fi(x)p Fo(\))31 b(for)f(short\))804 4308 y(sup)784
4386 y Fg(x)824 4395 y Fc(0)858 4386 y Fz(2)p Fg(B)977
4308 y Fo(min)o Fk(f)p Fo(1)21 b Fk(\000)f Fi(u)p Fo(\()p
Fi(x)1469 4322 y Fj(0)1509 4308 y Fo(\))p Fi(;)31 b Fo(1)21
b(+)f Fi(u)p Fo(\()p Fi(x)1896 4322 y Fj(0)1935 4308
y Fo(\))p Fk(g)26 b(\024)41 b Fo(const)30 b(dist)16 b(\()p
Fi(x;)f Fk(fj)p Fi(u)p Fk(j)27 b Fo(=)e(1)p Fk(g)p Fo(\))15
b Fi(:)357 b Fo(\(12.16\))150 4572 y(Also,)31 b(b)m(y)g(Lemma)f(2.4,)i
Fi(@)1051 4586 y Fg(i)1079 4572 y Fo(\()p Fi(a)1162 4586
y Fg(i;j)1243 4572 y Fi(@)1291 4586 y Fg(j)1328 4572
y Fo(\(1)21 b Fk(\006)e Fi(u)p Fo(\)\))26 b(=)f(0)31
b(in)f Fi(B)5 b Fo(;)30 b(therefore,)h(b)m(y)g(elliptic)h(estimates,)
1126 4828 y Fk(jr)p Fo(\(1)21 b Fk(\006)f Fi(u)p Fo(\)\()p
Fi(x)p Fo(\))p Fk(j)26 b(\024)41 b Fo(const)2138 4766
y(sup)2274 4788 y Fg(B)2350 4766 y Fk(j)p Fo(1)21 b Fk(\006)f
Fi(u)p Fk(j)p 2023 4807 701 4 v 2038 4890 a Fo(dist)c(\()p
Fi(x;)f Fk(fj)p Fi(u)p Fk(j)26 b Fo(=)f(1)p Fk(g)p Fo(\))2749
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Fi(D)23 b Fk(\\)c(N)13 b Fo(.)291 5243 y(This)30 b(and)f(standard)h
(elliptic)i(estimates)g(imply)e(that)h Fi(u)25 b Fk(2)g
Fo(C)2434 5210 y Fj(0)p Fg(;)p Fj(1)2528 5243 y Fo(\()p
Fi(S)2624 5210 y Fg(!)2619 5270 y(M)2698 5243 y Fo(\).)1905
5656 y(28)p eop
%%Page: 29 29
29 28 bop 291 399 a Fo(W)-8 b(e)32 b(no)m(w)e(pro)m(v)m(e)h(\(12.15\):)
44 b(T)-8 b(ak)m(e)31 b Fi(D)1523 366 y Fz(0)1577 399
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366 y Fz(0)2375 399 y Fk(\032\032)g Fi(S)2603 366 y Fg(!)2598
425 y(M)2676 399 y Fo(,)31 b(with)1263 630 y(dist)15
b(\()p Fi(D)1538 592 y Fz(0)1562 630 y Fi(;)g(@)5 b(S)1716
592 y Fg(!)1711 652 y(M)1790 630 y Fo(\))26 b Fk(\025)1979
568 y Fo(9)p 1957 609 91 4 v 1957 692 a(10)2073 630 y(dist)15
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630 y Fo(\))15 b Fi(:)150 846 y Fo(With)30 b(no)g(loss)h(of)f
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Fi(D)i Fo(and)e Fi(D)2229 813 y Fz(0)2282 846 y Fo(to)h(b)s(e)e
(connected.)42 b(Fix)30 b(an)m(y)g Fi(x)c Fk(2)f Fi(D)3551
813 y Fz(0)3604 846 y Fo(and)150 959 y(set)g Fi(r)327
973 y Fj(0)392 959 y Fo(:=)523 923 y Fj(1)p 523 938 36
4 v 523 990 a(5)584 959 y Fo(dist)15 b(\()p Fi(D)s(;)g(@)5
b(S)1013 926 y Fg(!)1008 986 y(M)1087 959 y Fo(\).)40
b(Consider)24 b(a)h(co)m(v)m(ering)i(of)e Fi(D)2158 926
y Fz(0)2206 959 y Fo(with)f(balls)i Fi(B)2684 973 y Fg(r)2716
982 y Fc(0)2754 959 y Fo(\()p Fi(x)2841 973 y Fg(j)2878
959 y Fo(\),)g(cen)m(tered)g(in)f Fi(D)3499 926 y Fz(0)3522
959 y Fo(,)h(with)150 1072 y Fi(x)202 1086 y Fj(0)267
1072 y Fo(:=)f Fi(x)h Fo(and)f Fi(x)690 1086 y Fg(j)752
1072 y Fk(2)g Fi(B)907 1090 y Fg(r)939 1099 y Fc(0)973
1090 y Fg(=)p Fj(2)1048 1072 y Fo(\()p Fi(x)1135 1086
y Fg(j)t Fz(\000)p Fj(1)1262 1072 y Fo(\).)40 b(Let)26
b Fi(k)k Fo(b)s(e)25 b(the)h(n)m(um)m(b)s(er)f(of)h(suc)m(h)g(balls:)39
b(ob)m(viously)-8 b(,)28 b Fi(k)i Fo(dep)s(ends)24 b(only)150
1185 y(on)30 b Fi(n)p Fo(,)g Fi(D)464 1152 y Fz(0)518
1185 y Fo(and)g Fi(r)736 1199 y Fj(0)775 1185 y Fo(,)h(hence)f(w)m(e)h
(ma)m(y)g(think)f Fi(k)j Fo(to)e(dep)s(end)e(only)i(on)f
Fi(n)p Fo(,)g Fi(D)j Fo(and)45 b(dist)15 b(\()p Fi(D)s(;)g(S)3273
1152 y Fg(!)3268 1212 y(M)3347 1185 y Fo(\).)291 1298
y(By)31 b(\(12.14\),)i(there)e(exists)g Fi(k)1301 1265
y Fz(\003)1366 1298 y Fk(\024)25 b Fi(k)33 b Fo(so)e(that)1445
1486 y Fi(B)1514 1500 y Fg(r)1546 1509 y Fc(0)1585 1486
y Fo(\()p Fi(x)1672 1501 y Fg(k)1711 1482 y Fq(\003)1751
1486 y Fo(\))21 b Fk(\\)e(fj)p Fi(u)p Fk(j)27 b Fo(=)e(1)p
Fk(g)h(6)p Fo(=)f Fk(;)15 b Fi(;)150 1675 y Fo(while)1259
1788 y Fi(B)1328 1802 y Fg(r)1360 1811 y Fc(0)1399 1788
