Content-Type: multipart/mixed; boundary="-------------0406170542281" This is a multi-part message in MIME format. ---------------0406170542281 Content-Type: text/plain; name="04-189.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-189.keywords" Ginzburg-Landau-Allen-Cahn phase transition models, $p$-Laplacian operator, sliding methods, geometric and qualitative properties of solutions. ---------------0406170542281 Content-Type: application/postscript; name="paper.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="paper.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: enricodino_fin.dvi %%Pages: 36 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips enricodino_fin -o paper.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.06.17:1105 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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Fw(1.)p eop %%Page: 5 5 5 4 bop 743 251 a Ft(MEAN)29 b(CUR)-7 b(V)g(A)i(TURE)29 b(PR)n(OPER)-5 b(TIES)28 b(F)n(OR)h Fq(p)p Ft(-LAPLA)n(CE)h(PHASE)e (TRANSITIONS)254 b(5)750 2609 y @beginspecial 0 @llx 0 @lly 563 @urx 526 @ury 2880 @rwi @setspecial %%BeginDocument: condition.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: condition.fig %%Creator: fig2dev Version 3.2 Patchlevel 4 %%CreationDate: Mon Jun 14 14:19:27 2004 %%For: enrico@mirsada (Enrico Valdinoci) %%BoundingBox: 0 0 563 526 %%Magnification: 1.0000 %%EndComments /MyAppDict 100 dict dup begin def /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 526 moveto 0 0 lineto 563 0 lineto 563 526 lineto closepath clip newpath -98.5 556.3 translate 1 -1 scale % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index oldshow % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proc char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % this is the pattern fill program from the Second edition Reference Manual % with changes to call the above pattern fill % left30 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P1 exch def /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /reencdict 12 dict def /ReEncode { reencdict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop 128 1 255 { newfont /Encoding get exch /.notdef put } for newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put } repeat newfontname newfont definefont pop end } def /isovec [ 8#055 /minus 8#200 /grave 8#201 /acute 8#202 /circumflex 8#203 /tilde 8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /hyphen 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 10 setmiterlimit 0 slj 0 slc 0.06299 0.06299 sc % % Fig objects follow % % % here starts figure with depth 50 % Ellipse 7.500 slw n 6007 4657 72 72 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Polyline n 3285 1935 m 8730 1935 l 8730 7380 l 3285 7380 l cp gs col0 s gr % Polyline 0.000 slw n 4995 6840 m 7020 6840 l 7020 7380 l 4995 7380 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 333.00 456.00] PATmp PATsp ef gr PATusp % Polyline 30.000 slw gs clippath 6067 4215 m 5992 4215 l 5992 4391 l 6030 4279 l 6067 4391 l cp eoclip n 6030 4680 m 6030 4230 l gs col0 s gr gr % arrowhead n 6067 4391 m 6030 4279 l 5992 4391 l 6030 4369 l 6067 4391 l cp gs 0.00 setgray ef gr col0 s % Polyline 7.500 slw [60] 0 sd n 6030 1035 m 6030 8370 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 3285 5265 m 8730 5265 l gs col0 s gr [] 0 sd % Polyline n 1575 8820 m 10485 8820 l 10485 495 l 1575 495 l cp gs col0 s gr % Polyline 2 slj 30.000 slw n 3285 4725 m 3287 4726 l 3292 4728 l 3300 4731 l 3313 4737 l 3332 4744 l 3355 4753 l 3383 4764 l 3416 4775 l 3451 4788 l 3490 4801 l 3529 4814 l 3570 4826 l 3610 4838 l 3650 4848 l 3689 4857 l 3727 4865 l 3763 4871 l 3797 4875 l 3831 4877 l 3863 4878 l 3894 4876 l 3924 4873 l 3954 4868 l 3983 4862 l 4013 4853 l 4043 4842 l 4073 4830 l 4096 4819 l 4120 4807 l 4145 4794 l 4170 4780 l 4195 4765 l 4221 4749 l 4248 4732 l 4276 4713 l 4304 4694 l 4333 4674 l 4362 4653 l 4393 4631 l 4423 4608 l 4455 4584 l 4487 4560 l 4520 4535 l 4553 4510 l 4586 4484 l 4620 4458 l 4654 4432 l 4688 4406 l 4723 4380 l 4757 4354 l 4792 4328 l 4826 4303 l 4860 4278 l 4894 4253 l 4929 4229 l 4962 4205 l 4996 4182 l 5030 4160 l 5063 4138 l 5097 4117 l 5130 4097 l 5164 4077 l 5198 4058 l 5233 4038 l 5270 4019 l 5307 4000 l 5344 3982 l 5382 3965 l 5421 3948 l 5460 3932 l 5500 3916 l 5541 3901 l 5582 3887 l 5623 3873 l 5665 3861 l 5707 3849 l 5750 3838 l 5792 3828 l 5834 3819 l 5876 3811 l 5918 3803 l 5959 3797 l 6000 3793 l 6039 3789 l 6078 3786 l 6116 3785 l 6153 3784 l 6189 3785 l 6223 3787 l 6257 3790 l 6289 3794 l 6320 3799 l 6350 3805 l 6378 3812 l 6406 3820 l 6432 3830 l 6458 3840 l 6487 3854 l 6515 3870 l 6543 3887 l 6570 3906 l 6595 3926 l 6621 3948 l 6646 3971 l 6670 3995 l 6694 4020 l 6718 4047 l 6741 4074 l 6764 4101 l 6787 4129 l 6809 4156 l 6831 4184 l 6853 4211 l 6875 4237 l 6897 4263 l 6918 4288 l 6940 4311 l 6961 4333 l 6983 4353 l 7004 4372 l 7026 4390 l 7048 4405 l 7071 4418 l 7094 4430 l 7118 4440 l 7141 4447 l 7165 4453 l 7190 4458 l 7216 4460 l 7243 4462 l 7271 4462 l 7301 4461 l 7331 4458 l 7362 4455 l 7395 4450 l 7428 4444 l 7462 4438 l 7496 4431 l 7531 4422 l 7566 4414 l 7602 4405 l 7637 4395 l 7672 4385 l 7707 4376 l 7742 4366 l 7776 4356 l 7809 4347 l 7842 4337 l 7873 4329 l 7905 4321 l 7935 4313 l 7964 4306 l 7993 4300 l 8020 4295 l 8048 4290 l 8078 4286 l 8108 4283 l 8137 4281 l 8167 4281 l 8196 4282 l 8226 4285 l 8256 4289 l 8287 4295 l 8319 4302 l 8352 4310 l 8386 4320 l 8422 4331 l 8458 4344 l 8495 4357 l 8532 4371 l 8567 4385 l 8601 4399 l 8632 4412 l 8660 4423 l 8683 4434 l 8701 4442 l 8714 4448 l 8723 4452 l 8728 4454 l 8730 4455 l gs col0 s gr % Polyline 7.500 slw gs clippath 4934 7228 m 4903 7159 l 4775 7216 l 4894 7205 l 4806 7285 l cp eoclip n 4905 7200 m 4867 7217 l 4844 7228 l 4814 7241 l 4780 7257 l 4741 7275 l 4700 7294 l 4658 7315 l 4615 7335 l 4574 7356 l 4534 7376 l 4497 7395 l 4463 7413 l 4433 7430 l 4405 7447 l 4381 7462 l 4360 7477 l 4342 7490 l 4326 7503 l 4314 7516 l 4304 7528 l 4296 7541 l 4290 7553 l 4285 7570 l 4284 7587 l 4286 7604 l 4290 7622 l 4298 7640 l 4308 7659 l 4319 7677 l 4332 7696 l 4346 7714 l 4359 7733 l 4373 7751 l 4386 7768 l 4397 7785 l 4406 7801 l 4413 7817 l 4418 7832 l 4419 7846 l 4418 7860 l 4413 7871 l 4407 7882 l 4397 7893 l 4384 7905 l 4368 7916 l 4348 7928 l 4324 7941 l 4296 7955 l 4265 7969 l 4230 7984 l 4191 7999 l 4151 8015 l 4109 8031 l 4067 8046 l 4028 8060 l 3993 8073 l 3964 8083 l 3942 8091 l 3927 8096 l 3919 8099 l 3916 8100 l 3915 8100 l gs col0 s gr gr % arrowhead 0 slj n 4806 7285 m 4894 7205 l 4775 7216 l 4811 7241 l 4806 7285 l cp gs 0.00 setgray ef gr col0 s /Times-Roman-iso ff 270.00 scf sf 3015 8370 m gs 1 -1 sc (here u<-1+) col0 sh gr /Symbol ff 270.00 scf sf 4275 8370 m gs 1 -1 sc (h) col0 sh gr /Times-Roman-iso ff 270.00 scf sf 7875 4230 m gs 1 -1 sc (u=0) col0 sh gr /Times-Roman-iso ff 270.00 scf sf 5715 4995 m gs 1 -1 sc (O) col0 sh gr /Symbol ff 270.00 scf sf 6165 4275 m gs 1 -1 sc (w) col0 sh gr /Times-Roman-iso ff 270.00 scf sf 6120 1890 m gs 1 -1 sc (l) col0 sh gr /Times-Roman-iso ff 270.00 scf sf 6120 5535 m gs 1 -1 sc (-l/10) col0 sh gr % here ends figure; $F2psEnd rs end showpage %%EndDocument @endspecial 1663 2779 a Fh(Condition)26 b(\(2.1\))555 3034 y Fw(Quan)n(tities)34 b(dep)r(ending)h(only)f(on)g FD(N)9 b Fw(,)36 b FD(p)e Fw(and)h(on)f(the)h(quan)n(tities)f(in)n(tro) r(duced)g(in)h(\(1.2\),)456 3134 y(\(1.3\))27 b(and)g(\(2.1\))h(will)g (b)r(e)g(often)f(referred)g(to)g(with)h(the)g(name)g(of)f(\\univ)n (ersal)f(constan)n(ts".)555 3234 y(With)j(these)e(con)n(v)n(en)n (tions,)f(w)n(e)h(are)g(ready)g(to)g(state)g(our)g(main)h(result:)456 3372 y FE(Theorem)j(2.1.)40 b Fg(L)l(et)30 b FD(u)24 b Fs(2)h FD(W)1438 3332 y Ft(1)p Fq(;p)1426 3397 y Ft(lo)r(c)1560 3372 y Fg(b)l(e)30 b(a)h(Sob)l(olev)h(we)l(ak)f(solution)f(of)i (\(1.5\))f(in)g(the)g(whole)g Fr(R)3375 3342 y Fq(N)456 3476 y Fg(satisfying)37 b(\(2.1\),)j(with)d Fs(j)p FD(u)p Fs(j)d(\024)g Fw(1)p Fg(.)58 b(L)l(et)36 b Fs(S)41 b Fw(=)35 b FD(@)5 b Fs(E)43 b Fg(b)l(e)36 b(a)h(c)l(ontinuous)e(hyp)l (ersurfac)l(e)j(in)e FD(R)3356 3445 y Fq(N)3419 3476 y Fg(.)456 3575 y(L)l(et)f FD(u)653 3587 y Fq(")688 3575 y Fw(\()p FD(x)p Fw(\))g(:=)f FD(u)p Fw(\()p FD(x=")p Fw(\))p Fg(.)57 b(Assume)35 b(that)h FD(u)1818 3587 y Fq(")1889 3575 y Fg(c)l(onver)l(ges)h(in)f FD(L)2431 3545 y Ft(1)2431 3599 y(lo)r(c)2553 3575 y Fg(to)g FD(\037)2711 3587 y Fo(E)2779 3575 y Fs(\000)23 b FD(\037)2919 3592 y Fq(R)2969 3575 y Fn(N)3022 3592 y Fo(nE)3137 3575 y Fg(and)36 b(that)456 3684 y Fs(f)p FD(u)546 3696 y Fq(")603 3684 y Fw(=)23 b(0)p Fs(g)29 b Fg(c)l(onver)l(ges)h(lo)l(c)l(al)t(ly)h (uniformly)g(to)f Fs(S)6 b Fg(,)30 b(i.e.,)i(that)e(for)g(any)h(c)l (omp)l(act)f(set)f FD(K)g Fs(\032)23 b Fr(R)3339 3654 y Fq(N)3407 3684 y Fg(,)456 3858 y Fw(\(2.2\))754 b(lim)1353 3909 y Fq(")p Fo(\000)-11 b(!)p Ft(0)1695 3858 y Fw(sup)1552 3931 y Fq(x)p Fo(2f)p Fq(u)1708 3939 y Fn(")1740 3931 y Ft(=0)p Fo(g\\)p Fq(K)1976 3858 y Fw(dist)15 b(\()p FD(x;)f Fs(S)6 b Fw(\))38 b(=)e(0)14 b FD(:)456 4094 y Fg(Then,)30 b Fs(S)37 b Fg(satis\014es)30 b(the)f(zer)l(o)i(me)l(an)e (curvatur)l(e)g(e)l(quation)h(in)g(the)g(visc)l(osity)h(sense.)555 4193 y(Mor)l(e)g(explicitly,)g(let)f FD(x)1308 4163 y Fq(?)1370 4193 y Fs(2)23 b(S)36 b Fg(b)l(e)30 b(so)g(that,)h(for)f(any) g FD(r)c(>)d Fw(0)456 4387 y(\(2.3\))362 b Ff(L)1044 4295 y Fp(\020)1094 4387 y FD(B)1157 4399 y Fq(r)1194 4387 y Fw(\()p FD(x)1273 4353 y Fq(?)1312 4387 y Fw(\))18 b Fs(\\)h Fw(\()p Fr(R)1522 4353 y Fq(N)1610 4387 y Fs(n)f(E)7 b Fw(\))1753 4295 y Fp(\021)1826 4387 y FD(>)23 b Fw(0)43 b Fg(and)h Ff(L)2229 4295 y Fp(\020)2279 4387 y FD(B)2342 4399 y Fq(r)2378 4387 y Fw(\()p FD(x)2457 4353 y Fq(?)2496 4387 y Fw(\))19 b Fs(\\)g(E)2672 4295 y Fp(\021)2745 4387 y FD(>)j Fw(0)14 b FD(;)456 4581 y Fg(wher)l(e)29 b Ff(L)g Fg(denotes)h(the)f FD(N)9 b Fg(-dimensional)30 b(L)l(eb)l(esgue)f(me)l(asur)l(e.)38 b(Assume)28 b(also)h(that)g Fs(S)36 b Fg(admits)456 4681 y(a)30 b(tangent)f(hyp)l(erplane)j(in)d FD(x)1388 4650 y Fq(?)1427 4681 y Fg(.)555 4780 y(Then:)661 4917 y Fs(\017)41 b Fg(if)35 b(a)g(p)l(ar)l(ab)l(oloid)i(with)e(vertex) f(in)g FD(x)1890 4887 y Fq(?)1963 4917 y Fg(touches)g Fs(S)41 b Fg(by)35 b(b)l(elow)g(at)f FD(x)2844 4887 y Fq(?)2882 4917 y Fg(,)i(then)e(its)g(me)l(an)744 5016 y(curvatur)l(e)c(at)f FD(x)1260 4986 y Fq(?)1329 5016 y Fg(must)f(b)l(e)i(non-p)l(ositive;)661 5116 y Fs(\017)41 b Fg(if)35 b(a)f(p)l(ar)l(ab)l(oloid)j(with)e(vertex)e(in)h FD(x)1888 5086 y Fq(?)1961 5116 y Fg(touches)g Fs(S)40 b Fg(by)35 b(ab)l(ove)g(at)f FD(x)2845 5086 y Fq(?)2883 5116 y Fg(,)i(then)e(its)f(me)l(an)744 5216 y(curvatur)l(e)d(at)f FD(x)1260 5185 y Fq(?)1329 5216 y Fg(must)f(b)l(e)i(non-ne)l(gative.)p eop %%Page: 6 6 6 5 bop 456 251 a Ft(6)681 b(BERARDINO)23 b(SCIUNZI)g(AND)f(ENRICO)h(V) -7 b(ALDINOCI)555 450 y Fg(In)25 b(p)l(articular,)j(if)f FD(x)1185 420 y Fq(?)1246 450 y Fs(2)d(S)32 b Fg(is)26 b(so)g(that)f Fs(S)32 b Fg(is)26 b FD(C)1991 420 y Ft(2)2054 450 y Fg(in)g(a)g(neighb)l(orho)l(o)l(d)i(of)e FD(x)2849 420 y Fq(?)2888 450 y Fg(,)g(then)g(the)f(me)l(an)456 550 y(curvatur)l(e)k(of)h Fs(S)37 b Fg(at)29 b FD(x)1155 520 y Fq(?)1224 550 y Fg(is)h(zer)l(o)g(in)g(the)g(classic)l(al)h (sense.)555 682 y Fw(The)25 b(pro)r(of)f(deeply)g(relies)g(on)g(the)h (ideas)f(of)h([17)o(],)g(whic)n(h)g(deals)f(with)h(the)g(case)e FD(p)g Fw(=)g(2,)i(and)456 781 y(it)20 b(is)g(geometric)f(in)h(nature.) 