y Fo(\()p Fi(x)1486 1802 y Fg(j)1522 1788 y Fo(\))21
b Fk(\\)f(fj)p Fi(u)p Fk(j)26 b Fo(=)f(1)p Fk(g)h Fo(=)f
Fk(;)15 b Fi(;)72 b Fk(8)p Fi(j)30 b(<)25 b(k)2561 1750
y Fz(\003)2616 1788 y Fi(:)150 1947 y Fo(It)30 b(follo)m(ws)i(that)1159
2075 y(dist)15 b(\()p Fi(x)1408 2090 y Fg(k)1447 2071
y Fq(\003)1487 2075 y Fi(;)g Fk(fj)p Fi(u)p Fk(j)26 b
Fo(=)f(1)p Fk(g)p Fo(\))i Fk(\024)2054 2014 y Fo(2)p
2054 2055 46 4 v 2054 2138 a(9)2124 2075 y(dist)15 b(\()p
Fi(x)2373 2090 y Fg(k)2412 2071 y Fq(\003)2452 2075 y
Fi(;)g(@)5 b(S)2606 2038 y Fg(!)2601 2098 y(M)2681 2075
y Fo(\))15 b Fi(;)150 2255 y Fo(so)31 b(w)m(e)f(can)h(apply)f(Lemma)h
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b Fk(\000)f Fi(u)p Fo(\()p Fi(x)1644 2459 y Fg(k)1683
2440 y Fq(\003)1723 2444 y Fo(\))p Fi(;)31 b Fo(1)20
b(+)g Fi(u)p Fo(\()p Fi(x)2109 2459 y Fg(k)2148 2440
y Fq(\003)2188 2444 y Fo(\))p Fk(g)26 b(\024)40 b Fo(const)16
b Fi(r)2669 2458 y Fj(0)2724 2444 y Fi(:)723 b Fo(\(12.17\))150
2633 y(Also,)29 b Fi(@)426 2647 y Fg(i)455 2633 y Fo(\()p
Fi(a)538 2647 y Fg(i;j)618 2633 y Fi(@)666 2647 y Fg(j)703
2633 y Fo(\(1)15 b Fk(\006)g Fi(u)p Fo(\)\))26 b(=)f(0)j(in)f
Fi(B)1373 2647 y Fg(r)1405 2656 y Fc(0)1444 2633 y Fo(\()p
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b(k)1981 2600 y Fz(\003)2048 2633 y Fo(b)s(ecause)i(of)h(Lemma)g(2.4.)
41 b(So,)28 b(w)m(e)g(deduce)f(from)150 2746 y(Harnac)m(k)k(inequalit)m
(y)h(that)1317 2859 y(const)16 b(\(1)21 b Fk(\006)f Fi(u)p
Fo(\()p Fi(x)1871 2873 y Fg(j)t Fj(+1)1998 2859 y Fo(\)\))26
b Fk(\025)f Fo(1)20 b Fk(\006)g Fi(u)p Fo(\()p Fi(x)2485
2873 y Fg(j)2522 2859 y Fo(\))15 b Fi(:)150 3018 y Fo(Iterating,)32
b(using)e(\(12.17\))j(and)c(recalling)j(that)f Fi(x)1887
3032 y Fj(0)1952 3018 y Fo(=)25 b Fi(x)p Fo(,)30 b(w)m(e)h(get)1154
3206 y(sup)1138 3286 y Fg(x)p Fz(2)p Fg(D)1285 3267 y
Fq(0)1322 3206 y Fo(min)o Fk(f)p Fo(1)21 b Fk(\000)f
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4268 y(th)m(us,)30 b(elliptic)i(estimates)g(imply)e(that)906
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3684 5407 4 62 v 3688 5349 55 4 v 3688 5407 V 3742 5407
4 62 v 1905 5656 a(29)p eop
%%Page: 30 30
30 29 bop 291 399 a Fo(W)-8 b(e)32 b(are)e(no)m(w)h(in)f(p)s(osition)g
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511 y(If)g(In)m(t\()p Fi(S)594 478 y Fg(!)589 538 y(M)668
511 y Fo(\))21 b Fk(\\)f(fj)p Fi(u)p Fk(j)26 b Fo(=)f(1)p
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624 y Fo(\))21 b Fk(\\)g(fj)p Fi(u)p Fk(j)29 b Fo(=)e(1)p
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1564 y(smo)s(oth)h(function)h(that)g(is)g(iden)m(tically)h(equal)f(to)h
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1876 y Fj(2)2668 1913 y Fo(+)k Fi(r)2803 1876 y Fg(n)p
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Fi(r)2706 2131 y Fg(n)p Fz(\000)p Fj(1)2842 2169 y Fo(\))26
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%%Page: 31 31
31 30 bop 150 412 a Fo(and)35 b(so)g Fk(L)511 379 y Fg(n)558
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3201 y Fi(\014)1026 3131 y Fc(2)p Fd(n)p 1002 3143 V
1002 3184 a(n)p Fq(\000)p Fc(1)1133 3100 y Fh(\021)1198
3095 y Fd(n)p Fq(\000)p Fc(1)p 1198 3108 V 1237 3149
a Fd(n)1354 3201 y Fk(\024)555 3449 y(\024)83 b Fi(C)796
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4 v 792 4023 a(A)886 3837 y Fh(Z)937 4043 y Fg(B)990
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4354 y(Z)971 4560 y Fg(B)1024 4568 y Fd(r)1078 4478 y
Fi(\014)1168 4408 y Fc(2)p Fd(n)p 1145 4420 117 3 v 1145
4461 a(n)p Fq(\000)p Fc(1)1275 4377 y Fh(\021)1340 4372
y Fd(n)p Fq(\000)p Fc(1)p 1340 4385 V 1379 4426 a Fd(n)1496
4478 y Fk(\024)697 4726 y(\024)83 b Fi(C)7 b(A)991 4625
y Fh(h)1049 4602 y(Z)1099 4808 y Fg(B)1152 4816 y Fd(r)1187
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5114 y Fh(Z)1079 5320 y Fg(B)1132 5328 y Fd(r)1167 5320
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Fg(\022)r Fz(g)1607 5238 y Fo(\()p Fi(u)h Fk(\000)f Fi(\033)s
Fo(\))1896 5200 y Fj(2)1951 5238 y Fi(:)1496 b Fo(\(13.21\))1905
5656 y(31)p eop
%%Page: 32 32
32 31 bop 150 399 a Fo(Since)30 b Fi(\033)f Fo(=)c Fi(h)g
Fo(=)g Fk(\000)p Fo(1)31 b(in)f Fi(B)1059 413 y Fg(r)r
Fz(\000)p Fj(1)1187 399 y Fo(,)h(the)f(left)h(hand)e(side)i(of)f
(\(13.21\))j(is)e(bigger)g(than)1368 659 y Fi(C)1454
558 y Fh(\020)1509 659 y Fk(L)1572 622 y Fg(n)1618 659
y Fo(\()p Fi(B)1722 673 y Fg(r)r Fz(\000)p Fj(1)1871
659 y Fk(\\)20 b(f)p Fi(u)26 b(>)f(\022)s Fk(g)p Fo(\))2297
558 y Fh(\021)2361 553 y Fd(n)p Fq(\000)p Fc(1)p 2361
566 117 3 v 2400 607 a Fd(n)2507 659 y Fi(:)150 886 y
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g(\(13.21\))i(in)d Fi(B)2195 900 y Fg(r)r Fz(\000)p Fj(1)2323
886 y Fo(,)i(in)e(whic)m(h)h Fi(\033)h Fo(=)d Fk(\000)p
Fo(1:)39 b(Notice)28 b(no)m(w)f(that,)150 999 y(if)j
Fi(\022)j Fo(is)d(close)i(to)f Fk(\000)p Fo(1)g(and)e
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y Fi(F)13 b Fo(\()p Fi(x;)i(u)p Fo(\))26 b Fk(\025)f
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1240 y Fk(\025)2215 1178 y Fi(C)p 2163 1219 177 4 v 2163
1302 a Fo(2)2208 1276 y Fj(2)p Fz(\000)p Fg(d)2349 1240
y Fo(\(1)i(+)e Fi(u)p Fo(\))2627 1202 y Fj(2)3472 1240
y Fo(\(13.22\))150 1465 y(so)31 b(that)894 1536 y Fi(C)p
894 1576 72 4 v 896 1660 a(A)990 1473 y Fh(Z)1041 1680
y Fg(B)1094 1689 y Fd(r)r Fq(\000)p Fc(1)1207 1680 y
Fz(\\f)p Fg(u)p Fz(\024)p Fg(\022)r Fz(g)1459 1597 y
Fo(\(1)21 b(+)f Fi(u)p Fo(\))1738 1560 y Fj(2)1798 1597
y Fk(\000)1899 1536 y Fi(C)7 b(A)p 1899 1576 140 4 v
1946 1660 a Fo(2)2064 1473 y Fh(Z)2114 1680 y Fg(B)2167
1689 y Fd(r)r Fq(\000)p Fc(1)2281 1680 y Fz(\\f)p Fg(u)p
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Fi(B)3164 1841 y Fg(r)r Fz(\000)p Fj(1)3326 1827 y Fo(is)h(negativ)m(e)
150 1940 y(and)30 b(less)g(than)h Fk(\000)p Fi(C)21 b
Fk(A)p Fo(\()p Fi(r)i Fk(\000)c Fo(1\).)291 2052 y(On)26
b(the)h(in)m(tegration)h(o)m(v)m(er)g Fi(B)1310 2066
y Fg(r)1361 2052 y Fk(\000)13 b Fi(B)1514 2066 y Fg(r)r
Fz(\000)p Fj(1)1641 2052 y Fo(,)28 b(w)m(e)f(split)g(the)g(domain)f(of)
h(in)m(tegration.)41 b(First)27 b(of)g(all,)i(since)150
2165 y Fi(\022)36 b Fo(is)e(close)h(to)f Fk(\000)p Fo(1)g(and)f
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b(if)g Fi(u)d Fk(\024)g Fi(\022)s Fo(.)150 2278 y(Th)m(us,)643
2428 y Fi(C)7 b(A)798 2304 y Fh(Z)849 2510 y Fg(B)902
2518 y Fd(r)937 2510 y Fz(\000)p Fg(B)1045 2519 y Fd(r)r
Fq(\000)p Fc(1)1177 2428 y Fi(F)13 b Fo(\()p Fi(x;)i(\033)s
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b Fk(\024)e Fi(C)7 b(A)15 b Fk(L)2203 2390 y Fg(n)2249
2428 y Fo(\()p Fk(f)p Fi(u)26 b(>)f(\022)s Fk(g)20 b(\\)g
Fo(\()p Fi(B)2799 2442 y Fg(r)2858 2428 y Fk(\000)f Fi(B)3017
2442 y Fg(r)r Fz(\000)p Fj(1)3146 2428 y Fo(\)\))c Fi(:)150
2657 y Fo(Also,)31 b(recalling)h(\(13.22\),)832 2842
y Fi(C)p 832 2882 72 4 v 834 2965 a(A)929 2779 y Fh(Z)979
2986 y Fg(B)1032 2994 y Fd(r)1067 2986 y Fz(\000)p Fg(B)1175
2995 y Fd(r)r Fq(\000)p Fc(1)1292 2903 y Fo(\()p Fi(u)21
b Fk(\000)f Fi(\033)s Fo(\))1581 2866 y Fj(2)1646 2903
y Fk(\024)668 3159 y(\024)832 3097 y Fi(C)p 832 3138
V 834 3221 a(A)929 3035 y Fh(Z)979 3241 y Fj(\()p Fg(B)1059
3249 y Fd(r)1095 3241 y Fz(\000)p Fg(B)1203 3250 y Fd(r)r
Fq(\000)p Fc(1)1316 3241 y Fj(\))p Fz(\\f)p Fg(u)p Fz(\024)p
Fg(\022)r Fz(g)1596 3159 y Fo(\()p Fi(u)g Fo(+)g(1\))1874