34 b(The)20 b(tec)n(hnique)g(presen)n(ted)f(seems)h(v)n(ery)f (\015exible,)i(and)f(it)h(ma)n(y)e(b)r(e)456 881 y(also)i(appropriate)g (for)h(further)g(in)n(teresting)g(extensions)g(\(suc)n(h)g(as)g(more)g (general)f(functionals)456 981 y(related)i(with)j(non)e(\015at)g (metrics)h(on)f(manifolds,)h(op)r(erators)d(in)j(non-div)n(ergence)d (form,)j(fully-)456 1080 y(nonlinear)34 b(equations,)i(etc.\).)60 b(In)36 b(the)f(case)g FD(p)g Fs(6)p Fw(=)h(2,)g(some)f(additional)g (care)f(is)h(needed)g(in)456 1180 y(the)23 b(sliding)g(pro)r(cedure,)g (since,)h(due)f(to)g(the)h(singularit)n(y)d(or)h(degeneracy)g(of)h(the) g FD(p)p Fw(-Laplacian)456 1279 y(op)r(erator)d(at)j(p)r(oin)n(ts)f (where)g Fs(r)p FD(u)g Fw(v)-5 b(anishes,)23 b(no)f(general)f(maxim)n (um)h(and)g(comparison)f(results)456 1379 y(are)28 b(a)n(v)-5 b(ailable)28 b(in)i(the)f(literature.)41 b(Also,)30 b(a)f(careful)g(c)n (hoice)f(of)h(parameters)f(is)h(necessary)f(to)456 1479 y(deal)f(with)h(the)g(more)f(sev)n(ere)f(nonlinearities)h(pro)n(vided)f (b)n(y)h(the)h FD(p)p Fw(-Laplace)e(equation.)555 1643 y(More)19 b(precisely)-7 b(,)20 b(Theorem)f(2.1)g(will)h(follo)n(ws)e (from)i(a)f(stronger)f(result)h(concerning)f(a)i(mean)456 1743 y(curv)-5 b(ature)33 b(prop)r(ert)n(y)-7 b(,)36 b(in)e(a)g(w)n(eak)g(viscosit)n(y)f(sense,)j(for)e(lev)n(el)g(sets)g (of)h(rescaled)e(solutions.)456 1842 y(Suc)n(h)27 b(result)h(is)f(the)h (follo)n(wing:)456 1974 y FE(Theorem)g(2.2.)38 b Fg(L)l(et)28 b FD(u)23 b Fs(2)g FD(W)1428 1944 y Ft(1)p Fq(;p)1519 1974 y Fw(\()p Fr(R)1605 1944 y Fq(N)1674 1974 y Fw(\))29 b Fg(b)l(e)f(a)g(Sob)l(olev)h(we)l(ak)g(solution)f(of)h(\(1.5\))h(in)e (the)g(whole)456 2074 y Fr(R)510 2044 y Fq(N)579 2074 y Fg(,)44 b(satisfying)e(\(2.1\),)k(so)41 b(that)g Fs(j)p FD(u)p Fs(j)j(\024)f Fw(1)d Fg(and)i FD(u)p Fw(\(0\))h(=)g(0)p Fg(.)72 b(L)l(et)41 b Ff(d)j Fs(2)g Fw(\(0)p FD(;)14 b Fw(1\))40 b Fg(and)i FD(M)52 b Fs(2)456 2173 y Fw(Mat\(\()p FD(N)27 b Fs(\000)18 b Fw(1\))h Fs(\002)f Fw(\()p FD(N)27 b Fs(\000)18 b Fw(1\)\))30 b Fg(with)1243 2339 y Fw(tr)13 b FD(M)32 b(>)23 b Ff(d)p Fs(k)p FD(M)9 b Fs(k)168 b Fg(and)j Fs(k)p FD(M)9 b Fs(k)21 b(\024)i Ff(d)2531 2305 y Fo(\000)p Ft(1)2634 2339 y FD(:)456 2503 y Fg(L)l(et)29 b FD(u)647 2515 y Fq(")682 2503 y Fw(\()p FD(x)p Fw(\))24 b(:=)f FD(u)p Fw(\()p FD(x=")p Fw(\))29 b Fg(and)933 2717 y Fw(\000)23 b(:=)1119 2600 y Fp(\032)1181 2717 y FD(x)h Fw(=)f(\()p FD(x)1419 2683 y Fo(0)1443 2717 y FD(;)14 b(x)1527 2729 y Fq(n)1572 2717 y Fw(\))23 b Fs(2)h Fr(R)1760 2683 y Fq(N)6 b Fo(\000)p Ft(1)1932 2717 y Fs(\002)18 b Fr(R)52 b Fg(s.t.)47 b FD(x)2327 2729 y Fq(N)2413 2717 y Fw(=)2511 2661 y(1)p 2511 2698 42 4 v 2511 2774 a(2)2562 2717 y FD(x)2609 2683 y Fo(0)2652 2717 y Fs(\001)18 b FD(M)9 b(x)2830 2683 y Fo(0)2854 2600 y Fp(\033)2943 2717 y FD(:)456 2931 y Fg(Then,)27 b(ther)l(e)g(exist)e(a)i(universal)f Ff(d)1548 2901 y Fq(?)1610 2931 y FD(>)c Fw(0)k Fg(and)g(a)h(function)f FD(\033)2361 2943 y Ft(0)2421 2931 y Fw(:)e(\(0)p FD(;)14 b Fw(1\))22 b Fs(\000)-14 b(!)23 b Fw(\(0)p FD(;)14 b Fw(1\))25 b Fg(such)i(that)e(if)456 3031 y FD(")f Fs(2)g Fw(\(0)p FD(;)14 b(\033)756 3043 y Ft(0)794 3031 y Fw(\()p Ff(d)p Fw(\)\))31 b Fg(and)g Ff(d)25 b Fs(2)g Fw(\(0)p FD(;)14 b Ff(d)1422 3001 y Fq(?)1460 3031 y Fw(\))p Fg(,)31 b(then)g Fw(\000)f Fg(c)l(annot)g(touch)h Fs(f)p FD(u)2402 3043 y Fq(")2461 3031 y Fw(=)24 b(0)p Fs(g)29 b Fg(by)i(b)l(elow)g(in)g FD(B)3159 3057 y Fe(d)p Fq(=)3228 3008 y Fo(p)p 3282 3008 133 3 v 3282 3057 a Ft(tr)11 b Fq(M)3419 3031 y Fg(:)456 3138 y(mor)l(e)30 b(explicitly,)456 3347 y Fw(\(2.4\))284 b Fs(f)p FD(u)1001 3359 y Fq(")1059 3347 y Fw(=)22 b(0)p Fs(g)32 b(\\)1349 3230 y Fp(\032)1412 3347 y FD(x)1459 3359 y Fq(N)1545 3347 y FD(<)1643 3291 y Fw(1)p 1643 3328 42 4 v 1643 3404 a(2)1694 3347 y FD(x)1741 3313 y Fo(0)1783 3347 y Fs(\001)19 b FD(M)9 b(x)1962 3313 y Fo(0)1985 3230 y Fp(\033)2080 3347 y Fs(\\)2167 3230 y Fp(\032)2230 3347 y Fs(j)p FD(x)p Fs(j)23 b FD(<)2542 3291 y Ff(d)p 2444 3328 238 4 v 2444 3345 a Fs(p)p 2513 3345 169 4 v 70 x Fw(tr)14 b FD(M)2691 3230 y Fp(\033)2800 3347 y Fs(6)p Fw(=)45 b Fs(;)14 b FD(:)555 3557 y Fw(W)-7 b(e)38 b(remark)d(that,)40 b(under)d(an)g(additional)f(h)n(yp)r (othesis)h(on)f(the)i(con)n(v)n(exit)n(y)d(of)i FD(h)3213 3569 y Ft(0)3287 3557 y Fw(near)456 3656 y Fs(\006)p Fw(1,)26 b(Class)g(A)i(minimizers)e(of)h Fs(F)35 b Fw(are)26 b(particular)g(solutions)g(satisfying)h(the)g(assumptions)g(of)456 3756 y(Theorems)f(2.1)h(and)g(2.2.)36 b(More)27 b(explicitly)-7 b(,)28 b(w)n(e)f(ha)n(v)n(e:)456 3888 y FE(Theorem)37 b(2.3.)44 b Fg(L)l(et)35 b FD(u)g Fg(b)l(e)h(a)g(Class)h(A)e (minimizers)h(of)h Fs(F)43 b Fg(with)36 b Fs(j)p FD(u)p Fs(j)e(\024)f Fw(1)p Fg(,)k(and)f(let)g FD(h)3300 3858 y Fo(0)3300 3909 y Ft(0)3372 3888 y Fg(b)l(e)456 3988 y(monotone)24 b(incr)l(e)l(asing)g(in)g Fw(\()p Fs(\000)p Fw(1)p FD(;)14 b Fs(\000)p Fw(1)5 b(+)g FD(\022)1714 3958 y Fo(\003)1750 3988 y Fw(\))g Fs([)g Fw(\(1)g Fs(\000)g FD(\022)2037 3958 y Fo(\003)2076 3988 y FD(;)14 b Fw(1\))p Fg(.)37 b(Then,)25 b(the)f(claims)h(of)f(The)l(or)l(ems)h(2.1)456 4087 y(and)30 b(2.2)h(hold)g(true.)555 4219 y Fw(The)c(pap)r(er)g(is)g (organized)f(as)g(follo)n(ws.)36 b(In)27 b Fs(x)h Fw(3)e(w)n(e)h (recall)f(some)h(standard)f(PDE)h(notions,)456 4319 y(suc)n(h)i(as)h (the)h(de\014nition)f(of)g(viscosit)n(y)f(solutions)h(and)g(some)f (comparison/maxim)n(um)f(prin-)456 4419 y(ciples)j(that)g(will)h(b)r(e) g(of)f(use)g(in)h(this)f(pap)r(er.)48 b(F)-7 b(or)31 b(making)f(the)i(pro)r(of)f(of)g(the)h(main)f(results)456 4518 y(more)25 b(readable,)h(w)n(e)g(collected)h(some)f(tec)n(hnical)g (lemmata,)h(mostly)f(elemen)n(tary)g(in)h(nature,)456 4618 y(in)36 b Fs(x)h Fw(4.)63 b(Of)36 b(course,)i(the)f(exp)r(ert)f (reader)f(ma)n(y)h(skip)g Fs(x)g Fw(3)g(and)h Fs(x)f Fw(4)g(and)g(dedicate)h(herself)456 4717 y(to)f(the)h(pro)r(of)f(of)h (the)g(main)f(results,)i(the)f(core)f(of)g(whic)n(h)h(is)f(con)n (tained)g(in)h Fs(x)g Fw(5)f(and)g Fs(x)h Fw(6.)456 4817 y(In)31 b(particular,)f Fs(x)h Fw(5)g(is)g(dev)n(oted)f(to)h(the)h (construction)e(of)h(suitable)g(barriers,)f(built)i(via)e(the)456 4917 y(one-dimensional)h(solution,)i(whic)n(h)f(will)h(b)r(e)g(used)f (in)h Fs(x)f Fw(6)g(for)g(the)h(sliding)f(metho)r(d.)52 b(Suc)n(h)456 5016 y(geometric)30 b(construction)i(is)g(an)f(extension) h(of)g(the)g(one)g(presen)n(ted)f(in)i([17)o(].)50 b(The)32 b(pro)r(of)g(of)456 5116 y(Theorem)e(2.2)g(will)h(b)r(e)h(completed)f (in)g Fs(x)g Fw(7,)h(while)f(the)h(one)e(of)i(Theorem)e(2.1)g(is)h(in)g Fs(x)g Fw(8.)47 b(In)456 5216 y Fs(x)27 b Fw(9)g(w)n(e)h(mak)n(e)e (commen)n(ts)i(on)f(the)h(assumption)f(in)h(\(2.1\))f(and)h(pro)n(v)n (e)e(Theorem)g(2.3.)p eop %%Page: 7 7 7 6 bop 743 251 a Ft(MEAN)29 b(CUR)-7 b(V)g(A)i(TURE)29 b(PR)n(OPER)-5 b(TIES)28 b(F)n(OR)h Fq(p)p Ft(-LAPLA)n(CE)h(PHASE)e (TRANSITIONS)254 b(7)1668 450 y Fw(3.)41 b Fv(PDE)30 b(tools)555 600 y Fw(W)-7 b(e)38 b(recall)f(here)g(the)h(de\014nition)g (of)f(viscosit)n(y)g(sup)r(ersolution)f(\(and)i(subsolution,)i(and)456 699 y(solution\))33 b(for)h FD(p)p Fw(-Laplacian)e(t)n(yp)r(e)i(op)r (erators.)54 b(Roughly)-7 b(,)35 b(the)f(notion)g(of)f(viscosit)n(y)g (sup)r(er-)456 799 y(solution)39 b(requires)f(a)i(p)r(oin)n(t)n(wise)f (ev)-5 b(aluation)40 b(of)f(the)i FD(p)p Fw(-Laplacian)d(of)i(smo)r (oth)f(functions)456 898 y(touc)n(hing)32 b(b)n(y)g(b)r(elo)n(w.)52 b(Ho)n(w)n(ev)n(er,)32 b(since)h(the)g FD(p)p Fw(-Laplacian)e(div)n (erges)g(at)h(critical)g(p)r(oin)n(ts)h(for)456 998 y(1)22 b FD(<)h(p)g(<)f Fw(2,)28 b(w)n(e)f(need)g(to)h Fg(exclude)i(this)g(c)l (ase)e Fw(from)f(the)h(follo)n(wing)f(de\014nition:)456 1117 y FE(De\014nition)k(3.1.)41 b Fw(Let)28 b(\012)d Fs(\022)f Fr(R)1470 1087 y Fq(N)1567 1117 y Fw(b)r(e)29 b(an)f(op)r(en)g(domain)g(and)g(let)h FD(u)24 b Fs(2)h FD(C)2794 1087 y Ft(0)2831 1117 y Fw(\(\012\).)40 b(If)29 b FD(p)24 b Fs(\025)g Fw(2,)k(w)n(e)456 1216 y(sa)n(y)22 b(that)i FD(u)g Fw(is)f(a)h(viscosit)n(y)e(sup)r(ersolution)h(of)h (\(1.5\))f(\(or)h(that)g(\001)2487 1228 y Fq(p)2525 1216 y FD(u)f Fs(\024)g FD(h)2732 1186 y Fo(0)2732 1237 y Ft(0)2769 1216 y Fw(\()p FD(u)p Fw(\))h(in)g(the)g(viscosit)n(y)456 1316 y(sense\))d(if,)j(whenev)n(er)d FD(x)1196 1328 y Ft(0)1256 1316 y Fs(2)j Fw(\012)e(and)f FD(\036)j Fs(2)f FD(C)1788 1286 y Ft(2)1826 1316 y Fw(\(\012\))f(are)f(suc)n(h)g(that)h FD(u)p Fw(\()p FD(x)2587 1328 y Ft(0)2625 1316 y Fw(\))h(=)g FD(\036)p Fw(\()p FD(x)2896 1328 y Ft(0)2934 1316 y Fw(\),)h FD(u)p Fw(\()p FD(x)p Fw(\))f Fs(\025)g FD(\036)p Fw(\()p FD(x)p Fw(\))456 1416 y(in)k(\012,)h(w)n(e)f(ha)n(v)n(e)1535 1518 y(\001)1604 1530 y Fq(p)1643 1518 y FD(\036)p Fw(\()p FD(x)1771 1530 y Ft(0)1809 1518 y Fw(\))37 b Fs(\024)g FD(h)2028 1484 y Fo(0)2028 1539 y Ft(0)2065 1518 y Fw(\()p FD(\036)p Fw(\()p FD(x)2225 1530 y Ft(0)2264 1518 y Fw(\)\))14 b FD(:)456 1639 y Fw(If)27 b(1)22 b FD(<)h(p)g(<)f Fw(2,)27 b(w)n(e)f(sa)n(y)f(that)i FD(u)g Fw(is)f(a)g(viscosit)n(y)g(sup)r (ersolution)f(of)i(\(1.5\))f(if,)i(whenev)n(er)d FD(x)3245 1651 y Ft(0)3306 1639 y Fs(2)e Fw(\012)456 1738 y(and)32 b FD(\036)h Fs(2)f FD(C)856 1708 y Ft(2)893 1738 y Fw(\(\012\))i(are)e (suc)n(h)g(that)h Fs(r)p FD(\036)p Fw(\()p FD(x)1769 1750 y Ft(0)1808 1738 y Fw(\))f Fs(6)p Fw(=)f(0,)j FD(u)p Fw(\()p FD(x)2194 1750 y Ft(0)2232 1738 y Fw(\))e(=)f FD(\036)p Fw(\()p FD(x)2520 1750 y Ft(0)2559 1738 y Fw(\),)j FD(u)p Fw(\()p FD(x)p Fw(\))f Fs(\025)e FD(\036)p Fw(\()p FD(x)p Fw(\))j(in)g(\012,)g(w)n(e)456 1838 y(ha)n(v)n(e)1535 1941 y(\001)1604 1953 y Fq(p)1643 1941 y FD(\036)p Fw(\()p FD(x)1771 1953 y Ft(0)1809 1941 y Fw(\))j Fs(\024)g FD(h)2028 1907 y Fo(0)2028 1961 y Ft(0)2065 1941 y Fw(\()p FD(\036)p Fw(\()p FD(x)2225 1953 y Ft(0)2264 1941 y Fw(\)\))14 b FD(:)456 2061 y Fw(Analogously)-7 b(,)36 b(if)h FD(p)g Fs(\025)g Fw(2,)h(w)n(e)e(sa)n(y)f(that)i FD(u)f Fw(is)g(a)g(viscosit)n (y)f(subsolution)g(of)i(\(1.5\))f(\(or)f(that)456 2161 y(\001)525 2173 y Fq(p)563 2161 y FD(u)29 b Fs(\025)f FD(h)781 2131 y Fo(0)781 2181 y Ft(0)818 2161 y Fw(\()p FD(u)p Fw(\))j(in)h(the)f(viscosit)n(y)f(sense\))h(if,)i(whenev)n(er)d FD(x)2315 2173 y Ft(0)2381 2161 y Fs(2)g Fw(\012)h(and)g FD(\036)e Fs(2)g FD(C)2949 2131 y Ft(2)2987 2161 y Fw(\(\012\))j(are)e (suc)n(h)456 2260 y(that)d FD(u)p Fw(\()p FD(x)762 2272 y Ft(0)800 2260 y Fw(\))c(=)g FD(\036)p Fw(\()p FD(x)1071 2272 y Ft(0)1109 2260 y Fw(\),)28 b FD(u)p Fw(\()p FD(x)p Fw(\))c Fs(\024)f FD(\036)p Fw(\()p FD(x)p Fw(\))29 b(in)f(\012,)f(w)n (e)h(ha)n(v)n(e)1535 2398 y(\001)1604 2410 y Fq(p)1643 2398 y FD(\036)p Fw(\()p FD(x)1771 2410 y Ft(0)1809 2398 y Fw(\))37 b Fs(\025)g FD(h)2028 2364 y Fo(0)2028 2419 y Ft(0)2065 2398 y Fw(\()p FD(\036)p Fw(\()p FD(x)2225 2410 y Ft(0)2264 2398 y Fw(\)\))14 b(;)456 2536 y(if)30 b(1)d FD(<)g(p)h(<)f Fw(2,)j(w)n(e)g(sa)n(y)f(that)i FD(u)f Fw(is)g(a)g(viscosit)n(y)f(subsolution)g(of)i(\(1.5\))f(if,)h (whenev)n(er)e FD(x)3236 2548 y Ft(0)3302 2536 y Fs(2)e Fw(\012)456 2635 y(and)32 b FD(\036)h Fs(2)f FD(C)856 2605 y Ft(2)893 2635 y Fw(\(\012\))i(are)e(suc)n(h)g(that)h Fs(r)p FD(\036)p Fw(\()p FD(x)1769 2647 y Ft(0)1808 2635 y Fw(\))f Fs(6)p Fw(=)f(0,)j FD(u)p Fw(\()p FD(x)2194 2647 y Ft(0)2232 2635 y Fw(\))e(=)f FD(\036)p Fw(\()p FD(x)2520 2647 y Ft(0)2559 2635 y Fw(\),)j FD(u)p Fw(\()p FD(x)p Fw(\))f Fs(\024)e FD(\036)p Fw(\()p FD(x)p Fw(\))j(in)g(\012,)g (w)n(e)456 2735 y(ha)n(v)n(e)1535 2838 y(\001)1604 2850 y Fq(p)1643 2838 y FD(\036)p Fw(\()p FD(x)1771 2850 y Ft(0)1809 2838 y Fw(\))j Fs(\025)g FD(h)2028 2804 y Fo(0)2028 2858 y Ft(0)2065 2838 y Fw(\()p FD(\036)p Fw(\()p FD(x)2225 2850 y Ft(0)2264 2838 y Fw(\)\))14 b FD(:)456 2958 y Fw(If)27 b FD(u)g Fw(is)h(b)r(oth)g(a)f(sup)r(ersolution)f(and)h(a)g (subsolution)g(in)h(the)g(viscosit)n(y)e(sense,)h(w)n(e)g(sa)n(y)f (that)i FD(u)456 3058 y Fw(is)f(a)g(viscosit)n(y)g(solution.)555 3253 y(Of)i(course,)e(if)i FD(u)23 b Fs(2)i FD(C)1245 3223 y Ft(2)1283 3253 y Fw(\(\012\))k(and)f FD(p)c Fs(\025)f Fw(2,)29 b(then)f FD(u)g Fw(is)h(a)e(viscosit)n(y)g(solution)h(of)h (\(1.5\))f(if,)h(and)456 3352 y(only)e(if,)456 3510 y(\(3.1\))370 b Fs(jr)p FD(u)p Fs(j)1160 3476 y Fq(p)p Fo(\000)p Ft(4)1283 3418 y Fp(\020)1332 3510 y Fs(jr)p FD(u)p Fs(j)1495 3476 y Ft(2)1532 3510 y Fw(\001)p FD(u)19 b Fw(+)f(\()p FD(p)g Fs(\000)g Fw(2\))c Fs(h)p FD(D)2117 3476 y Ft(2)2154 3510 y FD(u)g Fs(r)p FD(u)g(;)27 b Fs(r)p FD(u)p Fs(i)2546 3418 y Fp(\021)2619 3510 y Fw(=)22 b FD(h)2754 3476 y Fo(0)2754 3531 y Ft(0)2791 3510 y Fw(\()p FD(u)p Fw(\))456 3668 y(p)r(oin)n(t)n(wise.)38 b(Ho)n(w)n(ev)n(er,)27 b(if)i(1)24 b FD(<)g(p)g(<)g Fw(2,)k(the)h(expression)d(ab)r(o)n(v)n(e) h(ma)n(y)h(b)r(e)h(ill)f(de\014ned)h(ev)n(en)f(for)456 3768 y(smo)r(oth)d(functions,)h(due)g(to)g(the)g(v)-5 b(anishing)25 b(of)h(the)g(gradien)n(t.)35 b(Therefore,)25 b(if)h(1)c FD(<)h(p)g(<)f Fw(2,)k(for)456 3868 y(a)k(function)i FD(u)c Fs(2)h FD(C)1082 3838 y Ft(2)1119 3868 y Fw(\(\012\),)k(b)r (eing)e(a)f(viscosit)n(y)g(solution)g(of)h(\(1.5\))g(is)g(equiv)-5 b(alen)n(t)31 b(for)f(\(3.1\))h(to)456 3967 y(hold)c(at)h(p)r(oin)n(ts) f(where)g Fs(r)p FD(u)c Fs(6)p Fw(=)g(0.)555 4143 y(One)k(of)h(the)g (greatest)e(di\016cult)n(y)i(when)f(dealing)g(with)h FD(p)p Fw(-Laplace)e(equations)h(is)g(that)h(the)456 4243 y(solutions)19 b(b)r(elongs)h(generally)g(only)g(to)g(the)h(class) f FD(C)2091 4213 y Ft(1)p Fq(;\013)2212 4243 y Fw(with)h FD(\013)j(<)e Fw(1)e(\(see)h([8)o(])g(and)g([19)o(]\).)35 b(Also,)456 4342 y(the)30 b FD(p)p Fw(-Laplace)f(op)r(erator)f(is)i (singular)f(or)g(degenerate)g(elliptic)h(\(resp)r(ectiv)n(ely)g(if)h(1) 26 b FD(<)h(p)g(<)g Fw(2)456 4442 y(or)c FD(p)f(>)h Fw(2\).)36 b(A)24 b(consequence)f(of)h(suc)n(h)f(pathologies)g(is)g(that)i(there)e (is)h(no)g(general)e(comparison)456 4542 y(theorem)27 b(for)g(solutions)h(in)g(case)f FD(p)c Fs(6)p Fw(=)g(2.)37 b(Therefore,)27 b(no)h(complete)g(analogy)e(is)i(p)r(ossible,)f(in)456 4641 y(general,)f(b)r(et)n(w)n(een)h(the)h(cases)f FD(p)c Fw(=)f(2)28 b(and)f FD(p)c Fs(6)p Fw(=)f(2.)555 4817 y(In)38 b(this)h(pap)r(er,)h(w)n(e)d(will)h(need)g(to)g(compare)f(w)n (eak)g(Sob)r(olev)g(solutions)g(of)44 b(\(1.5\))38 b(with)456 4917 y(viscosit)n(y)28 b(sup)r(ersolutions)g(of)36 b(\(1.5\).)43 b(Ev)n(en)28 b(if)j(there)e(are)f(not)i(general)e(results)h(in)h(the)g (liter-)456 5016 y(ature)g(dealing)g(with)i(this)f(problem,)g(w)n(e)g (will)g(succeed)f(in)h(doing)g(this)g(b)n(y)g(exploiting)f(some)456 5116 y(results)h(obtained)g(in)h([6)o(],)h(together)d(with)i(Hopf)6 b('s)32 b(Lemma)g(for)f FD(p)p Fw(-Laplace)f(equations)g([21)o(],)456 5216 y(and)d(thanks)g(to)h(some)f(geometric)f(prop)r(erties)h(of)g (barriers)f(that)i(w)n(e)f(will)h(in)n(tro)r(duce.)p eop %%Page: 8 8 8 7 bop 456 251 a Ft(8)681 b(BERARDINO)23 b(SCIUNZI)g(AND)f(ENRICO)h(V) -7 b(ALDINOCI)555 450 y Fw(W)g(e)35 b(recall)f(no)n(w)h(the)g(maxim)n (um)g(and)f(comparison)f(principles)i(needed)g(in)g(our)f(pro)r(ofs.) 456 550 y(First)40 b(of)h(all,)i(in)e([6])f(\(see)h(in)g(particular)e (Theorem)h(1.4)f(there\))i(the)g(follo)n(wing)e(result)i(is)456 649 y(obtained:)456 767 y FE(Theorem)27 b(3.2)h Fw(\(Strong)d (Comparison)e(Principle)i(I\))p FE(.)39 b Fg(L)l(et)27 b Fw(\012)g Fg(b)l(e)h(an)f(op)l(en)h(\(not)f(ne)l(c)l(essarily)456 867 y(b)l(ounde)l(d)j(nor)g(c)l(onne)l(cte)l(d\))f(subset)g(of)i Fr(R)1733 837 y Fq(N)1802 867 y Fg(,)f Fw(\003)23 b Fs(2)g Fr(R)36 b Fg(and)30 b FD(u;)14 b(v)26 b Fs(2)d FD(C)2561 837 y Ft(1)2599 867 y Fw(\(\012\))30 b Fg(satisfy)456 1003 y Fw(\(3.2\))502 b Fs(\000)p Fw(\001)1263 1015 y Fq(p)1301 1003 y FD(u)18 b Fw(+)g(\003)p FD(u)k Fs(\024)h(\000)p Fw(\001)1800 1015 y Fq(p)1838 1003 y Fw(\()p FD(v)s Fw(\))c(+)f(\003)p FD(v)s(;)184 b(u)23 b Fs(\024)f FD(v)33 b Fg(in)d Fw(\012)p FD(:)456 1139 y Fg(De\014ne)k FD(Z)780 1151 y Fq(u;v)912 1139 y Fw(=)f Fs(f)p FD(x)g Fs(2)h Fw(\012)f(:)g Fs(j)p FD(D)r(u)p Fw(\()p FD(x)p Fw(\))p Fs(j)24 b Fw(+)e Fs(j)p FD(D)r(v)s Fw(\()p FD(x)p Fw(\))p Fs(j)35 b Fw(=)e(0)p Fs(g)h Fg(if)i FD(p)d Fs(6)p Fw(=)g(2)p Fg(,)k FD(Z)2699 1151 y Fq(u;v)2831 1139 y Fw(=)32 b Fs(;)j Fg(if)i FD(p)c Fw(=)g(2)p Fg(.)55 b(If)456 1238 y FD(x)503 1250 y Ft(0)563 1238 y Fs(2)24 b Fw(\012)10 b Fs(n)g FD(Z)821 1250 y Fq(u;v)944 1238 y Fg(and)26 b FD(u)p Fw(\()p FD(x)1228 1250 y Ft(0)1266 1238 y Fw(\))d(=)g FD(v)s Fw(\()p FD(x)1531 1250 y Ft(0)1569 1238 y Fw(\))p Fg(,)k(then)f FD(u)c Fw(=)h FD(v)29 b Fg(in)d(the)g(c)l(onne)l(cte)l(d)g(c)l(omp)l(onent)f (of)i Fw(\012)10 b Fs(n)g FD(Z)3347 1250 y Fq(u;v)456 1338 y Fg(c)l(ontaining)30 b FD(x)908 1350 y Ft(0)946 1338 y Fg(.)555 1456 y Fw(An)41 b(easy)e(consequence)g(of)h(the)h(ab)r (o)n(v)n(e)d(result)i(is)g(the)h(follo)n(wing)e(one,)k(whic)n(h)d(will) g(b)r(e)456 1555 y(suitable)27 b(for)g(our)g(applications)1475 1523 y Ft(4)1507 1555 y Fw(:)456 1673 y FE(Corollary)e(3.3)d Fw(\(Strong)e(Comparison)f(Principle)h(I)r(I\))p FE(.)36 b Fg(L)l(et)23 b Fw(\012)g Fg(b)l(e)g(an)h(op)l(en)f(\(not)g(ne)l(c)l (essarily)456 1773 y(b)l(ounde)l(d)30 b(nor)g(c)l(onne)l(cte)l(d\))f (subset)g(of)i Fr(R)1733 1743 y Fq(N)1802 1773 y Fg(,)f(and)g FD(u;)14 b(v)26 b Fs(2)d FD(C)2312 1743 y Ft(1)2350 1773 y Fw(\(\012\))30 b Fg(satisfy)456 1909 y Fw(\(3.3\))439 b Fs(\000)p Fw(\001)1200 1921 y Fq(p)1238 1909 y FD(u)18 b Fw(+)g FD(f)9 b Fw(\()p FD(u)p Fw(\))22 b Fs(\024)h(\000)p Fw(\001)1793 1921 y Fq(p)1831 1909 y Fw(\()p FD(v)s Fw(\))d(+)e FD(f)9 b Fw(\()p FD(v)s Fw(\))p FD(;)184 b(u)22 b Fs(\024)h FD(v)33 b Fg(in)d Fw(\012)14 b FD(;)456 2045 y Fg(with)28 b FD(f)36 b Fg(lo)l(c)l(al)t(ly)30 b(Lipschitz)f(c)l(ontinuous.)38 b(De\014ne)27 b FD(Z)2080 2057 y Fq(u;v)2201 2045 y Fw(=)c Fs(f)p FD(x)g Fs(2)g Fw(\012)g(:)g Fs(j)p FD(D)r(u)p Fw(\()p FD(x)p Fw(\))p Fs(j)15 b Fw(+)f Fs(j)p FD(D)r(v)s Fw(\()p FD(x)p Fw(\))p Fs(j)25 b Fw(=)d(0)p Fs(g)456 2144 y Fg(if)31 b FD(p)25 b Fs(6)p Fw(=)g(2)p Fg(,)31 b FD(Z)849 2156 y Fq(u;v)973 2144 y Fw(=)25 b Fs(;)30 b Fg(if)i FD(p)25 b Fw(=)g(2)p Fg(.)41 b(If)32 b FD(x)1618 2156 y Ft(0)1681 2144 y Fs(2)25 b Fw(\012)19 b Fs(n)g FD(Z)1958 2156 y Fq(u;v)2087 2144 y Fg(and)32 b FD(u)p Fw(\()p FD(x)2377 2156 y Ft(0)2414 2144 y Fw(\))26 b(=)f FD(v)s Fw(\()p FD(x)2684 2156 y Ft(0)2722 2144 y Fw(\))p Fg(,)32 b(then)e FD(u)25 b Fw(=)g FD(v)34 b Fg(in)d(the)456 2244 y(c)l(onne)l(cte)l(d)e(c)l(omp)l(onent)h(of)g Fw(\012)19 b Fs(n)f FD(Z)1538 2256 y Fq(u;v)1666 2244 y Fg(c)l(ontaining)30 b FD(x)2118 2256 y Ft(0)2156 2244 y Fg(.)456 2396 y(Pr)l(o)l(of.)43 b Fw(Let)18 b FD(\017)23 b(>)g Fw(0)18 b(b)r(e)g(so)g(that)h FD(B)1490 2408 y Fq(\017)1522 2396 y Fw(\()p FD(x)1601 2408 y Ft(0)1639 2396 y Fw(\))k Fs(\032)g Fw(\012)18 b(and)g(let)h FD(M)2204 2408 y Fq(u;v)2325 2396 y Fw(=)j(max)p Fs(fj)p FD(u)p Fs(j)2703 2411 y Fq(L)2749 2394 y Fm(1)2808 2411 y Ft(\()p Fq(B)2884 2419 y Fn(\017)2914 2411 y Ft(\()p Fq(x)2978 2419 y Fd(0)3010 2411 y Ft(\))3040 2396 y FD(;)28 b Fs(j)p FD(v)s Fs(j)3180 2411 y Fq(L)3226 2394 y Fm(1)3286 2411 y Ft(\()p Fq(B)3362 2419 y Fn(\017)3392 2411 y Ft(\()p Fq(x)3456 2419 y Fd(0)3488 2411 y Ft(\))3518 2396 y Fs(g)p Fw(.)456 2495 y(De\014ne)1371 2723 y(\003)23 b(=)165 b(sup)1678 2782 y Fn(U)5 b Fm(6)p Fd(=)p Fn(V)1550 2830 y Fm(j)p Fn(U)g Fm(j)p Fn(;)16 b Fm(j)p Fn(V)c Fm(j\024)p Fn(M)1849 2838 y(u;v)1973 2667 y Fs(j)p FD(f)d Fw(\()p FD(U)g Fw(\))19 b Fs(\000)f FD(f)9 b Fw(\()p FD(V)18 b Fw(\))p Fs(j)p 1973 2704 509 4 v 2087 2780 a(j)p FD(U)28 b Fs(\000)18 b FD(V)h Fs(j)2506 2723 y FD(:)456 2939 y Fw(Then,)1150 3075 y Fs(\000)p Fw(\001)1284 3087 y Fq(p)1322 3075 y FD(u)f Fw(+)g(\003)p FD(u)82 b Fs(\024)g(\000)p Fw(\001)1940 3087 y Fq(p)1979 3075 y FD(v)21 b Fw(+)d FD(f)9 b Fw(\()p FD(v)s Fw(\))19 b Fs(\000)f FD(f)9 b Fw(\()p FD(u)p Fw(\))18 b(+)g(\003)p FD(u)1659 3200 y Fs(\024)82 b(\000)p Fw(\001)1940 3212 y Fq(p)1979 3200 y FD(v)21 b Fw(+)d(\003)c Fs(j)p FD(v)22 b Fs(\000)c FD(u)p Fs(j)g Fw(+)g(\003)p FD(u)1659 3324 y Fw(=)82 b Fs(\000)p Fw(\001)1940 3336 y Fq(p)1979 3324 y FD(v)21 b Fw(+)d(\003)c(\()p FD(v)22 b Fs(\000)c FD(u)p Fw(\))g(+)g(\003)p FD(u)1659 3449 y Fw(=)82 b Fs(\000)p Fw(\001)1940 3461 y Fq(p)1979 3449 y FD(v)21 b Fw(+)d(\003)p FD(v)f(;)456 3585 y Fw(hence)27 b(the)h(result)g(follo)n(ws)e(from)h(Theorem)g(3.2.) 