3121 y Fj(2)1935 3159 y Fo(+)2035 3097 y Fi(C)p 2035
3138 V 2037 3221 a(A)2117 3159 y Fk(L)2180 3121 y Fg(n)2226
3159 y Fo(\()p Fk(f)p Fi(u)27 b(>)e(\022)s Fk(g)19 b(\\)h
Fo(\()p Fi(B)2776 3173 y Fg(r)2835 3159 y Fk(\000)g Fi(B)2995
3173 y Fg(r)r Fz(\000)p Fj(1)3123 3159 y Fo(\)\))26 b
Fk(\024)668 3415 y(\024)832 3353 y Fi(C)p 832 3394 V
834 3477 a(A)929 3291 y Fh(Z)979 3497 y Fj(\()p Fg(B)1059
3505 y Fd(r)1095 3497 y Fz(\000)p Fg(B)1203 3506 y Fd(r)r
Fq(\000)p Fc(1)1316 3497 y Fj(\))p Fz(\\f)p Fg(u)p Fz(\024)p
Fg(\022)r Fz(g)1611 3415 y Fi(F)13 b Fo(\()p Fi(x;)i(u)p
Fo(\))21 b(+)2018 3353 y Fi(C)p 2018 3394 V 2020 3477
a(A)2099 3415 y Fk(L)2162 3377 y Fg(n)2209 3415 y Fo(\()p
Fk(f)p Fi(u)26 b(>)f(\022)s Fk(g)20 b(\\)g Fo(\()p Fi(B)2759
3429 y Fg(r)2817 3415 y Fk(\000)g Fi(B)2977 3429 y Fg(r)r
Fz(\000)p Fj(1)3105 3415 y Fo(\)\))15 b Fi(:)150 3683
y Fo(Finally)-8 b(,)32 b(w)m(e)f(de\014ne,)f Fi(d)946
3650 y Fz(0)995 3683 y Fo(=)25 b(max)p Fk(f)p Fo(1)p
Fi(;)15 b(d)p Fk(g)32 b Fo(and,)e(for)g Fi(\032)c Fk(2)e
Fo([)p Fi(r)f Fk(\000)d Fo(1)p Fi(;)15 b(r)s Fo(],)1390
3933 y Fi(h)p Fo(\()p Fi(\032)p Fo(\))27 b(=)e(2\()p
Fi(\032)c Fk(\000)e Fi(r)k Fo(+)d(1\))2217 3838 y Fc(2)p
2166 3850 133 3 v 2166 3895 a(2)p Fq(\000)p Fd(d)2276
3881 y Fq(0)2155 3956 y Fj(+)2333 3933 y Fk(\000)g Fo(1)15
b Fi(:)963 b Fo(\(13.23\))150 4137 y(Notice)37 b(that)e(this)g(setting)
h(is)e(p)s(ossible)h(for)f(our)h(assumption)f(on)h Fi(d)p
Fo(.)54 b(With)35 b(this)g(de\014nition,)g(w)m(e)h(can)150
4250 y(complete)e(the)f(estimate)i(on)e(the)g(righ)m(t)g(hand)f(side)h
(of)g(\(13.21\):)48 b(Indeed,)33 b(from)f(a)h(direct)h(calculation)150
4363 y(one)d(sees)f(that,)i(in)e Fi(B)897 4377 y Fg(r)955
4363 y Fk(\000)20 b Fi(B)1115 4377 y Fg(r)r Fz(\000)p
Fj(1)1243 4363 y Fo(,)934 4612 y Fk(jr)p Fi(h)p Fk(j)h
Fo(+)f Fk(j)p Fi(D)1327 4574 y Fj(2)1366 4612 y Fi(h)p
Fk(j)26 b(\024)f Fi(C)7 b Fo(\()p Fi(\032)20 b Fk(\000)g
Fi(r)j Fo(+)c(1\))2074 4532 y Fc(2\()p Fd(d)2160 4511
y Fq(0)2184 4532 y(\000)p Fc(1\))p 2075 4550 212 3 v
2114 4595 a(2)p Fq(\000)p Fd(d)2224 4581 y Fq(0)2326
4612 y Fk(\024)25 b Fi(C)7 b Fo(\()p Fi(h)20 b Fo(+)g(1\))2772
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4612 y Fi(:)150 4816 y Fo(Also,)31 b(in)f(the)h(set)g
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Fk(j)p Fi(D)1303 4983 y Fj(2)1343 5020 y Fi(h)p Fk(j)p
Fo(\))i Fk(\001)g Fo(\()p Fi(u)f Fk(\000)g Fi(\033)s
Fo(\))84 b Fk(\024)e Fi(C)7 b Fo(\()p Fi(h)21 b Fo(+)f(1\))2398
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5020 y Fk(\001)g Fo(\()p Fi(u)h Fk(\000)f Fi(h)p Fo(\))26
b Fk(\024)1894 5158 y(\024)82 b Fi(C)22 b Fo(\()p Fi(u)f
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y Fi(;)956 b Fo(\(13.24\))1905 5656 y(32)p eop
%%Page: 33 33
33 32 bop 150 399 a Fo(in)32 b(\()p Fi(B)362 413 y Fg(r)421
399 y Fk(\000)21 b Fi(B)582 413 y Fg(r)r Fz(\000)p Fj(1)710
399 y Fo(\))h Fk(\\)f(f)p Fi(u)28 b(>)f(\033)s Fk(g)p
Fo(.)46 b(Therefore,)33 b(recalling)g(that)g Fi(\033)e
Fo(=)c Fk(\000)p Fo(1)32 b(in)g Fi(B)2753 413 y Fg(r)r
Fz(\000)p Fj(1)2881 399 y Fo(,)g(splitting)h(the)f(domain)150
511 y(of)i(in)m(tegration)h(in)m(to)f(the)g(sets)g Fk(f)p
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Fk(g)p Fo(,)k(using)f(\(13.24\))j(and)d(our)g(h)m(yp)s(otheses)g(on)h
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935 y Fg(B)832 943 y Fd(r)886 752 y Fh(\020)940 853 y
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875 y Fg(i;j)1197 853 y Fi(\033)18 b Fo(\()p Fi(u)j Fk(\000)f
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853 y Fi(a)1827 867 y Fg(i;j)1907 853 y Fo(\))p Fi(@)1990
867 y Fg(j)2027 853 y Fi(\033)f Fo(\()p Fi(u)h Fk(\000)g
Fi(\033)s Fo(\))2386 752 y Fh(\021)2466 853 y Fo(=)728
1101 y(=)824 977 y Fh(Z)875 1183 y Fj(\()p Fg(B)955 1191
y Fd(r)990 1183 y Fz(\000)p Fg(B)1098 1192 y Fd(r)r Fq(\000)p
Fc(1)1211 1183 y Fj(\))p Fz(\\f)p Fg(u>\033)r Fz(g)1514
1000 y Fh(\020)1568 1101 y Fi(a)1616 1115 y Fg(i;j)1696
1101 y Fi(@)1749 1063 y Fj(2)1744 1123 y Fg(i;j)1825
1101 y Fi(\033)e Fo(\()p Fi(u)j Fk(\000)f Fi(\033)s Fo(\))g(+)g(\()p
Fi(@)2378 1115 y Fg(i)2407 1101 y Fi(a)2455 1115 y Fg(i;j)2535
1101 y Fo(\))p Fi(@)2618 1115 y Fg(j)2655 1101 y Fi(\033)f
Fo(\()p Fi(u)h Fk(\000)g Fi(\033)s Fo(\))3014 1000 y
Fh(\021)3094 1101 y Fk(\024)728 1357 y(\024)25 b Fi(C)7
b Fk(L)959 1319 y Fg(n)1005 1256 y Fh(\020)1060 1357
y Fo(\()p Fi(B)1164 1371 y Fg(r)1222 1357 y Fk(\000)20
b Fi(B)1382 1371 y Fg(r)r Fz(\000)p Fj(1)1510 1357 y
Fo(\))h Fk(\\)e(f)p Fi(u)26 b(>)f(\022)s Fk(g)1956 1256
y Fh(\021)2030 1357 y Fo(+)20 b Fi(C)2208 1233 y Fh(Z)2258
1439 y Fj(\()p Fg(B)2338 1447 y Fd(r)2374 1439 y Fz(\000)p
Fg(B)2482 1448 y Fd(r)r Fq(\000)p Fc(1)2595 1439 y Fj(\))p
Fz(\\f)p Fg(u)p Fz(\024)p Fg(\022)r Fz(g)2875 1357 y
Fo(\()p Fi(u)g Fo(+)g(1\))3153 1319 y Fg(d)3189 1296
y Fq(0)3242 1357 y Fk(\024)728 1613 y(\024)25 b Fi(C)7
b Fk(L)959 1575 y Fg(n)1005 1512 y Fh(\020)1060 1613
y Fo(\()p Fi(B)1164 1627 y Fg(r)1222 1613 y Fk(\000)20
b Fi(B)1382 1627 y Fg(r)r Fz(\000)p Fj(1)1510 1613 y
Fo(\))h Fk(\\)e(f)p Fi(u)26 b(>)f(\022)s Fk(g)1956 1512
y Fh(\021)2030 1613 y Fo(+)20 b Fi(C)2208 1489 y Fh(Z)2258
1695 y Fj(\()p Fg(B)2338 1703 y Fd(r)2374 1695 y Fz(\000)p
Fg(B)2482 1704 y Fd(r)r Fq(\000)p Fc(1)2595 1695 y Fj(\))p
Fz(\\f)p Fg(u)p Fz(\024)p Fg(\022)r Fz(g)2890 1613 y
Fi(F)13 b Fo(\()p Fi(x;)i(u)p Fo(\))g Fi(:)150 1867 y
Fo(Collecting)32 b(all)f(these)f(estimates,)i(the)e(pro)s(of)g(of)g
(\(13.19\))j(follo)m(ws.)41 b(This)30 b(\014nishes)e(the)j(pro)s(of)e
(of)h(\(ii\))h(in)150 1980 y(the)g(\014rst)e(case.)291
2139 y Ff(Case)40 b(2:)50 b Fo(if)35 b Fi(d)f Fo(=)f(2.)55
b(W)-8 b(e)37 b(use)d(suitable)i(p)s(ositiv)m(e)g(parameters)g(\002)e
(and)h Fi(T)13 b Fo(:)50 b(w)m(e)36 b(will)f(\014x)g(\002)f(small)150
2251 y(enough)c(and)g(then)g(c)m(ho)s(ose)h Fi(T)43 b
Fo(suitably)31 b(large.)291 2364 y(Let)g Fi(k)e Fk(2)d
Ff(N)p Fo(.)41 b(W)-8 b(e)32 b(use)e(a)h(barrier)f(function)h
Fi(h)26 b Fo(=)f Fi(h)2037 2379 y Fg(k)2106 2364 y Fk(2)g
Fo(C)2258 2331 y Fj(2)2298 2364 y Fo(\()p Fi(B)2402 2383
y Fj(\()p Fg(k)r Fj(+1\))p Fg(T)2640 2364 y Fo(\))31
b(so)g(that)h Fk(\000)p Fo(1)26 b Fk(\024)f Fi(h)h Fk(\024)g
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b(B)398 2496 y Fj(\()p Fg(k)r Fj(+1\))p Fg(T)637 2477
y Fo(,)1064 2604 y(\()p Fi(h)21 b Fo(+)e(1\))i(+)f Fk(jr)p
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2604 y Fi(h)p Fk(j)25 b(\024)g Fi(C)7 b Fo(\()p Fi(h)21
b Fo(+)f(1\))26 b Fk(\024)f Fi(C)7 b(e)2672 2566 y Fz(\000)p
Fj(\002)p Fg(T)3472 2604 y Fo(\(13.25\))150 2764 y(in)30
b Fi(B)325 2779 y Fg(k)r(T)419 2764 y Fo(,)g(and)1424
2877 y Fk(jr)p Fi(h)p Fk(j)20 b Fo(+)g Fk(j)p Fi(D)1816
2840 y Fj(2)1856 2877 y Fi(h)p Fk(j)26 b(\024)f Fi(C)7
b Fo(\002\()p Fi(h)20 b Fo(+)g(1\))996 b(\(13.26\))150
3038 y(in)33 b Fi(B)328 3056 y Fj(\()p Fg(k)r Fj(+1\))p
Fg(T)567 3038 y Fo(.)50 b(F)-8 b(rom)34 b(the)f(last)i(form)m(ula)e
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(if)f(\002)g(is)h(suitably)150 3151 y(small,)1402 3264
y Fk(jr)p Fi(h)p Fk(j)21 b Fo(+)f Fk(j)p Fi(D)1795 3226
y Fj(2)1834 3264 y Fi(h)p Fk(j)26 b(\024)2033 3182 y(p)p
2109 3182 71 4 v 82 x Fo(\002)o Fi(F)2237 3278 y Fg(u)2283
3264 y Fo(\()p Fi(x;)15 b(h)p Fo(\))975 b(\(13.27\))150
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4 v 876 4970 a(A)970 4784 y Fh(Z)1021 4990 y Fg(B)1074