1370 b Fc(\003)555 3736 y Fw(W)-7 b(e)35 b(recall)f(that)h(a)g(similar) f(result)h(had)f(b)r(een)i(previously)d(pro)n(v)n(ed)g(b)n(y)i(P)-7 b(.)35 b(T)-7 b(olksdorf)33 b(in)456 3836 y([18)o(],)27 b(under)h(stronger)e(assumptions.)555 3936 y(W)-7 b(e)35 b(no)n(w)f(state)g(a)g(result)h(whic)n(h)f(will)h(allo)n(w)e(us)h(of)h (tak)n(e)f(care)f(of)h(the)h(p)r(oin)n(ts)g(in)g(whic)n(h)456 4035 y(solutions)19 b(of)26 b(\(1.5\))20 b(ma)n(y)f(ha)n(v)n(e)g(a)g(v) -5 b(anishing)20 b(gradien)n(t.)33 b(T)-7 b(o)19 b(this)h(aim,)i(w)n(e) d(recall)g(the)i(follo)n(wing)456 4135 y(v)n(ersion)26 b(of)h(a)g(more)g(general)f(result)i(pro)n(v)n(ed)1871 4103 y Ft(5)1930 4135 y Fw(b)n(y)f(J.)h(L.)f(V)-7 b(azquez)27 b(in)h([21)o(]:)p 456 4197 499 4 v 555 4270 a Ft(4)588 4296 y FC(As)e(a)g(tec)n(hnical)h(remark,)d(w)n(e)j(p)r(oin)n(t)f(out)h (that)g(w)n(e)f(will)e(use)i(Corollary)f(3.3)h(on)g(pages)h(25)f(and)h (30)f(here)456 4379 y(b)r(elo)n(w,)f(with)g FA(f)30 b FC(:=)21 b FA(h)1040 4355 y Fx(0)1040 4399 y Fz(0)1075 4379 y FC(.)35 b(In)26 b(our)f(case,)h FA(h)1559 4388 y Fz(0)1618 4379 y FC(is)f(not)h(assumed)e(to)i(b)r(e)g FA(C)2343 4355 y Fz(1)p Fy(;)p Fz(1)2452 4379 y FC(in)f(the)h(closed)g (in)n(tev)l(al)f([)p Fl(\000)p FC(1)p FA(;)12 b FC(1],)25 b(but)456 4462 y(only)d(in)h(the)g(op)r(en)h(one)f(\()p Fl(\000)p FC(1)p FA(;)12 b FC(1\).)31 b(Nev)n(ertheless,)23 b(w)n(e)g(will)e(b)r(e)i(able)g(to)h(exclude)f(touc)n(hing)h(p)r(oin)n (ts)f(at)h Fl(\006)p FC(1)e(with)456 4545 y(a)h(direct)h(argumen)n(t,)g (hence)h(w)n(e)f(will)e(apply)i(Corollary)f(3.3)g(in)h(the)g(domain)f (where)h FA(h)2802 4554 y Fz(0)2860 4545 y FC(is)f FA(C)2986 4521 y Fz(1)p Fy(;)p Fz(1)3070 4545 y FC(.)555 4607 y Ft(5)588 4633 y FC(Although)e(w)n(e)e(do)h(not)g(explicitly)f(assume)g Fl(j)p FA(u)p Fl(j)g FA(<)g FC(1)h(here,)g(but)g(only)f Fl(j)p FA(u)p Fl(j)g(\024)g FC(1,)h(w)n(e)g(think)g(it)f(is)g (appropriate)456 4716 y(to)25 b(notice)g(that,)h(in)e(man)n(y)g(cases)h (of)f(in)n(terest,)h(the)h(t)n(w)n(o)f(conditions)g(are)g(equiv)l(alen) n(t,)h(thanks)f(to)g(the)h(results)456 4799 y(men)n(tioned)g(in)g (these)h(pages.)40 b(Indeed,)28 b(the)f(condition)g Fl(j)p FA(u)p Fl(j)c FA(<)h FC(1,)j(under)f(suitable)h(assumptions,)f(is)f (ful\014lled)456 4882 y(b)n(y)30 b(an)n(y)h(solution)f FA(u)g FC(suc)n(h)h(that)g Fl(j)p FA(u)p Fl(j)f(\024)g FC(1)g(with)h Fl(j)p FA(u)p Fl(j)e FC(not)i(iden)n(tically)f(equal)h (to)g Fl(\006)p FC(1.)50 b(F)-6 b(or)30 b(instance,)j(let)d(us)456 4965 y(supp)r(ose)23 b(that)h(for)f(an)n(y)g FA(\022)f Fl(2)31 b FC([0)p FA(;)11 b(\022)1366 4942 y Fx(\003)1402 4965 y FC(\),)23 b FA(h)1513 4942 y Fx(0)1513 4986 y Fz(0)1547 4965 y FC(\()p Fl(\000)p FC(1)15 b(+)f FA(\022)r FC(\))20 b Fl(\024)f FA(c)1935 4942 y Fx(0)1957 4965 y FA(\022)1992 4942 y Fy(p)p Fx(\000)p Fz(1)2130 4965 y FC(and)k FA(h)2307 4942 y Fx(0)2307 4986 y Fz(0)2341 4965 y FC(\(1)15 b Fl(\000)f FA(\022)r FC(\))20 b Fl(\025)g(\000)p FA(c)2730 4942 y Fx(0)2752 4965 y FA(\022)2787 4942 y Fy(p)p Fx(\000)p Fz(1)2924 4965 y FC(\(this)j(condition)h(is)456 5050 y(in)d(particular)g(v)n(eri\014ed)h(b)n(y)f FA(h)1240 5059 y Fz(0)1275 5050 y FC(\()p FA(\020)5 b FC(\))20 b(:=)f(\(1)11 b Fl(\000)g FA(\020)1654 5026 y Fz(2)1689 5050 y FC(\))1716 5026 y Fy(p)1752 5050 y FC(\).)31 b(Let)22 b(us)f(assume)g(that)h(there)g(are)g(p)r(oin)n(ts)g(where)f FA(u)f FC(=)f(1)j(and)456 5133 y(let)k(us)f(sho)n(w)h(that)h(this)f(is) f(not)i(p)r(ossible)e(\(the)i(case)g FA(u)c FC(=)g Fl(\000)p FC(1)i(follo)n(ws)g(in)h(the)g(same)f(w)n(a)n(y\).)38 b(Since)27 b Fl(j)p FA(u)p Fl(j)d FC(is)h(not)456 5216 y(iden)n(tically)g(equal)g(to)g(1,)g(there)h(exists)f(a)g(ball)f FA(B)1781 5224 y Fy(r)1816 5216 y FC(\()p FA(x)p FC(\))h(with)g FA(u)c(<)g FC(1)k(in)g(the)g(in)n(terior)f(of)h FA(B)2886 5224 y Fy(r)2921 5216 y FC(\()p FA(x)p FC(\))g(and)g FA(u)p FC(\()p FA(y)r FC(\))e(=)e(1)p eop %%Page: 9 9 9 8 bop 743 251 a Ft(MEAN)29 b(CUR)-7 b(V)g(A)i(TURE)29 b(PR)n(OPER)-5 b(TIES)28 b(F)n(OR)h Fq(p)p Ft(-LAPLA)n(CE)h(PHASE)e (TRANSITIONS)254 b(9)456 450 y FE(Theorem)27 b(3.4)h Fw(\(Strong)c(Maxim)n(um)h(Principle)f(and)h(Hopf)6 b('s)25 b(Lemma\))p FE(.)39 b Fg(L)l(et)27 b Fw(\012)g Fg(b)l(e)h(an)f(op)l(en) 456 550 y(c)l(onne)l(cte)l(d)j(\(not)h(ne)l(c)l(essarily\))g(b)l(ounde) l(d)g(set)g(in)g Fr(R)2068 520 y Fq(N)2167 550 y Fg(and)h(supp)l(ose)f (that)g FD(u)24 b Fs(2)h FD(C)3022 520 y Ft(1)3060 550 y Fw(\(\012\))p Fg(,)32 b FD(u)24 b Fs(\025)h Fw(0)456 649 y Fg(in)k Fw(\012)p Fg(,)i(we)l(akly)g(solves)1411 769 y Fs(\000)p Fw(\001)1545 781 y Fq(p)1583 769 y FD(u)18 b Fw(+)g FD(cu)1816 735 y Fq(q)1875 769 y Fw(=)23 b FD(g)i Fs(\025)e Fw(0)84 b Fg(in)115 b Fw(\012)456 906 y Fg(with)28 b FD(q)e Fs(\025)c FD(p)14 b Fs(\000)g Fw(1)p Fg(,)27 b FD(c)c Fs(\025)g Fw(0)k Fg(and)h FD(g)d Fs(2)f FD(L)1589 876 y Fo(1)1589 929 y Fq(loc)1676 906 y Fw(\(\012\))p Fg(.)38 b(If)28 b FD(u)f Fg(is)h(not)f(identic)l(al)t(ly)j(zer)l(o,)f (then)e FD(u)c(>)f Fw(0)27 b Fg(in)h Fw(\012)p Fg(.)456 1006 y(Mor)l(e)l(over,)37 b(for)f(any)f(p)l(oint)g FD(x)1416 1018 y Ft(0)1486 1006 y Fs(2)e FD(@)5 b Fw(\012)34 b Fg(wher)l(e)h(the)g(interior)h(spher)l(e)g(c)l(ondition)f(is)h (satis\014e)l(d,)456 1105 y(and)c(such)f(that)h FD(u)f Fg(is)h FD(C)1216 1075 y Ft(1)1285 1105 y Fg(in)f(a)h(neighb)l(orho)l (o)l(d)i(of)f Fw(\012)20 b Fs([)g(f)p FD(x)2301 1117 y Ft(0)2338 1105 y Fs(g)31 b Fg(and)h FD(u)p Fw(\()p FD(x)2701 1117 y Ft(0)2738 1105 y Fw(\))27 b(=)f(0)p Fg(,)32 b(we)g(have)g(that)466 1175 y Fq(@)t(u)p 466 1189 79 4 v 470 1236 a(@)t(s)577 1207 y FD(>)23 b Fw(0)29 b Fg(for)h(any)g(inwar)l(d)h(dir)l(e)l(ctional)g(derivative.)456 1334 y(R)l(emark)j Fw(3.5)p Fg(.)j Fw(The)23 b(pro)r(of)g(of)g(Theorem) f(3.4)h(follo)n(ws)f(at)h(once)f(exploiting)h(Theorem)f(5)h(in)g([21)o (])456 1434 y(with)k FD(\014)t Fw(\()p FD(s)p Fw(\))d(=)e FD(cs)984 1404 y Fq(q)1021 1434 y Fw(,)27 b FD(q)f Fs(\025)c FD(p)17 b Fs(\000)f Fw(1)26 b(and)h FD(c)c Fs(\025)f Fw(0)k(.)37 b(In)27 b(particular)e(the)i(condition)g FD(q)f Fs(\025)c FD(p)17 b Fs(\000)f Fw(1)26 b(ensure)456 1534 y(that)33 b(condition)h(\(13\))f(and)g(\(13'\))g(in)h([21)o(])g (are)f(ful\014lled.)55 b(Moreo)n(v)n(er,)32 b(the)i(condition)g FD(c)e Fs(\025)h Fw(0)456 1633 y(causes)26 b FD(\014)32 b Fw(to)c(b)r(e)g(nondecreasing)d(with)k FD(\014)t Fw(\(0\))23 b(=)g(0.)1139 1859 y(4.)42 b Fv(Technical)31 b(and)g(element)-6 b(ar)g(y)33 b(lemma)-6 b(t)g(a)555 2009 y Fw(This)27 b(section,)g(whic)n(h)g(ma)n(y)g(b)r(e)g(skipp)r(ed)g(b)n(y)g(the)h (exp)r(ert)f(reader,)f(collects)g(some)h(elemen-)456 2108 y(tary)f(lemmata)i(that)g(will)f(b)r(e)h(of)g(use)f(in)h(the)g (course)f(of)g(the)h(pro)r(ofs)f(of)g(the)h(main)g(results.)456 2235 y FE(Lemma)h(4.1.)40 b Fg(Ther)l(e)31 b(exists)e(a)h(p)l(ositive)h (c)l(onstant)e FD(C)6 b Fg(,)31 b(dep)l(ending)g(only)f(on)g FD(p)p Fg(,)g(such)g(that)1412 2429 y FD(a)1456 2394 y Ft(1)p Fq(=p)1580 2429 y Fs(\000)18 b FD(b)1699 2394 y Ft(1)p Fq(=p)1827 2429 y Fs(\024)23 b FD(C)2132 2372 y(a)18 b Fs(\000)g FD(b)p 2004 2409 437 4 v 2004 2511 a(a)2058 2447 y Fn(p)p Fm(\000)p Fd(1)p 2058 2461 104 3 v 2094 2494 a Fn(p)2194 2511 y Fw(+)g FD(b)2323 2447 y Fn(p)p Fm(\000)p Fd(1)p 2323 2461 V 2359 2494 a Fn(p)2465 2429 y FD(;)456 2639 y Fg(for)30 b(any)g FD(a)23 b Fs(\025)g FD(b)g Fs(\025)f Fw(0)p Fg(,)30 b FD(a)23 b Fs(6)p Fw(=)f(0)p Fg(.)456 2816 y(Pr)l(o)l(of.)43 b Fw(F)-7 b(or)27 b FD(t)c Fs(2)g Fw([0)p FD(;)14 b Fw(1\),)27 b(set)1311 3030 y FD(f)9 b Fw(\()p FD(t)p Fw(\))83 b(:=)1718 2974 y(\(1)19 b Fs(\000)f FD(t)1924 2944 y Ft(1)p Fq(=p)2029 2974 y Fw(\)\(1)h(+)f FD(t)2267 2944 y Ft(\()p Fq(p)p Fo(\000)p Ft(1\))p Fq(=p)2510 2974 y Fw(\))p 1718 3011 825 4 v 2044 3087 a(1)g Fs(\000)g FD(t)2566 3030 y(:)456 3225 y Fw(Notice)27 b(that)h FD(f)k Fs(2)23 b FD(C)1112 3195 y Ft(0)1150 3225 y Fw(\([0)p FD(;)14 b Fw(1\)\),)27 b FD(f)9 b Fw(\(0\))23 b(=)g(1)k(and)1709 3418 y(lim)1684 3470 y Fq(t)p Fo(\000)-11 b(!)p Ft(1)1863 3418 y FD(f)9 b Fw(\()p FD(t)p Fw(\))23 b(=)2128 3362 y(2)p 2127 3399 42 4 v 2127 3475 a FD(p)2193 3418 y(:)456 3617 y Fw(Hence,)k FD(f)9 b Fw(\()p FD(t)p Fw(\))24 b Fs(\024)e FD(C)34 b Fw(for)27 b(an)n(y)g FD(t)c Fs(2)h Fw([0)p FD(;)14 b Fw(1\))26 b(and)i(so)1109 3792 y(\()p FD(a)1185 3762 y Ft(1)p Fq(=p)1309 3792 y Fs(\000)18 b FD(b)1428 3762 y Ft(1)p Fq(=p)1533 3792 y Fw(\)\()p FD(a)1651 3728 y Fn(p)p Fm(\000)p Fd(1)p 1651 3742 104 3 v 1688 3775 a Fn(p)1788 3792 y Fw(+)g FD(b)1917 3728 y Fn(p)p Fm(\000)p Fd(1)p 1916 3742 V 1953 3775 a Fn(p)2034 3792 y Fw(\))p 1109 3829 959 4 v 1497 3905 a FD(a)g Fs(\000)g FD(b)2100 3848 y Fw(=)k FD(f)9 b Fw(\()p FD(b=a)p Fw(\))23 b Fs(\024)f FD(C)e(:)3380 4031 y Fc(\003)456 4209 y FE(Lemma)29 b(4.2.)40 b Fg(F)-6 b(or)30 b(any)g Fw(0)23 b Fs(\024)f FD(s)h Fs(\024)g FD(t)g Fs(\024)g FD(\022)1782 4179 y Fo(\003)1820 4209 y Fg(,)1260 4364 y FD(h)1308 4376 y Ft(0)1345 4364 y Fw(\()p Fs(\000)p Fw(1)18 b(+)g FD(t)p Fw(\))h Fs(\000)f FD(h)1797 4376 y Ft(0)1834 4364 y Fw(\()p Fs(\000)p Fw(1)g(+)g FD(s)p Fw(\))23 b Fs(\025)f FD(c)p Fw(\()p FD(t)2353 4329 y Fq(p)2410 4364 y Fs(\000)d FD(s)2533 4329 y Fq(p)2571 4364 y Fw(\))14 b FD(;)456 4518 y Fg(for)30 b(a)g(suitable)h(universal) f(c)l(onstant)f FD(c)23 b(>)g Fw(0)p Fg(.)p 456 4628 499 4 v 456 4720 a FC(with)j FA(y)f Fl(2)f FA(@)t(B)844 4728 y Fy(r)879 4720 y FC(\()p FA(x)p FC(\).)38 b(If)26 b(no)n(w)h(w)n(e)f(consider)g FA(w)37 b FC(:=)d(1)18 b Fl(\000)f FA(u)26 b FC(it)f(follo)n(ws)g(that)j FA(w)f FC(is)e(not)i(iden)n(tically)f(zero)h(in)e(the)456 4803 y(in)n(terior)e(of)g FA(B)838 4811 y Fy(r)873 4803 y FC(\()p FA(x)p FC(\))h(and)g FA(w)r FC(\()p FA(y)r FC(\))c(=)g(0.)31 b(Moreo)n(v)n(er,)24 b(w)n(e)f(ha)n(v)n(e)1239 4972 y Fl(\000)p FC(\001)1353 4980 y Fy(p)1389 4972 y FA(w)e FC(=)f FA(h)1577 4944 y Fx(0)1577 4989 y Fz(0)1611 4972 y FC(\()p FA(u)p FC(\))g(=)1811 4924 y FA(h)1852 4901 y Fx(0)1852 4945 y Fz(0)1886 4924 y FC(\(1)c Fl(\000)g FA(w)r FC(\))p 1811 4957 305 3 v 1879 5022 a FA(w)1932 5003 y Fy(p)p Fx(\000)p Fz(1)2125 4972 y FA(w)2178 4944 y Fy(p)p Fx(\000)p Fz(1)2311 4972 y Fl(\025)k(\000)p FA(c)2472 4944 y Fx(0)2494 4972 y FA(w)2547 4944 y Fy(p)p Fx(\000)p Fz(1)456 5131 y FC(i.e.)46 b Fl(\000)p FC(\001)707 5139 y Fy(p)743 5131 y FA(w)20 b FC(+)f FA(c)919 5108 y Fx(0)941 5131 y FA(w)994 5108 y Fy(p)p Fx(\000)p Fz(1)1137 5131 