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34 33 bop 291 399 a Fo(Let)32 b(us)e(no)m(w)h(estimate)i(the)e(righ)m
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%%Page: 35 35
35 34 bop 150 399 a Fo(Let)25 b(us)f(no)m(w)g(de\014ne)g
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y Fz(\000)p Fj(\002)p Fg(T)429 3922 y Fo(.)64 b(This)37
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Fi(d)26 b Fo(=)f(2.)291 4197 y Ff(Case)43 b(3:)57 b Fo(if)38
b Fi(d)h Fo(=)f(0.)65 b(This)37 b(will)i(b)s(e)e(treated)i(similarly)g
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4310 y(to)d(tak)m(e)h(deriv)-5 b(ativ)m(es)36 b(of)g(the)f(co)s
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Fi(a)716 4437 y Fg(i;j)796 4423 y Fo(\).)42 b(De\014ning)30
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4536 y(ellipticit)m(y)h(of)e(the)f(co)s(e\016cien)m(ts)i(and)e(the)h
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4691 y Fh(\020)886 4668 y(Z)937 4874 y Fg(B)990 4882
y Fd(r)1044 4792 y Fi(\014)1134 4722 y Fc(2)p Fd(n)p
1110 4734 117 3 v 1110 4775 a(n)p Fq(\000)p Fc(1)1241
4691 y Fh(\021)1305 4685 y Fd(n)p Fq(\000)p Fc(1)p 1305
4698 V 1345 4739 a Fd(n)1462 4792 y Fk(\024)663 5039
y(\024)83 b Fi(C)904 4916 y Fh(Z)954 5122 y Fg(B)1007
5130 y Fd(r)1042 5122 y Fz(\\f)p Fg(u)p Fz(\000)p Fg(\033)r
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y(\024)83 b Fi(C)7 b(A)957 5194 y Fh(\020)1026 5172 y(Z)1077
5378 y Fg(B)1130 5386 y Fd(r)1165 5378 y Fz(\\f)p Fj(0)p
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Fz(g)1695 5295 y Fo(\()p Fk(jr)p Fi(u)p Fk(j)1908 5258
y Fj(2)1968 5295 y Fk(\000)20 b(jr)p Fi(\033)s Fk(j)2240
5258 y Fj(2)2300 5295 y Fk(\000)g Fo(2)p Fk(r)p Fo(\()p
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Fo(\))3033 5194 y Fh(\021)3108 5295 y Fo(+)1905 5656
y(35)p eop
%%Page: 36 36
36 35 bop 817 431 a Fo(+)898 369 y Fi(C)p 898 410 72
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898 922 V 900 1005 a(A)994 819 y Fh(Z)1045 1025 y Fg(B)1098
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Fh(h)1015 1075 y(Z)1065 1281 y Fg(B)1118 1289 y Fd(r)1153
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Fj(1+)p Fg(\022)r Fz(g)1699 1199 y Fk(jr)p Fi(\033)s
Fk(j)1880 1161 y Fj(2)1940 1199 y Fo(+)20 b(\()p Fi(F)13
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b Fo(\()p Fi(x;)i(u)p Fo(\)\))22 b(+)817 1455 y(+2)948
1331 y Fh(Z)999 1537 y Fg(B)1052 1545 y Fd(r)1087 1537
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y Fo(+)2126 1393 y Fi(C)p 2126 1434 V 2128 1517 a(A)2223
1331 y Fh(Z)2273 1537 y Fg(B)2326 1545 y Fd(r)2361 1537
y Fz(\\f)p Fg(u)p Fz(\000)p Fg(\033)r Fz(\024)p Fj(1+)p
Fg(\022)r Fz(g)2801 1455 y Fo(\()p Fi(u)21 b Fk(\000)f
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k(w)m(e)d(imme-)150 1835 y(diately)i(get)816 1952 y Fh(Z)866
2158 y Fg(B)919 2166 y Fd(r)954 2158 y Fz(\\f)p Fg(u>\033)r
Fz(g)1230 2076 y Fk(jr)p Fi(h)p Fk(j)1408 2038 y Fj(2)1473
2076 y Fk(\024)25 b Fi(C)d Fk(L)1719 2038 y Fg(n)1765
1975 y Fh(\020)1820 2076 y Fo(\()p Fi(B)1924 2090 y Fg(r)1982
2076 y Fk(\000)e Fi(B)2142 2090 y Fg(r)r Fz(\000)p Fj(1)2270
2076 y Fo(\))h Fk(\\)e(f)p Fi(u)26 b(>)f(\033)s Fk(g)2725
1975 y Fh(\021)2805 2076 y Fk(\024)662 2332 y(\024)83
b Fi(C)22 b Fk(L)966 2294 y Fg(n)1012 2231 y Fh(\020)1067
2332 y Fo(\()p Fi(B)1171 2346 y Fg(r)1229 2332 y Fk(\000)e
Fi(B)1389 2346 y Fg(r)r Fz(\000)p Fj(1)1517 2332 y Fo(\))h
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2332 y Fo(+)20 b Fi(C)2215 2208 y Fh(Z)2265 2414 y Fj(\()p
Fg(B)2345 2422 y Fd(r)2381 2414 y Fz(\000)p Fg(B)2489
2423 y Fd(r)r Fq(\000)p Fc(1)2602 2414 y Fj(\))p Fz(\\f)p
Fg(u)p Fz(\024)p Fg(\022)r Fz(g)2897 2332 y Fi(\037)2954
2350 y Fj(\()p Fz(\000)p Fj(1)p Fg(;)p Fj(1\))3158 2332
y Fo(\()p Fi(u)p Fo(\))15 b Fi(:)150 2598 y Fo(F)-8 b(rom)40
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(Theorems)g(3.1)i(and)d(10.1)j(is)f(completed.)150 3149
y Fn(Bibliograph)l(y)189 3331 y Fx([A)n(C)45 b(])35 b(Alt,)i(H.)f(W.;)j
(Ca\013arelli,)c(L.)g(A.)g({)g Fe(Existenc)l(e)h(and)g(r)l(e)l
(gularity)h(for)g(a)g(minimum)f(pr)l(oblem)i(with)f(fr)l(e)l(e)377
3430 y(b)l(oundary)p Fx(,)29 b(J.)e(Reine)h(Angew.)37
b(Math.)g(325)26 b(\(1981\),)g(105{144.)93 3580 y([A)n(CF1)45
b(])37 b(Alt,)i(H.)e(W.;)42 b(Ca\013arelli,)c(L.)e(A.;)42
b(F)-7 b(riedman,)39 b(A.)d({)h Fe(Asymmetric)h(jet)g(\015ows)p
Fx(,)h(Comm.)64 b(Pure)35 b(Appl.)377 3679 y(Math.)i(35)27
b(\(1982\),)f(no.)37 b(1,)27 b(29{68.)93 3829 y([A)n(CF2)45
b(])26 b(Alt,)h(H.)f(W.;)h(Ca\013arelli,)e(L.)h(A.;)h(F)-7
b(riedman,)26 b(A.)g({)f Fe(A)j(fr)l(e)l(e)g(b)l(oundary)h(pr)l(oblem)g
(for)g(quasiline)l(ar)g(el)t(liptic)377 3928 y(e)l(quations)p
Fx(,)f(Ann.)38 b(Scuola)27 b(Norm.)36 b(Sup.)h(Pisa)27
b(Cl.)37 b(Sci.)g(\(4\))28 b(11)e(\(1984\),)h(no.)36
b(1,)28 b(1{44.)93 4078 y([A)n(CF3)45 b(])35 b(Alt,)j(H.)d(W.;)k
(Ca\013arelli,)d(L.)f(A.;)j(F)-7 b(riedman,)37 b(A.)e({)g
Fe(Jets)g(with)i(two)g(\015uids.)59 b(I.)37 b(One)e(fr)l(e)l(e)i(b)l
(oundary)p Fx(,)377 4177 y(Indiana)27 b(Univ.)38 b(Math.)f(J.)27
b(33)g(\(1984\),)f(no.)37 b(2,)27 b(213{247.)208 4327
y([Ba)45 b(])c(Bangert,)h(V.)f({)f Fe(L)l(aminations)i(of)g
Fx(3)p Fe(-tori)f(by)h(le)l(ast)g(ar)l(e)l(a)g(surfac)l(es)p
Fx(,)i(Analysis,)f(et)d(cetera,)j(85{114,)377 4426 y(Academic)28
b(Press,)e(Boston,)h(MA,)h(1990.)199 4576 y([BZ)45 b(])21
b(Birkho\013,)h(G.;)i(Zaran)n(tonello,)c(E.)g(H.)i({)e
Fe(Jets,)25 b(wakes,)h(and)e(c)l(avities)p Fx(,)g(Academic)d(Press)f
(Inc.,)i(Publishers,)377 4675 y(New)28 b(Y)-7 b(ork,)27
b(1957.)131 4825 y([BPV)45 b(])29 b(Bellettini,)h(G.;)g(P)n(aolini,)d
(M.;)j(V)-7 b(en)n(turini)29 b(S.)g({)f Fe(Some)j(R)l(esults)f(on)h
(Surfac)l(e)g(Me)l(asur)l(es)g(in)g(Calculus)g(of)377
4925 y(V)-6 b(ariations)31 b Fx(,)c(Ann.)38 b(Mat.)f(Pura)26
b(Appl.)38 b(\(4\),)28 b(170)e(\(1996\),)g(329{359.)250
5074 y([B)45 b(])40 b(Bouc)n(hitt)n(\022)-39 b(e,)42
b(G.)e({)g Fe(Singular)h(p)l(erturb)l(ations)g(of)h(variational)h(pr)l
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y(tr)l(ansition)30 b(mo)l(del)p Fx(,)f(Appl.)37 b(Math.)g(Optim.)g(21)
27 b(\(1990\),)f(no.)37 b(3,)27 b(289{314.)249 5323 y([C)45
b(])28 b(Ca\013arelli,)f(L.)g(A.)h({)g Fe(The)i(obstacle)h(pr)l(oblem)p
Fx(,)e(preprin)n(t,)e(1998.)1905 5656 y Fo(36)p eop
%%Page: 37 37
37 36 bop 189 399 a Fx([CC)45 b(])36 b(Ca\013arelli,)i(L.)e(A.;)k
(C\023)-42 b(ordoba,)37 b(A.)f({)g Fe(Uniform)i(c)l(onver)l(genc)l(e)f
(of)i(a)e(singular)h(p)l(erturb)l(ation)g(pr)l(oblem)p
Fx(,)377 498 y(Comm.)f(Pure)27 b(Appl.)37 b(Math.)g(48)27
b(\(1995\),)f(no.)37 b(1,)27 b(1{12.)197 646 y([CL)45
b(])27 b(Ca\013arelli,)e(L.)i(A.;)g(de)f(la)g(Lla)n(v)n(e,)g(R.)g({)g
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746 y(00-368.)135 894 y([DEF)45 b(])30 b(Dolcini,)g(A.;)h(Esp)r(osito,)
e(L.;)i(F)-7 b(usco,)29 b(N.)h({)f Fv(C)1878 864 y Fs(0)p
Ft(;\013)2010 894 y Fe(r)l(e)l(gularity)j(of)g Fv(!)s
Fe(-minima)p Fx(,)f(Boll.)42 b(Un.)h(Mat.)f(Ital.)h(A)377
994 y(\(7\))28 b(10)f(\(1996\),)f(no.)37 b(1,)27 b(113{125.)244
1142 y([G)45 b(])30 b(Giaquin)n(ta,)g(M.)g({)f Fe(Intr)l(o)l(duction)i
(to)h(r)l(e)l(gularity)g(the)l(ory)g(for)h(nonline)l(ar)f(el)t(liptic)i
(systems)p Fx(,)c(Birkh\177)-42 b(auser)377 1242 y(V)-7
b(erlag,)27 b(Basel,)g(1993.)178 1390 y([GG)46 b(])33
b(Giaquin)n(ta,)h(M.;)i(Giusti,)f(E.)d({)h Fe(On)h(the)g(r)l(e)l