y Fl(\025)28 b FC(0)h(w)n(eakly)-6 b(.)48 b(W)-6 b(e)29 b(can)h(therefore)f(exploit)h(the)g(Strong)f(Maxim)n(um)e (Principle)h(for)456 5216 y(p-Laplace)c(equations)h(and)f(pro)n(v)n(e)g (that)h FA(u)20 b(<)f FC(1.)32 b(In)24 b(the)g(same)f(w)n(a)n(y)h(w)n (e)g(also)g(get)g FA(u)c(>)f Fl(\000)p FC(1.)p eop %%Page: 10 10 10 9 bop 456 251 a Ft(10)648 b(BERARDINO)23 b(SCIUNZI)g(AND)f(ENRICO)h (V)-7 b(ALDINOCI)456 450 y Fg(Pr)l(o)l(of.)43 b Fw(F)-7 b(rom)27 b(\(1.3\),)1057 644 y FD(h)1105 656 y Ft(0)1142 644 y Fw(\()p Fs(\000)p Fw(1)18 b(+)g FD(t)p Fw(\))h Fs(\000)f FD(h)1594 656 y Ft(0)1631 644 y Fw(\()p Fs(\000)p Fw(1)f(+)h FD(s)p Fw(\))84 b(=)2172 531 y Fp(Z)2255 552 y Fq(t)2218 720 y(s)2298 644 y FD(h)2346 610 y Fo(0)2346 665 y Ft(0)2383 644 y Fw(\()p Fs(\000)p Fw(1)18 b(+)g FD(\022)r Fw(\))c FD(d\022)2025 881 y Fs(\025)82 b Fw(const)2390 768 y Fp(Z)2473 789 y Fq(t)2436 957 y(s)2516 881 y FD(\022)2557 847 y Fq(p)p Fo(\000)p Ft(1)2694 881 y FD(d\022)44 b(;)456 1060 y Fw(whic)n(h)27 b(implies)h(the)g(desired)f(result.)1753 b Fc(\003)456 1214 y FE(Lemma)29 b(4.3.)40 b Fg(Ther)l(e)31 b(exists)e(a)h(p)l(ositive)h(c)l(onstant)2171 1193 y Fw(~)2152 1214 y FD(C)7 b Fg(,)30 b(dep)l(ending)h(only)f(on)g FD(p)p Fg(,)g(so)g(that)1259 1292 y Fp(Z)1342 1312 y Ft(0)1305 1481 y Fo(\000)p Ft(1+)p Fe(b)1799 1349 y FD(d\020)p 1505 1386 675 4 v 1505 1422 a Fp(\020)1554 1514 y Fw(\(1)19 b(+)f FD(\020)6 b Fw(\))1804 1490 y Fq(p)1861 1514 y Fs(\000)18 b Ff(a)1986 1490 y Fq(p)2024 1422 y Fp(\021)2074 1439 y Ft(1)p Fq(=p)2226 1405 y Fs(\024)2346 1384 y Fw(~)2327 1405 y FD(C)34 b Fw(log)2552 1349 y(1)p 2551 1386 43 4 v 2551 1462 a Ff(b)2618 1405 y FD(;)456 1671 y Fg(for)c(any)g Fw(0)23 b FD(<)g Ff(a)f Fs(\024)h Ff(b)f Fs(\024)h Fw(1)p Fg(.)456 1824 y(Pr)l(o)l(of.)43 b Fw(Let)1566 1971 y Ff(x)24 b Fw(:=)1732 1854 y Fp(\032)1795 1920 y Fw(1)110 b(if)28 b Ff(b)23 b FD(<)f Fw(1)p FD(=)p Fw(2,)1795 2020 y(0)110 b(if)28 b Ff(b)23 b Fs(\025)f Fw(1)p FD(=)p Fw(2.)456 2140 y(Using)27 b(the)h(substitution)g FD(\034)33 b Fw(:=)23 b(\(1)18 b(+)g FD(\020)6 b Fw(\))p FD(=)p Ff(b)p Fw(,)27 b(w)n(e)h(b)r(ound)g(the)g(in)n(tegral)e(ab)r(o)n(v)n(e)g(b)n(y)769 2229 y Fp(Z)852 2249 y Ft(1)p Fq(=)p Fe(b)815 2417 y Ft(1)1207 2285 y FD(d\034)p 983 2322 538 4 v 983 2363 a Fp(\000)1021 2431 y FD(\034)1066 2407 y Fq(p)1123 2431 y Fs(\000)1206 2363 y Fp(\000)1255 2398 y Fe(a)p 1254 2412 36 4 v 1254 2459 a(b)1300 2363 y Fp(\001)1338 2381 y Fq(p)1377 2363 y Fp(\001)1415 2376 y Ft(1)p Fq(=p)1613 2342 y Fs(\024)1761 2229 y Fp(Z)1844 2249 y Ft(1)p Fq(=)p Fe(b)1807 2417 y Ft(1)2128 2285 y FD(d\034)p 1974 2322 398 4 v 1974 2418 a Fw(\()q FD(\034)2052 2394 y Fq(p)2109 2418 y Fs(\000)18 b Fw(1\))2266 2376 y Ft(1)p Fq(=p)1613 2629 y Fs(\024)1761 2516 y Fp(Z)1844 2536 y Ft(2)1807 2705 y(1)2059 2573 y FD(d\034)p 1905 2610 V 1905 2705 a Fw(\()p FD(\034)1982 2681 y Fq(p)2039 2705 y Fs(\000)h Fw(1\))2196 2663 y Ft(1)p Fq(=p)2330 2629 y Fw(+)f Ff(x)2473 2516 y Fp(Z)2556 2536 y Ft(1)p Fq(=)p Fe(b)2519 2705 y Ft(2)2841 2573 y FD(d\034)p 2687 2610 V 2687 2705 a Fw(\()p FD(\034)2764 2681 y Fq(p)2822 2705 y Fs(\000)g Fw(1\))2979 2663 y Ft(1)p Fq(=p)3108 2629 y FD(:)456 2829 y Fw(Noticing)27 b(that)1611 2967 y FD(\034)1656 2932 y Fq(p)1713 2967 y Fs(\000)18 b Fw(1)23 b Fs(\025)1958 2910 y Fw(2)2000 2880 y Fq(p)2057 2910 y Fs(\000)18 b Fw(1)p 1958 2948 223 4 v 2030 3024 a(2)2072 3000 y Fq(p)2205 2967 y FD(\034)2250 2932 y Fq(p)456 3109 y Fw(if)28 b FD(\034)k Fs(\025)23 b Fw(2)k(and)h(that)1687 3211 y FD(\034)1732 3176 y Fq(p)1789 3211 y Fs(\000)18 b Fw(1)23 b Fs(\025)g FD(\034)28 b Fs(\000)18 b Fw(1)456 3330 y(if)28 b FD(\034)k Fs(\025)23 b Fw(1,)k(w)n(e)h(b)r(ound)g(the)g(quan)n(tit)n (y)f(here)g(ab)r(o)n(v)n(e)f(b)n(y)878 3537 y(const)1095 3395 y Fp( )1161 3424 y(Z)1244 3444 y Ft(2)1207 3612 y(1)1440 3480 y FD(d\034)p 1305 3517 359 4 v 1305 3595 a Fw(\()p FD(\034)i Fs(\000)18 b Fw(1\))1558 3571 y Ft(1)p Fq(=p)1692 3537 y Fw(+)1775 3424 y Fp(Z)1858 3444 y Ft(1)p Fq(=)p Fe(b)1821 3612 y Ft(2)1989 3480 y FD(d\034)p 1989 3517 89 4 v 2010 3594 a(\034)2088 3395 y Fp(!)2176 3537 y Fs(\024)23 b Fw(const)2481 3420 y Fp(\022)2542 3537 y Fw(1)18 b(+)g(log)2838 3480 y(1)p 2816 3517 85 4 v 2816 3594 a(2)p Ff(b)2910 3420 y Fp(\023)2999 3537 y FD(;)456 3743 y Fw(whic)n(h)27 b(pro)n(v)n(es)f(the)i(desired)f (result.)1778 b Fc(\003)456 3896 y FE(Lemma)27 b(4.4.)39 b Fg(L)l(et)27 b FD(U)37 b Fg(b)l(e)29 b(an)f(op)l(en)g(subset)g(of)h Fr(R)p Fg(.)44 b(L)l(et)28 b FD(g)e Fs(2)d FD(C)2442 3866 y Ft(2)2479 3896 y Fw(\()p FD(U)9 b Fw(\))29 b Fg(and)g(assume)f (that)g FD(g)j Fg(has)456 3995 y(no)e(critic)l(al)i(p)l(oints.)39 b(De\014ne)456 4132 y Fw(\(4.1\))877 b(\011)1569 4098 y Fq(y)r(;l)1649 4132 y Fw(\()p FD(x)p Fw(\))38 b(:=)f FD(g)s Fw(\()p Fs(j)p FD(x)18 b Fs(\000)g FD(y)s Fs(j)h(\000)f FD(l)r Fw(\))456 4268 y Fg(Then,)30 b(for)h FD(t)23 b Fw(=)g Fs(j)p FD(x)c Fs(\000)f FD(y)s Fs(j)g(\000)g FD(l)24 b Fs(2)g FD(U)38 b Fg(and)31 b FD(x)23 b Fs(6)p Fw(=)g FD(y)s Fg(,)30 b(we)g(have)456 4447 y Fw(\(4.2\))327 b(\001)1023 4459 y Fq(p)1061 4447 y Fw(\(\011)1158 4413 y Fq(y)r(;l)1239 4447 y Fw(\()p FD(x)p Fw(\)\))25 b(=)d(\()p FD(p)d Fs(\000)f Fw(1\))p FD(g)1787 4413 y Fo(00)1829 4447 y Fw(\()p FD(t)p Fw(\))p FD(g)1966 4413 y Fo(0)p Ft(\()p Fq(p)p Fo(\000)p Ft(2\))2160 4447 y Fw(\()p FD(t)p Fw(\))h(+)f FD(g)2399 4413 y Fo(0)p Ft(\()p Fq(p)p Fo(\000)p Ft(1\))2593 4447 y Fw(\()p FD(t)p Fw(\))2708 4391 y FD(N)27 b Fs(\000)18 b Fw(1)p 2697 4428 239 4 v 2697 4504 a Fs(j)p FD(x)h Fs(\000)g FD(y)s Fs(j)456 4644 y Fg(Pr)l(o)l(of.)43 b Fw(Let)27 b(us)h(note)g(that)456 4824 y(\(4.3\))886 b Fs(r)p Fw(\011)1647 4789 y Fq(y)r(;l)1727 4824 y Fw(\()p FD(x)p Fw(\))24 b(=)f FD(g)1993 4789 y Fo(0)2016 4824 y Fw(\()p FD(t)p Fw(\))2120 4767 y(\()p FD(x)d Fs(\000)e FD(y)s Fw(\))p 2120 4805 258 4 v 2129 4881 a Fs(j)p FD(x)i Fs(\000)e FD(y)s Fs(j)456 5004 y Fw(and)456 5170 y(\(4.4\))102 b(\011)794 5130 y Fq(y)r(;l)794 5193 y(ij)874 5170 y Fw(\()p FD(x)p Fw(\))24 b(=)f FD(g)1140 5136 y Fo(00)1182 5170 y Fw(\()p FD(t)p Fw(\))1286 5114 y(\()p FD(x)1365 5126 y Fq(i)1412 5114 y Fs(\000)18 b FD(y)1536 5126 y Fq(i)1564 5114 y Fw(\)\()p FD(x)1675 5126 y Fq(j)1729 5114 y Fs(\000)g FD(y)1853 5126 y Fq(j)1888 5114 y Fw(\))p 1286 5151 634 4 v 1465 5227 a Fs(j)p FD(x)h Fs(\000)f FD(y)s Fs(j)1704 5203 y Ft(2)1949 5170 y Fw(+)g FD(g)2075 5136 y Fo(0)2098 5170 y Fw(\()p FD(t)p Fw(\))2206 5053 y Fp(\022)2349 5114 y FD(\016)2386 5126 y Fq(ij)p 2277 5151 239 4 v 2277 5227 a Fs(j)p FD(x)h Fs(\000)f FD(y)s Fs(j)2544 5170 y(\000)2637 5114 y Fw(\()p FD(x)2716 5126 y Fq(i)2763 5114 y Fs(\000)g FD(y)2887 5126 y Fq(i)2914 5114 y Fw(\)\()p FD(x)3025 5126 y Fq(j)3080 5114 y Fs(\000)g FD(y)3204 5126 y Fq(j)3238 5114 y Fw(\))p 2637 5151 634 4 v 2816 5227 a Fs(j)p FD(x)h Fs(\000)f FD(y)s Fs(j)3055 5203 y Ft(3)3281 5053 y Fp(\023)p eop %%Page: 11 11 11 10 bop 743 251 a Ft(MEAN)29 b(CUR)-7 b(V)g(A)i(TURE)29 b(PR)n(OPER)-5 b(TIES)28 b(F)n(OR)h Fq(p)p Ft(-LAPLA)n(CE)h(PHASE)e (TRANSITIONS)221 b(11)456 450 y Fw(Since)456 608 y(\001)525 620 y Fq(p)563 608 y Fw(\(\011)660 574 y Fq(y)r(;l)741 608 y Fw(\()p FD(x)p Fw(\)\))24 b(=)f Fs(jr)p Fw(\011)1153 574 y Fq(y)r(;l)1234 608 y Fw(\()p FD(x)p Fw(\))p Fs(j)1368 574 y Fq(p)p Fo(\000)p Ft(2)1492 608 y Fw(\001\011)1626 574 y Fq(y)r(;l)1707 608 y Fw(\()p FD(x)p Fw(\))11 b(+)g(\()p FD(p)g Fs(\000)g Fw(2\))p Fs(jr)p Fw(\011)2297 574 y Fq(y)r(;l)2376 608 y Fw(\()p FD(x)p Fw(\))p Fs(j)2510 574 y Fq(p)p Fo(\000)p Ft(4)2635 608 y Fw(\011)2700 568 y Fq(y)r(;l)2700 631 y(ij)2780 608 y Fw(\()p FD(x)p Fw(\)\011)2956 568 y Fq(y)r(;l)2956 631 y(i)3038 608 y Fw(\()p FD(x)p Fw(\)\011)3214 568 y Fq(y)r(;l)3214 631 y(j)3295 608 y Fw(\()p FD(x)p Fw(\))j FD(;)456 766 y Fw(w)n(e)27 b(get)570 1078 y(\001)639 1090 y Fq(p)678 1078 y Fw(\(\011)775 1043 y Fq(y)r(;l)856 1078 y Fw(\()p FD(x)p Fw(\)\))d(=)f FD(g)1154 1043 y Fo(0)p Ft(\()p Fq(p)p Fo(\000)p Ft(2\))1348 1078 y Fw(\()p FD(t)p Fw(\))p FD(g)1485 1043 y Fo(00)1527 1078 y Fw(\()p FD(t)p Fw(\))c(+)f(\()p FD(p)h Fs(\000)f Fw(2\))1987 999 y Fp(X)2019 1175 y Fq(ij)2120 1078 y FD(g)2163 1043 y Fo(0)p Ft(\()p Fq(p)p Fo(\000)p Ft(2\))2357 1078 y FD(g)2400 1043 y Fo(00)2442 1078 y Fw(\()p FD(t)p Fw(\))2546 1021 y(\()p FD(x)2625 1033 y Fq(i)2673 1021 y Fs(\000)g FD(y)2797 1033 y Fq(i)2824 1021 y Fw(\))2856 991 y Ft(2)2894 1021 y Fw(\()p FD(x)2973 1033 y Fq(j)3027 1021 y Fs(\000)g FD(y)3151 1033 y Fq(j)3185 1021 y Fw(\))3217 991 y Ft(2)p 2547 1058 709 4 v 2763 1134 a Fs(j)p FD(x)h Fs(\000)f FD(y)s Fs(j)3002 1111 y Ft(4)3265 1078 y Fw(+)1019 1341 y(+)g FD(g)1145 1307 y Fo(0)p Ft(\()p Fq(p)p Fo(\000)p Ft(1\))1430 1285 y FD(N)p 1349 1322 239 4 v 1349 1398 a Fs(j)p FD(x)h Fs(\000)f FD(y)s Fs(j)1616 1341 y Fw(+)g(\()p FD(p)g Fs(\000)g Fw(2\))p FD(g)1991 1307 y Fo(0)p Ft(\()p Fq(p)p Fo(\000)p Ft(1\))2294 1285 y Fw(1)p 2195 1322 V 2195 1398 a Fs(j)p FD(x)h Fs(\000)f FD(y)s Fs(j)2444 1341 y Fw(+)1019 1576 y Fs(\000)g FD(g)1145 1542 y Fo(0)p Ft(\()p Fq(p)p Fo(\000)p Ft(1\))1447 1520 y Fw(1)p 1349 1557 V 1349 1633 a Fs(j)p FD(x)h Fs(\000)f FD(y)s Fs(j)1616 1576 y(\000)g Fw(\()p FD(p)g Fs(\000)g Fw(2\))1962 1497 y Fp(X)1995 1674 y Fq(ij)2096 1576 y FD(g)2139 1542 y Fo(0)p Ft(\()p Fq(p)p Fo(\000)p Ft(1\))2343 1520 y Fw(\()p FD(x)2422 1532 y Fq(i)2468 1520 y Fs(\000)g FD(y)2592 1532 y Fq(i)2620 1520 y Fw(\))2652 1490 y Ft(2)2689 1520 y Fw(\()p FD(x)2768 1532 y Fq(j)2822 1520 y Fs(\000)g FD(y)2946 1532 y Fq(j)2981 1520 y Fw(\))3013 1490 y Ft(2)p 2343 1557 709 4 v 2559 1633 a Fs(j)p FD(x)h Fs(\000)f FD(y)s Fs(j)2798 1609 y Ft(5)3084 1576 y Fw(=)1023 1840 y(=)23 b(\()p FD(p)18 b Fs(\000)g Fw(1\))p FD(g)1403 1805 y Fo(0)p Ft(\()p Fq(p)p Fo(\000)p Ft(2\))1597 1840 y Fw(\()p FD(t)p Fw(\))p FD(g)1734 1805 y Fo(00)1777 1840 y Fw(\()p FD(t)p Fw(\))h(+)f FD(g)2016 1805 y Fo(0)p Ft(\()p Fq(p)p Fo(\000)p Ft(1\))2210 1840 y Fw(\()2262 1783 y FD(N)27 b Fs(\000)18 b Fw(1)p 2252 1820 239 4 v 2252 1896 a Fs(j)p FD(x)h Fs(\000)f FD(y)s 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FD(x)p Fw(\)\))p Fs(j)1668 2589 y Fq(p)p Fo(\000)p Ft(2)1793 2623 y FD(g)1836 2589 y Fo(00)1878 2623 y Fw(\()p FD(d)1953 2635 y Ft(\000)1999 2623 y Fw(\()p FD(x)p Fw(\)\))i(+)e Fs(j)p FD(g)2311 2589 y Fo(0)2334 2623 y Fw(\()p FD(d)2409 2635 y Ft(\000)2454 2623 y Fw(\()p FD(x)p Fw(\)\))p Fs(j)2620 2589 y Fq(p)p Fo(\000)p Ft(2)2745 2623 y FD(g)2788 2589 y Fo(0)2811 2623 y Fw(\()p FD(d)2886 2635 y Ft(\000)2932 2623 y Fw(\()p FD(x)p Fw(\)\)\001)p FD(d)3187 