(gularity)h(of)h(the)f(minima)g(of)h(variational)g(inte)l(gr)l(als)p
Fx(,)f(Acta)377 1490 y(Math.)i(148)26 b(\(1982\),)h(31{46.)184
1638 y([GT)45 b(])26 b(Gilbarg,)f(D.;)i(T)-7 b(rudinger,)25
b(N.)h(S.)g({)f Fe(El)t(liptic)30 b(p)l(artial)g(di\013er)l(ential)f(e)
l(quations)f(of)h(se)l(c)l(ond)f(or)l(der.)39 b(Se)l(c)l(ond)377
1738 y(e)l(dition)p Fx(,)29 b(Springer-V)-7 b(erlag,)25
b(Berlin-New)i(Y)-7 b(ork,)27 b(1983.)192 1886 y([GL)45
b(])27 b(Ginzburg,)f(V.;)i(Landau,)e(L.)g({)g Fe(On)i(the)h(the)l(ory)h
(of)f(sup)l(er)l(c)l(onductivity)p Fx(,)f(Zh.)2897 1865
y(\022)2890 1886 y(Eksp)r(er.)35 b(T)-7 b(eoret.)36 b(Fiz.)h(20)377
1985 y(\(1950\),)27 b(1064{1082.)187 2134 y([GP)45 b(])34
b(Ginzburg,)g(V.)f(L.;)j(Pitaevski)-9 b(\025)-32 b(\020,)33
b(L.)g(P)-7 b(.)33 b({)f Fe(On)i(the)h(the)l(ory)h(of)g(sup)l
(er\015uidity)p Fx(,)f(So)n(viet)d(Ph)n(ysics.)52 b(JETP)377
2233 y(34/7)26 b(\(1958\),)g(858{861.)202 2382 y([G1)45
b(])34 b(Giusti,)i(E.)d({)g Fe(Minimal)k(surfac)l(es)f(and)g(functions)
f(of)h(b)l(ounde)l(d)g(variation)p Fx(,)h(Monographs)31
b(in)j(Mathe-)377 2481 y(matics,)28 b(80.)36 b(Birkh\177)-42
b(auser)25 b(V)-7 b(erlag,)27 b(Basel-Boston,)e(Mass.,)i(1984.)202
2629 y([G2)45 b(])24 b(Giusti,)h(E.)e({)h Fe(Meto)l(di)j(dir)l(etti)g
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(Matematica)f(Italiana,)h(Bologna,)377 2729 y(1994.)197
2877 y([Gu)46 b(])35 b(Gurtin,)h(M.)f(E.)f({)f Fe(On)j(a)g(the)l(ory)g
(of)h(phase)h(tr)l(ansitions)d(with)i(interfacial)h(ener)l(gy)p
Fx(,)e(Arc)n(h.)57 b(Rational)377 2977 y(Mec)n(h.)37
b(Anal.)g(87)27 b(\(1985\),)f(no.)37 b(3,)27 b(187{212.)195
3125 y([HL)45 b(])35 b(Han,)g(Q.;)i(Lin,)f(F.)f({)f Fe(El)t(liptic)j(p)
l(artial)g(di\013er)l(ential)g(e)l(quations)p Fx(,)f(Couran)n(t)d
(Lecture)h(Notes)g(in)g(Math-)377 3225 y(ematics,)39
b(1.)64 b(New)38 b(Y)-7 b(ork)36 b(Univ)n(ersit)n(y)-7
b(,)38 b(Couran)n(t)e(Institute)h(of)g(Mathematical)g(Sciences,)i(New)e
(Y)-7 b(ork,)377 3324 y(1997.)246 3473 y([H)46 b(])30
b(Hedlund,)h(G.)e(A.)h(-)f Fe(Ge)l(o)l(desics)k(on)f(a)g
(two-dimensional)h(R)n(iemannian)f(manifold)h(with)f(p)l(erio)l(dic)i
(c)l(o)l(ef-)377 3572 y(\014cients)p Fx(,)28 b(Ann.)37
b(of)28 b(Math.)37 b(33)26 b(\(1932\),)h(719{739.)205
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b(Studies,)g(Amsterdam,)f(1984.)191 4117 y([Ma)45 b(])33
b(Mather,)h(J.)e(N.)h({)f Fe(Invariant)j(subsets)f(for)h(ar)l(e)l(a)g
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Fx(,)e(Math-)377 4216 y(ematical)e(analysis)f(and)h(applications,)h(P)n
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(Hamiltonian)h(systems)p Fx(,)d(T)-7 b(ransition)29 b(to)377
4564 y(c)n(haos)d(in)i(classical)e(and)h(quan)n(tum)g(mec)n(hanics)g
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4812 y([Md)45 b(])31 b(Mo)r(dica,)h(L.)f({)f Fe(The)k(gr)l(adient)f
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(interfac)l(e)g(criterion)p Fx(,)377 4911 y(Arc)n(h.)j(Rational)27
b(Mec)n(h.)36 b(Anal.)h(98)27 b(\(1987\),)f(no.)37 b(2,)27
b(123{142.)191 5060 y([Mo)45 b(])25 b(Morse,)g(H.)g(M.)h({)e
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5159 y(one)p Fx(,)g(T)-7 b(rans.)36 b(Amer.)h(Math.)g(So)r(c.)g(26)27
b(\(1924\),)f(no.)36 b(1,)28 b(25{60.)233 5308 y([M)45
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b(Lin.)37 b(3)27 b(\(1986\),)f(no.)37 b(3,)27 b(229{272.)1905
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%%Page: 38 38
38 37 bop 198 399 a Fx([OS)45 b(])34 b(Ow)n(en,)h(N.;)j(Stern)n(b)r
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end
userdict /end-hook known{end-hook}if
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