2635 y Ft(\000)3233 2623 y Fw(\()p FD(x)p Fw(\))c FD(:)456 2806 y Fg(Pr)l(o)l(of.)43 b Fw(Easy)26 b(calculations)g(sho)n (ws)h(that)456 2964 y(\(4.7\))761 b Fs(r)p FD(g)s Fw(\()p FD(d)1575 2976 y Ft(\000)1620 2964 y Fw(\()p FD(x)p Fw(\)\))24 b(=)f FD(g)1918 2930 y Fo(0)1941 2964 y Fw(\()p FD(d)2016 2976 y Ft(\000)2062 2964 y Fw(\()p FD(x)p Fw(\)\))p Fs(r)p FD(d)2317 2976 y Ft(\000)2364 2964 y Fw(\()p FD(x)p Fw(\))14 b FD(:)456 3122 y Fw(and)456 3280 y(\(4.8\))218 b FD(g)885 3292 y Fq(ij)943 3280 y Fw(\()p FD(d)1018 3292 y Ft(\000)1064 3280 y Fw(\()p FD(x)p Fw(\)\))25 b(=)d FD(g)1362 3246 y Fo(00)1404 3280 y Fw(\()p FD(d)1479 3292 y Ft(\000)1525 3280 y Fw(\()p FD(x)p Fw(\)\)\()p FD(d)1743 3292 y Ft(\000)1790 3280 y Fw(\))1822 3292 y Fq(i)1850 3280 y Fw(\()p FD(x)p Fw(\)\()p FD(d)2036 3292 y Ft(\000)2083 3280 y Fw(\))2115 3292 y Fq(j)2150 3280 y Fw(\()p FD(x)p Fw(\))d(+)f FD(g)2406 3246 y Fo(0)2429 3280 y Fw(\()p FD(d)2504 3292 y Ft(\000)2550 3280 y Fw(\()p FD(x)p Fw(\)\)\()p FD(d)2768 3292 y Ft(\000)2815 3280 y Fw(\))2847 3292 y Fq(ij)2906 3280 y Fw(\()p FD(x)p Fw(\))c FD(:)456 3438 y Fw(By)31 b(the)i(rotation)e(in)n(v)-5 b(ariance,)32 b(it)g(is)g(not)g(restrictiv)n(e)f(to)h(consider)f(a)h (particular)f(system)h(of)456 3538 y(co)r(ordinates)26 b(for)h(whic)n(h)g(w)n(e)h(ha)n(v)n(e)1515 3696 y Fs(r)p FD(d)1627 3708 y Ft(\000)1673 3696 y Fw(\()p FD(x)p Fw(\))c(=)f(\(0)p FD(;)14 b Fw(0)p FD(;)g(:)g(:)g(:)f(;)h Fw(0)p FD(;)g Fw(1\))456 3854 y(and)816 4006 y FD(D)887 3972 y Ft(2)925 4006 y FD(d)968 4018 y Ft(\000)1036 4006 y Fw(=)23 b(diag)1305 3889 y Fp(\022)1459 3950 y Fs(\000)p FD(k)1567 3962 y Ft(1)p 1376 3987 312 4 v 1376 4063 a Fw(1)18 b Fs(\000)g FD(d)1562 4075 y Ft(\000)1607 4063 y FD(k)1650 4075 y Ft(1)1698 4006 y FD(;)c(:)g(:)g(:)f(;)1975 3950 y Fs(\000)p FD(k)2083 3962 y Fq(N)6 b Fo(\000)p Ft(1)p 1892 3987 423 4 v 1892 4063 a Fw(1)18 b Fs(\000)g FD(d)2078 4075 y Ft(\000)2123 4063 y FD(k)2166 4075 y Fq(N)6 b Fo(\000)p Ft(1)2325 4006 y FD(;)14 b Fw(0)2404 3889 y Fp(\023)2487 4006 y Fs(2)24 b Fw(Mat)13 b(\()p FD(N)28 b Fs(\002)18 b FD(N)9 b Fw(\))14 b FD(;)456 4192 y Fw(where)28 b(the)i FD(k)885 4204 y Fq(i)912 4192 y Fw('s)f(are)f(the)i(principal)e(curv)-5 b(atures)28 b(of)h(\000)g(at)g(the)h(p)r(oin)n(t)f(where)f(the)i (distance)f(is)456 4292 y(realized)g(\(see,)i(for)f(instance,)h Fs(x)p Fw(14.6)e(of)h([9])g(for)g(details\).)45 b(Therefore,)30 b(taking)g(in)n(to)g(accoun)n(t)456 4392 y(\(4.7\))d(and)g(\(4.8\),)h (w)n(e)f(get)518 4551 y(\001)587 4563 y Fq(p)625 4551 y FD(g)s Fw(\()p FD(d)743 4563 y Ft(\000)788 4551 y Fw(\()p FD(x)p Fw(\)\))e(=)d(\()p FD(p)d Fs(\000)f Fw(1\))p Fs(j)p FD(g)1359 4517 y Fo(0)1382 4551 y Fw(\()p FD(d)1457 4563 y Ft(\000)1502 4551 y Fw(\()p FD(x)p Fw(\)\))p Fs(j)1668 4517 y Fq(p)p Fo(\000)p Ft(2)1793 4551 y FD(g)1836 4517 y Fo(00)1878 4551 y Fw(\()p FD(d)1953 4563 y Ft(\000)1999 4551 y Fw(\()p FD(x)p Fw(\)\))i(+)e Fs(j)p FD(g)2311 4517 y Fo(0)2334 4551 y Fw(\()p FD(d)2409 4563 y Ft(\000)2454 4551 y Fw(\()p FD(x)p Fw(\)\))p Fs(j)2620 4517 y Fq(p)p Fo(\000)p Ft(2)2745 4551 y FD(g)2788 4517 y Fo(0)2811 4551 y Fw(\()p FD(d)2886 4563 y Ft(\000)2932 4551 y Fw(\()p FD(x)p Fw(\)\)\001)p FD(d)3187 4563 y Ft(\000)3233 4551 y Fw(\()p FD(x)p Fw(\))c FD(:)3380 4709 y Fc(\003)456 4892 y FE(Lemma)31 b(4.6.)40 b Fg(L)l(et)31 b FD(I)h Fs(3)25 b Fw(0)30 b Fg(b)l(e)h(an)g(interval)h(of)f Fr(R)37 b Fg(and)31 b(let)g FD(h)25 b Fs(2)g FD(C)2567 4862 y Ft(1)2605 4892 y Fw(\()p FD(I)7 b Fw(\))31 b Fg(satisfy)h FD(h)p Fw(\()p FD(s)p Fw(\))26 b FD(>)e Fw(0)30 b Fg(for)456 4992 y(any)g FD(s)23 b Fs(2)g FD(I)7 b Fg(.)39 b(L)l(et)1233 5170 y FD(H)7 b Fw(\()p FD(s)p Fw(\))37 b(:=)1574 5057 y Fp(Z)1657 5077 y Fq(s)1620 5245 y Ft(0)1716 5113 y Fw(\()p FD(p)18 b Fs(\000)g Fw(1\))1965 5083 y Ft(1)p Fq(=p)2085 5113 y FD(d\020)p 1716 5150 455 4 v 1753 5228 a Fw(\()p FD(p)c(h)p Fw(\()p FD(\020)6 b Fw(\)\))2027 5204 y Ft(1)p Fq(=p)2194 5170 y FD(;)184 b Fs(8)p FD(s)22 b Fs(2)h FD(I)e(:)p eop %%Page: 12 12 12 11 bop 456 251 a Ft(12)648 b(BERARDINO)23 b(SCIUNZI)g(AND)f(ENRICO)h (V)-7 b(ALDINOCI)456 450 y Fg(De\014ne)30 b(also)i FD(g)h Fg(as)e(the)g(inverse)h(of)f FD(H)7 b Fg(,)32 b(that)e(is)i FD(g)s Fw(\()p FD(t)p Fw(\))25 b(:=)f FD(H)2338 420 y Fo(\000)p Ft(1)2427 450 y Fw(\()p FD(t)p Fw(\))32 b Fg(for)f(any)h FD(t)25 b Fs(2)g FD(H)7 b Fw(\()p FD(I)g Fw(\))p Fg(.)42 b(Then,)456 550 y FD(g)25 b Fs(2)f FD(C)665 520 y Ft(2)716 550 y Fw(\()p FD(H)7 b Fw(\()p FD(I)g Fw(\)\))31 b Fg(and)1268 735 y FD(g)1311 700 y Fo(0)1334 735 y Fw(\()p FD(t)p Fw(\))84 b(=)1659 642 y Fp(\020)1790 678 y FD(p)p 1719 715 185 4 v 1719 792 a(p)18 b Fs(\000)g Fw(1)1928 735 y FD(h)p Fw(\()p FD(g)s Fw(\()p FD(t)p Fw(\)\))2177 642 y Fp(\021)2227 660 y Ft(1)p Fq(=p)1249 1046 y FD(g)1292 1011 y Fo(00)1334 1046 y Fw(\()p FD(t)p Fw(\))84 b(=)1669 867 y Fp(\020)1719 959 y FD(p)14 b(h)p Fw(\()p FD(g)s Fw(\()p FD(t)p Fw(\)\))2024 867 y Fp(\021)2074 884 y Ft(\(2)p Fo(\000)p Fq(p)p Ft(\))p Fq(=p)p 1669 1027 648 4 v 1816 1104 a Fw(\()p FD(p)k Fs(\000)g Fw(1\))2065 1080 y Ft(2)p Fq(=p)2341 1046 y FD(h)2389 1011 y Fo(0)2412 1046 y Fw(\()p FD(g)s Fw(\()p FD(t)p Fw(\)\))c FD(;)456 1233 y Fg(for)30 b(any)g FD(t)23 b Fs(2)h FD(H)7 b Fw(\()p FD(I)g Fw(\))p Fg(.)456 1386 y(Pr)l(o)l(of.)43 b Fw(The)d(\014rst)h (iden)n(tit)n(y)f(easily)g(follo)n(ws)g(b)n(y)g(di\013eren)n(tiating)g FD(H)7 b Fw(\()p FD(g)s Fw(\()p FD(t)p Fw(\)\))45 b(=)f FD(t)p Fw(.)76 b(F)-7 b(or)40 b(the)456 1485 y(second)27 b(claim,)g(notice)g(that,)h(using)g(the)g(\014rst)f(iden)n(tit)n(y)h(t) n(wice,)1232 1665 y FD(g)1275 1631 y Fo(00)1317 1665 y Fw(\()p FD(t)p Fw(\))84 b(=)1656 1609 y FD(d)p 1653 1646 74 4 v 1653 1722 a(dt)1736 1573 y Fp(\020)1867 1609 y FD(p)p 1795 1646 185 4 v 1795 1722 a(p)19 b Fs(\000)f Fw(1)2004 1665 y FD(h)p Fw(\()p FD(g)s Fw(\()p FD(t)p Fw(\)\))2253 1573 y Fp(\021)2303 1590 y Ft(1)p Fq(=p)1495 1982 y Fw(=)1653 1804 y Fp(\020)1702 1896 y FD(p)c(h)p Fw(\()p FD(g)s Fw(\()p FD(t)p Fw(\)\))2007 1804 y Fp(\021)2068 1798 y Fd(1)p 2067 1807 31 3 v 2067 1841 a Fn(p)2108 1821 y Fo(\000)p Ft(1)p 1653 1963 545 4 v 1747 2040 a Fw(\()p FD(p)19 b Fs(\000)f Fw(1\))1997 2016 y Ft(1)p Fq(=p)2220 1982 y FD(h)2268 1948 y Fo(0)2292 1982 y Fw(\()p FD(g)s Fw(\()p FD(t)p Fw(\)\))c FD(g)2550 1948 y Fo(0)2573 1982 y Fw(\()p FD(t)p Fw(\))1495 2305 y(=)1653 2126 y Fp(\020)1702 2218 y FD(p)g(h)p Fw(\()p FD(g)s Fw(\()p FD(t)p Fw(\)\))2007 2126 y Fp(\021)2068 2121 y Fd(2)p 2067 2130 31 3 v 2067 2163 a Fn(p)2108 2143 y Fo(\000)p Ft(1)p 1653 2286 545 4 v 1747 2363 a Fw(\()p FD(p)19 b Fs(\000)f Fw(1\))1997 2339 y Ft(2)p Fq(=p)2220 2305 y FD(h)2268 2271 y Fo(0)2292 2305 y Fw(\()p FD(g)s Fw(\()p FD(t)p Fw(\)\))c FD(:)3380 2486 y Fc(\003)456 2638 y FE(Lemma)35 b(4.7.)43 b Fg(L)l(et)35 b Fw(\012)f Fg(b)l(e)h(an)g(op)l (en)g(domain)h(in)e Fr(R)2129 2608 y Fq(N)2233 2638 y Fg(and)h(let)g FD(x)2568 2650 y Ft(0)2637 2638 y Fs(2)e Fw(\012)p Fg(.)53 b(L)l(et)34 b FD(w)h Fs(2)d FD(C)3257 2608 y Ft(1)3294 2638 y Fw(\(\012\))p Fg(.)456 2738 y(Assume)d(that)g (ther)l(e)h(exist)g FD(v)c Fs(2)d Fr(R)1534 2708 y Fq(N)1603 2738 y Fg(,)30 b FD($)25 b Fs(2)f Fr(R)1885 2708 y Fq(N)1972 2738 y Fs(n)18 b(f)p Fw(0)p Fs(g)28 b Fg(such)i(that)1451 2875 y FD(w)r Fw(\()p FD(x)1591 2887 y Ft(0)1648 2875 y Fw(+)18 b FD(x)p Fw(\))38 b Fs(\024)e FD(v)22 b Fs(\001)c FD(x)h Fw(+)f FD(w)r Fw(\()p FD(x)2341 2887 y Ft(0)2380 2875 y Fw(\))c FD(;)456 3012 y Fg(for)35 b(any)f FD(x)d Fs(2)h Fr(R)975 2982 y Fq(N)1078 3012 y Fg(so)i(that)g FD(x)22 b Fw(+)f FD(x)1564 3024 y Ft(0)1633 3012 y Fs(2)31 b Fw(\012)j Fg(and)h FD($)23 b Fs(\001)f FD(x)31 b Fs(\025)g Fw(0)p Fg(.)51 b(If)34 b FD(P)43 b Fs(2)31 b FD(C)2746 2982 y Ft(2)2784 3012 y Fw(\(\012\))j Fg(is)h(a)f(quadr)l(atic)456 3111 y(function)i(touching)h FD(w)i Fg(fr)l(om)d(b)l(elow)i(at)e FD(x)1812 3123 y Ft(0)1849 3111 y Fg(,)j(then)d Fw(\001)2173 3123 y Fq(p)2211 3111 y FD(P)12 b Fw(\()p FD(x)2355 3123 y Ft(0)2393 3111 y Fw(\))35 b Fs(\024)g Fw(0)h Fg(in)g(the)g(visc)l (osity)i(sense.)456 3211 y(A)n(nalo)l(gously,)31 b(if)1451 3313 y FD(w)r Fw(\()p FD(x)1591 3325 y Ft(0)1648 3313 y Fw(+)18 b FD(x)p Fw(\))38 b Fs(\025)e FD(v)22 b Fs(\001)c FD(x)h Fw(+)f FD(w)r Fw(\()p FD(x)2341 3325 y Ft(0)2380 3313 y Fw(\))c FD(;)456 3432 y Fg(for)33 b(any)f FD(x)c Fs(2)g Fr(R)964 3402 y Fq(N)1065 3432 y Fg(so)k(that)g FD(x)21 b Fw(+)f FD(x)1545 3444 y Ft(0)1610 3432 y Fs(2)27 b Fw(\012)32 b Fg(and)h FD($)22 b Fs(\001)e FD(x)28 b Fs(\025)f Fw(0)p Fg(,)33 b(and)f FD(P)40 b Fs(2)27 b FD(C)2752 3402 y Ft(2)2790 3432 y Fw(\(\012\))32 b Fg(is)h(a)f(quadr)l (atic)456 3532 y(function)d(touching)i FD(w)h Fg(fr)l(om)e(ab)l(ove)h (at)f FD(x)1777 3544 y Ft(0)1814 3532 y Fg(,)h(then)e Fw(\001)2123 3544 y Fq(p)2162 3532 y FD(P)12 b Fw(\()p FD(x)2306 3544 y Ft(0)2344 3532 y Fw(\))23 b Fs(\025)g Fw(0)29 b Fg(in)h(the)g(visc)l(osity)h(sense.)456 3685 y(Pr)l(o)l(of.)43 b Fw(W)-7 b(e)29 b(pro)n(v)n(e)f(the)h(\014rst)g (claim,)g(the)h(second)e(one)h(b)r(eing)g(analogous.)39 b(Since)29 b FD(P)41 b Fw(touc)n(hes)456 3784 y FD(w)30 b Fw(at)d FD(x)693 3796 y Ft(0)731 3784 y Fw(,)h FD(w)e Fs(2)d FD(C)1010 3754 y Ft(1)1048 3784 y Fw(\(\012\))28 b(and)f Fs(r)p FD(w)r Fw(\()p FD(x)1570 3796 y Ft(0)1609 3784 y Fw(\))c(=)g FD(v)s Fw(,)28 b(then)963 3960 y FD(P)12 b Fw(\()p FD(x)p Fw(\))37 b(=)1288 3904 y(1)p 1288 3941 42 4 v 1288 4017 a(2)1339 3960 y FD(M)9 b Fw(\()p FD(x)19 b Fs(\000)f FD(x)1657 3972 y Ft(0)1695 3960 y Fw(\))h Fs(\001)f Fw(\()p FD(x)h Fs(\000)f FD(x)2015 3972 y Ft(0)2053 3960 y Fw(\))h(+)f FD(v)j Fs(\001)e Fw(\()p FD(x)g Fs(\000)f FD(x)2518 3972 y Ft(0)2556 3960 y Fw(\))h(+)f FD(w)r Fw(\()p FD(x)2830 3972 y Ft(0)2868 3960 y Fw(\))c FD(;)456 4125 y Fw(for)27 b(some)g FD(M)32 b Fs(2)24 b Fw(Mat\()p FD(N)k Fs(\002)18 b FD(N)9 b Fw(\).)38 b(Notice)27 b(that)i FD(M)36 b Fw(m)n(ust)28 b(b)r(e)g(non-p)r(ositiv)n(e)f(de\014nite:)38 b(indeed,)456 4225 y(if)f FD(M)46 b Fw(had)36 b(a)h(p)r(ositiv)n(e)f (eigen)n(v)-5 b(alue)36 b FD(\025)i Fw(with)f(corresp)r(onding)e(eigen) n(v)n(ector)g FD(e)i Fw(and)f Fs(j)p FD(e)p Fs(j)j Fw(=)f(1,)456 4324 y(p)r(ossibly)22 b(c)n(hanging)g FD(e)h Fw(in)n(to)f Fs(\000)p FD(e)h Fw(w)n(e)f(ma)n(y)h(assume)f(that)h FD($)12 b Fs(\001)d FD(e)23 b Fs(\025)f Fw(0)h(and)g(therefore,)g(for)f (a)h(small)456 4424 y FD(")f(>)h Fw(0,)1301 4561 y FD("v)e Fs(\001)e FD(e)f Fw(+)g FD(r)86 b Fs(\025)c FD(w)r Fw(\()p FD(x)1993 4573 y Ft(0)2050 4561 y Fw(+)18 b FD("e)p Fw(\))1706 4685 y Fs(\025)82 b FD(P)12 b Fw(\()p FD(x)1997 4697 y Ft(0)2053 4685 y Fw(+)18 b FD("e)p Fw(\))1706 4863 y(=)1863 4807 y FD(")1902 4777 y Ft(2)p 1863 4844 76 4 v 1880 4920 a Fw(2)1949 4863 y FD(M)9 b(e)18 b Fs(\001)g FD(e)g Fw(+)g FD("v)k Fs(\001)c FD(e)g Fw(+)g FD(r)1706 5077 y Fs(\025)1863 5021 y FD(")1902 4991 y Ft(2)1939 5021 y FD(\025)p 1863 5058 125 4 v 1905 5134 a Fw(2)2016 5077 y(+)g FD("v)j Fs(\001)e FD(e)f Fw(+)g FD(r)1706 5216 y(>)82 b("v)22 b Fs(\001)c FD(e)g Fw(+)g FD(r)f(;)p eop %%Page: 13 13 13 12 bop 743 251 a Ft(MEAN)29 b(CUR)-7 b(V)g(A)i(TURE)29 b(PR)n(OPER)-5 b(TIES)28 b(F)n(OR)h Fq(p)p Ft(-LAPLA)n(CE)h(PHASE)e (TRANSITIONS)221 b(13)456 450 y Fw(whic)n(h)25 b(is)h(a)f(con)n (tradiction.)35 b(Hence,)27 b FD(M)34 b Fw(is)26 b(non-p)r(ositiv)n(e)f (de\014nite)h(and)g(th)n(us)g(\001)p FD(P)35 b Fw(=)23 b(tr)13 b FD(M)32 b Fs(\024)456 550 y Fw(0.)k(No)n(w,)27 b(if)h FD(p)23 b Fs(\025)g Fw(2,)897 762 y(\001)966 774 y Fq(p)1005 762 y FD(P)94 b Fw(=)83 b Fs(jr)p FD(P)12 b Fs(j)1480 727 y Fq(p)p Fo(\000)p Ft(4)1604 669 y Fp(\020)1653 762 y Fs(jr)p FD(P)g Fs(j)1833 727 y Ft(2)1871 762 y Fw(\001)p FD(P)30 b Fw(+)18 b(\()p FD(p)h Fs(\000)f Fw(2\))c Fs(h)p FD(D)2473 727 y Ft(2)2510 762 y FD(P)25 b Fs(r)p FD(P)h(;)i Fs(r)p FD(P)12 b Fs(i)2953 669 y Fp(\021)1152 944 y Fw(=)83 b Fs(j)p FD(v)s Fs(j)1389 910 y Fq(p)p Fo(\000)p Ft(4)1513 852 y Fp(\020)1562 944 y Fs(j)p FD(v)s Fs(j)1651 910 y Ft(2)1689 944 y Fw(tr)14 b FD(M)27 b Fw(+)18 b(\()p FD(p)g Fs(\000)g Fw(2\))c FD(M)9 b(v)21 b Fs(\001)e FD(v)2458 852 y Fp(\021)2530 944 y Fs(\024)k Fw(0)14 b FD(;)456 1156 y Fw(at)27 b FD(x)604 1168 y Ft(0)642 1156 y Fw(,)h(whic)n(h)f(pro)n(v)n(es)f(the)i(claim)f(for)g FD(p)c Fs(\025)g Fw(2.)456 1256 y(Let)30 b(no)n(w)f(assume)h(1)d FD(<)g(p)g(<)g Fw(2.)44 b(If)31 b FD(v)f Fw(=)d(0,)j(b)n(y)g (De\014nition)h(3.1,)f(there)g(is)g(nothing)g(to)g(c)n(hec)n(k;)456 1355 y(w)n(e)d(ma)n(y)g(then)h(assume)f FD(v)f Fs(6)p Fw(=)d(0.)36 b(Let)28 b FD(\025)1686 1367 y Ft(1)1724 1355 y FD(;)14 b(:)g(:)g(:)f(;)h(\025)1956 1367 y Fq(N)2042 1355 y Fs(\024)23 b Fw(0)k(b)r(e)h(the)g(eigen)n(v)-5 b(alues)27 b(of)g FD(M)9 b Fw(.)37 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Ft(1)2053 2022 y Fw(+)g FD(\025)2184 2034 y Ft(2)2240 2022 y Fw(+)g Fs(\001)c(\001)g(\001)19 b Fw(+)f FD(\025)2570 2034 y Fq(N)2633 2022 y Fw(\))37 b Fs(\024)g Fw(0)14 b FD(;)456 2258 y Fw(at)27 b FD(x)604 2270 y Ft(0)642 2258 y Fw(,)h(whic)n(h)f(ends)h(the)g(pro)r(of)f(for)g(1)22 b FD(<)h(p)g(<)f Fw(2.)1404 b Fc(\003)1557 2590 y Fw(5.)41 b Fv(Useful)31 b(barriers)555 2740 y Fw(The)i(core)f(of)h(the)g(pro)r (of)g(of)f(our)h(results)f(b)r(egins)h(here.)52 b(Before)32 b(going)g(in)n(to)g(the)i(details)456 2839 y(of)e(the)h(argumen)n(ts,)g (whic)n(h)f(will)h(b)r(e)g(quite)g(tec)n(hnical,)h(w)n(e)e(w)n(ould)g (lik)n(e)g(to)h(p)r(oin)n(t)f(out)h(some)456 2939 y(heuristic)40 b(ideas)g(that)h(are)e(underneath)h(the)h(construction)f(here)g(b)r (elo)n(w.)75 b(Roughly)-7 b(,)43 b(the)456 3039 y(crucial)32 b(idea,)j(whic)n(h)e(go)r(es)f(bac)n(k)h(to)g(De)g(Giorgi,)h(is)f(that) h(one-dimensional)e(solutions)g(are)456 3138 y(the)g(ones)f(whic)n(h)h 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Ft(0)1043 778 y Fw(\()p FD(s)p Fw(\))24 b Fs(\024)e FD(H)1326 790 y Fq(l)1352 778 y Fw(\()p FD(s)p Fw(\))d Fs(\000)p 1567 656 66 4 v 1567 722 a FD(C)1632 734 y Ft(1)p 1567 759 103 4 v 1605 835 a FD(l)1693 778 y Fw(log\(1)g Fs(\000)f(j)p FD(s)p Fs(j)p Fw(\))254 b Fs(8j)p FD(s)p Fs(j)23 b FD(<)f Fw(1)c Fs(\000)g FD(e)2771 744 y Fo(\000)p 2833 689 27 3 v 2833 713 a Fn(c)2860 725 y Fd(1)2892 713 y Fn(l)p 2833 731 80 3 v 2858 764 a Fd(2)2940 778 y Fw(;)456 1041 y(\(5.5\))963 b FD(H)1659 1053 y Fq(l)1685 1041 y Fw(\(1)18 b Fs(\000)g FD(e)1899 1006 y Fo(\000)p 1960 951 27 3 v 1960 975 a Fn(c)1987 987 y Fd(1)2020 975 y Fn(l)p 1960 993 80 3 v 1986 1026 a Fd(2)2054 1041 y Fw(\))23 b Fs(\025)i Fw(\026)-44 b FD(c)14 b(l)h Fw(;)456 1267 y(\(5.6\))931 b FD(H)1627 1279 y Fq(l)1652 1267 y Fw(\()p FD(e)1723 1233 y Fo(\000)p 1785 1177 27 3 v 1785 1201 a Fn(c)1812 1213 y Fd(1)1844 1201 y Fn(l)p 1785 1219 80 3 v 1811 1253 a Fd(2)1897 1267 y Fs(\000)18 b Fw(1\))23 b Fs(\024)g(\000)r Fw(\026)-44 b FD(c)13 b(l)i(:)456 1440 y Fg(Pr)l(o)l(of.)43 b Fw(The)27 b(idea)f(of)h(the)g(pro)r(of)f(is)h(that,)g(once)f FD(H)2054 1452 y Fq(l)2107 1440 y Fw(is)h(w)n(ell)f(de\014ned,)h (estimates)g(\(5.2\),)g(\(5.3\),)456 1539 y(\(5.4\))j(and)g(the)h (viscosit)n(y)e(sup)r(ersolution)h(prop)r(ert)n(y)f(for)h(\011)2356 1509 y Fq(y)r(;l)2467 1539 y Fw(are)g(the)h(core)e(of)i(the)g(matter.) 456 1639 y(Indeed,)23 b(\(5.3\))e(sa)n(ys,)g(in)h(particular,)g(that,)h (b)n(y)e(construction,)h FD(g)2464 1651 y Fq(l)2511 1639 y Fw(is)f(constan)n(t)g(in)g(\()p Fs(\0001)p FD(;)14 b Fs(\000)p FD(l)r(=)p Fw(2].)456 1739 y(Also,)27 b(estimates)g (\(5.4\))h(and)f(\(1.2\))g(imply)h(that)1040 1995 y FD(H)1109 2007 y Fq(l)1135 1995 y Fw(\(1)18 b Fs(\000)g FD(e)1349 1960 y Fo(\000)p 1410 1905 27 3 v 1410 1929 a Fn(c)1437 1941 y Fd(1)1470 1929 y Fn(l)p 1410 1947 80 3 v 1436 1980 a Fd(2)1504 1995 y Fw(\))83 b Fs(\025)g Fw(const)1984 1882 y Fp(Z)2067 1902 y Ft(1)p Fo(\000)p Fq(e)2183 1874 y Fm(\000)p 2238 1824 27 3 v 2238 1848 a 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Fw(\).)555 2957 y(Also,)28 b(if)g FD(T)891 2969 y Fq(l)939 2957 y Fw(:=)22 b FD(H)1118 2969 y Fq(l)1144 2957 y Fw(\(1\),)28 b(b)n(y)f(\(5.2\))g (and)h(\(5.5\),)f(w)n(e)g(ha)n(v)n(e)g(that)g FD(T)2539 2969 y Fq(l)2588 2957 y Fs(2)c Fw([)r(\026)-44 b FD(cl)r(;)14 b(l)r(=)p Fw(2].)35 b(Some)27 b(careful)456 3057 y(computation)32 b(will)g(b)r(e)h(needed)g(to)f(sho)n(w)g(\011)1888 3027 y Fq(y)r(;l)2001 3057 y Fw(to)g(b)r(e)h(a)f(strict)h(viscosit)n(y)e (sup)r(ersolution)h(of)456 3156 y(\(1.5\))d(at)h(an)n(y)g(p)r(oin)n(t)g (where)f(it)i(is)f(de\014ned,)h(except)f(p)r(ossibly)f(on)h(the)h (sphere)e Fs(fj)p FD(x)20 b Fs(\000)g FD(y)s Fs(j)27 b Fw(=)f FD(l)r Fs(g)456 3256 y Fw(\(the)33 b(fact)g(that)h FD(h)1039 3268 y Fq(l)1097 3256 y Fw(ma)n(y)e(b)r(e)i(discon)n(tin)n (uous)d(at)i(0)g(mak)n(e)f FD(g)2367 3268 y Fq(l)2425 3256 y Fw(not)h(necessarily)e(smo)r(oth)i(at)g(0)456 3356 y(and)26 b(depriv)n(e)h(us)f(of)h(information)g(on)f(the)i(v)-5 b(alue)26 b(of)h(\001)2189 3368 y Fq(p)2228 3356 y Fw(\011)2293 3325 y Fq(y)r(;l)2400 3356 y Fw(on)g(the)g(aforesaid)f(sphere\).)36 b(W)-7 b(e)456 3455 y(also)27 b(remark)f(that)j(the)f(extension)g(in)g (\()p FD(ii)p Fw(\))g(is)g(a)g FD(C)2065 3425 y Ft(1)p Fq(;)p Ft(1)2155 3455 y Fw(-extension,)g(since,)g(using)f(Lemma)h(4.6,) 456 3555 y(if)g FD(t)23 b Fw(=)f FD(H)741 3567 y Fq(l)767 3555 y Fw(\()p FD(s)838 3567 y Fq(l)882 3555 y Fs(\000)c Fw(1\),)1370 3754 y FD(g)1413 3719 y Fo(0)1410 3774 y Fq(l)1436 3754 y Fw(\()p FD(t)p Fw(\))84 b(=)1761 3662 y Fp(\020)1892 3698 y FD(p)p 1821 3735 185 4 v 1821 3811 a(p)18 b Fs(\000)g Fw(1)2029 3754 y FD(h)2077 3766 y Fq(l)2103 3754 y Fw(\()p FD(g)2175 3766 y Fq(l)2200 3754 y Fw(\()p FD(t)p Fw(\)\))2326 3662 y Fp(\021)2377 3679 y Ft(1)p Fq(=p)1614 3979 y Fw(=)1761 3886 y Fp(\020)1892 3922 y FD(p)p 1821 3959 V 1821 4036 a(p)g Fs(\000)g Fw(1)2029 3979 y FD(h)2077 3991 y Fq(l)2103 3979 y Fw(\()p FD(s)2174 3991 y Fq(l)2218 3979 y Fs(\000)g Fw(1\))2375 3886 y Fp(\021)2424 3904 y Ft(1)p Fq(=p)1614 4138 y Fw(=)82 b(0)14 b FD(:)555 4492 y Fw(W)-7 b(e)38 b(no)n(w)e(deal)h(with)g(the)g (actual)g(pro)r(of)f(of)h(Lemma)g(5.1:)55 b(in)37 b(ligh)n(t)g(of)f (the)i(argumen)n(ts)456 4591 y(ab)r(o)n(v)n(e,)32 b(w)n(e)g(will)g(fo)r (cus)g(on)g(pro)n(ving)f(\(5.2\),)i(\(5.3\),)h(\(5.4\))e(and)g(the)g (viscosit)n(y)f(sup)r(ersolution)456 4691 y(prop)r(ert)n(y)26 b(for)h(\011)987 4661 y Fq(y)r(;l)1068 4691 y Fw(.)555 4893 y(The)g(pro)r(of)f(will)h(consider)e(separately)h(the)h(cases)e (\()p FD(s)2234 4905 y Fq(l)2276 4893 y Fs(\000)17 b Fw(1\))22 b FD(<)h(s)g(<)g Fw(0)j(and)g(0)d Fs(\024)g FD(s)g(<)f Fw(1.)36 b(Let)456 4993 y(us)27 b(\014rst)g(consider)g(the)h (case)f(\()p FD(s)1447 5005 y Fq(l)1491 4993 y Fs(\000)18 b Fw(1\))23 b FD(<)f(s)h(<)g Fw(0.)36 b(F)-7 b(rom)28 b(\(1.4\),)456 5192 y(\(5.7\))983 b FD(h)1658 5204 y Fq(l)1684 5192 y Fw(\()p FD(s)p Fw(\))23 b Fs(\025)1908 5136 y FD(h)1956 5148 y Ft(0)1993 5136 y Fw(\()p FD(\022)2066 5106 y Fo(\003)2123 5136 y Fs(\000)18 b Fw(1\))p 1908 5173 373 4 v 2073 5249 a(2)p eop %%Page: 16 16 16 15 bop 456 251 a Ft(16)648 b(BERARDINO)23 b(SCIUNZI)g(AND)f(ENRICO)h (V)-7 b(ALDINOCI)456 450 y Fw(if)28 b FD(\022)573 420 y Fo(\003)630 450 y Fs(\000)18 b Fw(1)k Fs(\024)h FD(s)g(<)f Fw(0,)28 b(pro)n(vided)e FD(l)j Fw(is)f(suitably)f(large.)36 b(Also,)27 b(in)h(ligh)n(t)f(of)h(Lemma)f(4.2,)g(w)n(e)g(get)456 640 y(\(5.8\))619 b FD(h)1294 652 y Ft(0)1331 640 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f FD(h)1584 652 y Ft(0)1622 640 y Fw(\()p FD(s)1693 652 y Fq(l)1737 640 y Fs(\000)g Fw(1\))23 b Fs(\025)f FD(c)2054 547 y Fp(\020)2104 640 y Fw(\(1)c(+)g FD(s)p Fw(\))2350 605 y Fq(p)2407 640 y Fs(\000)g FD(s)2529 600 y Fq(p)2529 665 y(l)2567 547 y Fp(\021)2631 640 y FD(;)456 834 y Fw(for)27 b(\()p FD(s)654 846 y Fq(l)698 834 y Fs(\000)18 b Fw(1\))23 b FD(<)f(s)h(<)g(\022)1156 804 y Fo(\003)1213 834 y Fs(\000)18 b Fw(1,)27 b(therefore)456 1042 y(\(5.9\))1340 986 y FD(p)18 b Fs(\000)g Fw(1)p 1340 1023 185 4 v 1411 1099 a FD(p)1534 1042 y Fw(\()p FD(h)1614 1054 y Ft(0)1652 1042 y Fw(\()p FD(s)p Fw(\))h Fs(\000)f FD(h)1905 1054 y Ft(0)1942 1042 y Fw(\()p FD(s)2013 1054 y Fq(l)2057 1042 y Fs(\000)g Fw(1\)\))23 b Fs(\024)g FD(h)2405 1054 y Fq(l)2430 1042 y Fw(\()p FD(s)p Fw(\))14 b FD(;)456 1255 y Fw(if)23 b(\()p FD(s)598 1267 y Fq(l)633 1255 y Fs(\000)9 b Fw(1\))22 b FD(<)h(s)g(<)f(\022)1081 1225 y Fo(\003)1129 1255 y Fs(\000)9 b Fw(1,)23 b(pro)n(vided)f FD(l)i Fw(is)f(su\016cien)n(tly)g(large.)34 b(This,)24 b(\(1.4\))e(and)h(\(5.7\))f(sa)n(y)g(that)456 1355 y FD(h)504 1367 y Fq(l)529 1355 y Fw(\()p FD(s)p Fw(\))h FD(>)g Fw(0)h(in)g(\()p FD(s)973 1367 y Fq(l)1010 1355 y Fs(\000)11 b Fw(1\))23 b FD(<)g(s)g(<)f Fw(0,)j(sho)n(wing)e(that)h FD(H)2068 1367 y Fq(l)2118 1355 y Fw(is)g(w)n(ell)g(de\014ned)g(and)g (strictly)g(increasing)456 1455 y(in)j(this)h(case.)36 b(Also,)28 b(from)f(the)h(de\014nition)g(of)f FD(H)2004 1467 y Fq(l)2030 1455 y Fw(,)h(\(5.7\),)f(\(5.9\))g(and)h(\(5.8\),)f(w) n(e)g(gather)781 1683 y Fs(\000)p FD(H)915 1695 y Fq(l)940 1683 y Fw(\()p FD(s)1011 1695 y Fq(l)1055 1683 y Fs(\000)18 b Fw(1\))83 b(=)g(const)1660 1570 y Fp(Z)1743 1591 y Ft(0)1706 1759 y Fq(s)1737 1768 y Fn(l)1762 1759 y Fo(\000)p Ft(1)2007 1627 y FD(d\020)p 1875 1664 351 4 v 1875 1742 a Fw(\()p FD(h)1955 1754 y Fq(l)1980 1742 y Fw(\()p FD(\020)6 b Fw(\)\))2118 1718 y Ft(1)p Fq(=p)1295 1946 y Fw(=)83 b(const)1660 1804 y Fp( )1726 1833 y(Z)1809 1854 y Ft(0)1772 2022 y Fq(\022)1806 2005 y Fm(\003)1840 2022 y Fo(\000)p Ft(1)2085 1890 y FD(d\020)p 1953 1927 V 1953 2005 a Fw(\()p FD(h)2033 2017 y Fq(l)2059 2005 y Fw(\()p FD(\020)6 b Fw(\)\))2197 1981 y Ft(1)p Fq(=p)2332 1946 y Fw(+)2415 1833 y Fp(Z)2498 1854 y Fq(\022)2532 1829 y Fm(\003)2566 1854 y Fo(\000)p Ft(1)2461 2022 y Fq(s)2492 2031 y Fn(l)2516 2022 y Fo(\000)p Ft(1)2811 1890 y FD(d\020)p 2679 1927 V 2679 2005 a Fw(\()p FD(h)2759 2017 y Fq(l)2784 2005 y Fw(\()p FD(\020)g Fw(\)\))2922 1981 y Ft(1)p Fq(=p)3039 1804 y Fp(!)1295 2228 y Fs(\024)83 b Fw(const)1660 2087 y Fp( )1726 2228 y Fw(1)18 b(+)1869 2115 y Fp(Z)1952 2136 y Fq(\022)1986 2111 y Fm(\003)2020 2136 y Fo(\000)p Ft(1)1915 2304 y Fq(s)1946 2313 y Fn(l)1970 2304 y Fo(\000)p Ft(1)2265 2172 y FD(d\020)p 2133 2209 V 2133 2287 a Fw(\()p FD(h)2213 2299 y Fq(l)2238 2287 y Fw(\()p FD(\020)6 b Fw(\)\))2376 2263 y Ft(1)p Fq(=p)2493 2087 y Fp(!)1295 2561 y Fs(\024)83 b Fw(const)1660 2369 y Fp(0)1660 2515 y(B)1660 2568 y(@)1733 2561 y Fw(1)18 b(+)1876 2448 y Fp(Z)1959 2468 y Fq(\022)1993 2443 y Fm(\003)2027 2468 y Fo(\000)p Ft(1)1922 2636 y Fq(s)1953 2645 y Fn(l)1977 2636 y Fo(\000)p Ft(1)2433 2504 y FD(d\020)p 2140 2541 672 4 v 2140 2578 a Fp(\020)2189 2670 y Fw(\(1)h(+)f FD(\020)6 b Fw(\))2439 2646 y Fq(p)2496 2670 y Fs(\000)18 b FD(s)2618 2630 y Fq(p)2618 2695 y(l)2657 2578 y Fp(\021)2706 2595 y Ft(1)p Fq(=p)2822 2369 y Fp(1)2822 2515 y(C)2822 2568 y(A)1295 2952 y Fs(\024)83 b Fw(const)1660 2760 y Fp(0)1660 2907 y(B)1660 2960 y(@)1733 2952 y Fw(1)18 b(+)1876 2839 y Fp(Z)1959 2860 y Ft(0)1922 3028 y Fq(s)1953 3037 y Fn(l)1977 3028 y Fo(\000)p Ft(1)2383 2896 y FD(d\020)p 2090 2933 V 2090 2969 a Fp(\020)2140 3062 y Fw(\(1)g(+)g FD(\020)6 b Fw(\))2389 3038 y Fq(p)2447 3062 y Fs(\000)18 b FD(s)2569 3022 y Fq(p)2569 3087 y(l)2607 2969 y Fp(\021)2656 2987 y Ft(1)p Fq(=p)2772 2760 y Fp(1)2772 2907 y(C)2772 2960 y(A)2872 2952 y FD(;)456 3251 y Fw(hence,)27 b(from)h(Lemma)f (4.3,)g(w)n(e)g(get)1630 3459 y FD(H)1699 3471 y Fq(l)1724 3459 y Fw(\()p FD(s)1795 3471 y Fq(l)1840 3459 y Fs(\000)18 b Fw(1\))k Fs(\025)h(\000)2189 3403 y FD(l)p 2182 3440 42 4 v 2182 3516 a Fw(2)2247 3459 y FD(;)456 3656 y Fw(pro)n(vided)p 797 3610 36 4 v 26 w FD(c)833 3668 y Ft(1)898 3656 y Fw(is)k(suitably)h(small,)f(pro)n(ving)f(\(5.3\).)555 3895 y(W)-7 b(e)25 b(no)n(w)e(sho)n(w)g(that)i(\011)1308 3865 y Fq(y)r(;l)1413 3895 y Fw(is)f(a)f(viscosit)n(y)g(sup)r (ersolution)h(of)g(\(1.5\))f(when)i Fs(j)p FD(x)12 b Fs(\000)g FD(y)s Fs(j)22 b FD(<)g(l)k Fw(\(i.e.,)456 3995 y(when)20 b FD(s)j Fw(=)g FD(g)855 4007 y Fq(l)880 3995 y Fw(\()p FD(t)p Fw(\))h FD(<)e Fw(0;)h(here)d(and)g(in)g(what)h (follo)n(ws,)g(w)n(e)f(often)h(use)f(the)h(notation)f FD(t)j Fw(=)f Fs(j)p FD(x)t Fs(\000)t FD(y)s Fs(j)t(\000)t FD(l)456 4094 y Fw(and)27 b FD(s)c Fw(=)g FD(g)807 4106 y Fq(l)832 4094 y Fw(\()p FD(t)p Fw(\))g(=)g(\011)1102 4064 y Fq(y)r(;l)1183 4094 y Fw(\()p FD(x)p Fw(\)\).)456 4194 y(Of)k(course,)g(if)h Fs(j)p FD(x)19 b Fs(\000)f FD(y)s Fs(j)23 b FD(<)f(l)r(=)p Fw(2,)k(then)i(\011)1690 4164 y Fq(y)r(;l)1771 4194 y Fw(\()p FD(x)p Fw(\))c(=)f FD(s)2033 4206 y Fq(l)2077 4194 y Fs(\000)18 b Fw(1)27 b(b)n(y)g(de\014nition)h(of)g FD(g)2848 4206 y Fq(l)2873 4194 y Fw(,)f(and)h(therefore)456 4367 y(\(5.10\))511 b(\001)1249 4379 y Fq(p)1288 4367 y Fw(\011)1353 4333 y Fq(y)r(;l)1433 4367 y Fw(\()p FD(x)p Fw(\))24 b(=)f(0)f FD(<)h(h)1856 4333 y Fo(0)1856 4388 y Ft(0)1893 4367 y Fw(\()p FD(s)1964 4379 y Fq(l)2008 4367 y Fs(\000)18 b Fw(1\))23 b(=)g FD(h)2324 4333 y Fo(0)2324 4388 y Ft(0)2361 4367 y Fw(\(\011)2458 4333 y Fq(y)r(;l)2539 4367 y Fw(\()p FD(x)p Fw(\)\))14 b FD(;)456 4539 y Fw(sho)n(wing)31 b(that)h(the)h(viscosit)n(y)e(sup)r(ersolution)h(prop)r(ert)n(y)f(of)h (\011)2473 4509 y Fq(y)r(;l)2586 4539 y Fw(holds)g(for)g Fs(j)p FD(x)22 b Fs(\000)g FD(y)s Fs(j)30 b FD(<)h(l)r(=)p Fw(2.)456 4639 y(Hence,)24 b(w)n(e)e(can)g(no)n(w)g(concen)n(trate)g (on)g(the)h(case)f FD(l)r(=)p Fw(2)g Fs(\024)g(j)p FD(x)9 b Fs(\000)g FD(y)s Fs(j)22 b FD(<)h(l)r Fw(.)35 b(In)23 b(ligh)n(t)f(of)h(Lemma)f(4.6,)1272 4887 y FD(g)1315 4852 y Fo(0)1312 4907 y Fq(l)1338 4887 y Fw(\()p FD(t)p Fw(\))83 b(=)1663 4770 y Fp(\022)1806 4830 y FD(p)p 1734 4867 185 4 v 1734 4944 a(p)18 b Fs(\000)g Fw(1)1943 4887 y FD(h)1991 4899 y Fq(l)2016 4887 y Fw(\()p FD(g)s Fw(\()p FD(t)p Fw(\)\))2217 4770 y Fp(\023)2290 4764 y Fd(1)p 2289 4773 31 3 v 2289 4807 a Fn(p)2513 4887 y Fw(and)1253 5151 y FD(g)1296 5117 y Fo(00)1293 5172 y Fq(l)1338 5151 y Fw(\()p FD(t)p Fw(\))83 b(=)1673 5095 y(\()p FD(p)14 b(h)1809 5107 y Fq(l)1834 5095 y Fw(\()p FD(g)s Fw(\()p FD(t)p Fw(\)\)\))2078 5026 y Fd(2)p Fm(\000)p Fn(p)p 2078 5040 104 3 v 2115 5073 a(p)p 1673 5132 524 4 v 1783 5228 a Fw(\()p FD(p)k Fs(\000)g Fw(1\))2043 5169 y Fd(2)p 2042 5178 31 3 v 2042 5211 a Fn(p)2220 5151 y FD(h)2268 5117 y Fo(0)2268 5172 y Fq(l)2293 5151 y Fw(\()p FD(g)2365 5163 y Fq(l)2391 5151 y Fw(\()p FD(t)p Fw(\)\))c FD(:)p eop %%Page: 17 17 17 16 bop 743 251 a Ft(MEAN)29 b(CUR)-7 b(V)g(A)i(TURE)29 b(PR)n(OPER)-5 b(TIES)28 b(F)n(OR)h Fq(p)p Ft(-LAPLA)n(CE)h(PHASE)e (TRANSITIONS)221 b(17)456 450 y Fw(Th)n(us,)27 b(b)n(y)g(Lemma)h(4.4,)e (w)n(e)i(ha)n(v)n(e)926 666 y(\001)995 678 y Fq(p)1034 666 y Fw(\(\011)1131 631 y Fq(y)r(;l)1211 666 y Fw(\()p FD(x)p Fw(\)\))d(=)d(\()p FD(p)d Fs(\000)f Fw(1\))p FD(g)1759 631 y Fo(00)1801 666 y Fw(\()p FD(t)p Fw(\))p FD(g)1938 631 y Fo(0)p Ft(\()p Fq(p)p Fo(\000)p Ft(2\))2132 666 y Fw(\()p FD(t)p Fw(\))h(+)f FD(g)2371 631 y Fo(0)p Ft(\()p Fq(p)p Fo(\000)p Ft(1\))2565 666 y Fw(\()p FD(t)p Fw(\))2680 609 y FD(N)27 b Fs(\000)18 b Fw(1)p 2669 646 239 4 v 2669 722 a Fs(j)p FD(x)i Fs(\000)e FD(y)s Fs(j)1379 886 y(\024)k FD(h)1514 852 y Fo(0)1514 907 y Fq(l)1540 886 y Fw(\()p FD(g)1612 898 y Fq(l)1637 886 y Fw(\()p FD(t)p Fw(\)\))d(+)f FD(K)6 b Fw(\()p FD(N)27 b Fs(\000)18 b Fw(1\)\()p FD(h)2305 898 y Fq(l)2331 886 y Fw(\()p FD(g)2403 898 y Fq(l)2428 886 y Fw(\()p FD(t)p Fw(\)\)\))2596 822 y Fn(p)p Fm(\000)p Fd(1)p 2597 836 104 3 v 2634 869 a Fn(p)2824 830 y Fw(1)p 2725 867 239 4 v 2725 943 a Fs(j)p FD(x)h Fs(\000)f FD(y)s Fs(j)1379 1145 y(\024)k FD(h)1514 1110 y Fo(0)1514 1165 y Fq(l)1540 1145 y Fw(\()p FD(g)1612 1157 y Fq(l)1637 1145 y Fw(\()p FD(t)p Fw(\)\))d(+)1875 1089 y(2)p FD(K)6 b Fw(\()p FD(N)27 b Fs(\000)18 b Fw(1\)\()p FD(h)2357 1101 y Fq(l)2382 1089 y Fw(\()p FD(g)2454 1101 y Fq(l)2480 1089 y Fw(\()p FD(t)p Fw(\)\)\))2648 1025 y Fn(p)p Fm(\000)p Fd(1)p 2649 1039 104 3 v 2685 1072 a Fn(p)p 1875 1126 892 4 v 2308 1202 a FD(l)456 898 y Fw(\(5.11\))456 1361 y(for)27 b Fs(j)p FD(x)19 b Fs(\000)f FD(y)s Fs(j)k(\025)948 1328 y Fq(l)p 942 1342 34 4 v 942 1390 a Ft(2)985 1361 y Fw(,)28 b(pro)n(vided)e FD(K)j(>)23 b Fw(0)k(is)g(suitably)h(large.)456 1461 y(Hence,)f(b)n(y)h (de\014nition)g(of)f FD(h)1352 1473 y Fq(l)1405 1461 y Fw(w)n(e)g(get)h(\(using)f(again)g(the)h(notation)f FD(s)c Fw(=)f FD(g)2801 1473 y Fq(l)2827 1461 y Fw(\()p FD(t)p Fw(\)\))28 b(that)1416 1638 y FD(h)1464 1650 y Fq(l)1489 1638 y Fw(\()p FD(s)p Fw(\))84 b Fs(\024)f FD(h)1872 1650 y Ft(0)1909 1638 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f FD(h)2162 1650 y Ft(0)2199 1638 y Fw(\()p FD(s)2270 1650 y Fq(l)2314 1638 y Fs(\000)g Fw(1\))23 b Fs(\024)f FD(h)2629 1650 y Ft(0)2666 1638 y Fw(\()p FD(s)p Fw(\))1116 1822 y(and)166 b FD(h)1464 1788 y Fo(0)1464 1842 y Fq(l)1489 1822 y Fw(\()p FD(s)p Fw(\))84 b(=)f FD(h)1872 1788 y Fo(0)1872 1842 y Ft(0)1909 1822 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)2124 1766 y FD(p)p 2166 1699 66 4 v(C)2231 1778 y Ft(2)p 2124 1803 145 4 v 2182 1879 a FD(l)2278 1822 y Fw(\(1)f(+)g FD(s)p Fw(\))2524 1788 y Ft(\()p Fq(p)p Fo(\000)p Ft(1\))456 2028 y Fw(in)27 b(\()p FD(s)623 2040 y Fq(l)668 2028 y Fs(\000)18 b Fw(1\))k FD(<)h(s)g(<)g Fw(0,)k(hence)456 2264 y(\(5.12\))209 b(\001)947 2276 y Fq(p)985 2264 y Fw(\(\011)1082 2230 y Fq(y)r(;l)1163 2264 y Fw(\()p FD(x)p Fw(\)\))24 b FD(<)f(h)1466 2230 y Fo(0)1466 2285 y Ft(0)1503 2264 y Fw(\()p FD(s)p Fw(\))c Fs(\000)1718 2208 y FD(p)p 1760 2141 66 4 v(C)1825 2220 y Ft(2)p 1718 2245 145 4 v 1777 2321 a FD(l)1872 2264 y Fw(\(1)f(+)g FD(s)p Fw(\))2118 2230 y Ft(\()p Fq(p)p Fo(\000)p Ft(1\))2312 2264 y Fw(+)2405 2208 y(2)p FD(K)6 b Fw(\()p FD(N)27 b Fs(\000)18 b Fw(1\))p 2405 2245 402 4 v 2593 2321 a FD(l)2817 2264 y Fw(\()p FD(h)2897 2276 y Ft(0)2934 2264 y Fw(\()p FD(s)p Fw(\)\))3079 2200 y Fn(p)p Fm(\000)p Fd(1)p 3080 2214 104 3 v 3116 2247 a Fn(p)3212 2264 y FD(;)456 2481 y Fw(for)27 b(\()p FD(s)654 2493 y Fq(l)698 2481 y Fs(\000)18 b Fw(1\))23 b FD(<)f(s)h(<)g Fw(0.)36 b(By)28 b(\(1.2\),)f(w)n(e)g(get,)h(for)p 1979 2414 66 4 v 27 w FD(C)2044 2493 y Ft(2)2109 2481 y Fw(suitably)f (large,)g(that)456 2718 y(\(5.13\))1210 2661 y FD(p)p 1252 2595 V(C)1318 2673 y Ft(2)p 1210 2698 145 4 v 1269 2774 a FD(l)1365 2718 y Fw(\(1)18 b(+)g FD(s)p Fw(\))1611 2683 y Ft(\()p Fq(p)p Fo(\000)p Ft(1\))1810 2718 y Fs(\025)1907 2661 y Fw(2)p FD(K)6 b Fw(\()p FD(N)27 b Fs(\000)18 b Fw(1\))p 1907 2698 402 4 v 2095 2774 a FD(l)2318 2718 y Fw(\()p FD(h)2398 2730 y Ft(0)2436 2718 y Fw(\()p FD(s)p Fw(\)\))2581 2654 y Fn(p)p Fm(\000)p Fd(1)p 2581 2668 104 3 v 2618 2701 a Fn(p)456 2919 y Fw(and)27 b(therefore)456 3096 y(\(5.14\))807 b(\001)1545 3108 y Fq(p)1584 3096 y Fw(\(\011)1681 3062 y Fq(y)r(;l)1762 3096 y Fw(\()p FD(x)p Fw(\)\))24 b FD(<)f(h)2065 3062 y Fo(0)2065 3117 y Ft(0)2102 3096 y Fw(\(\011)2199 3062 y Fq(y)r(;l)2279 3096 y Fw(\()p FD(x)p Fw(\)\))456 3279 y(for)k FD(s)622 3291 y Fq(l)666 3279 y Fs(\000)18 b Fw(1)k FD(<)h(g)941 3291 y Fq(l)966 3279 y Fw(\()p FD(t)p Fw(\))28 b(and)f Fs(j)p FD(x)19 b Fs(\000)f FD(y)s Fs(j)23 b(\025)1615 3247 y Fq(l)p 1609 3261 34 4 v 1609 3308 a Ft(2)1652 3279 y Fw(.)456 3387 y(Estimates)h(\(5.10\))h(and)g(\(5.14\))g(sho)n(w) f(\011)1741 3357 y Fq(y)r(;l)1847 3387 y Fw(to)i(b)r(e)f(a)g(strict)h (viscosit)n(y)e(sup)r(ersolution)g(of)i(\(1.5\))456 3486 y(at)h(an)n(y)g(p)r(oin)n(ts)g FD(x)i Fw(so)d(that)i Fs(j)p FD(x)19 b Fs(\000)f FD(y)s Fs(j)23 b FD(<)g(l)r Fw(,)k(pro)n(vided)f(\011)2153 3456 y Fq(y)r(;l)2234 3486 y Fw(\()p FD(x)p Fw(\))e Fs(6)p Fw(=)f FD(s)2496 3498 y Fq(l)2540 3486 y Fs(\000)18 b Fw(1.)456 3586 y(Hence,)26 b(w)n(e)g(no)n(w)f(deal)h(with)h(the)f(case)g(in)g(whic)n(h)g Fs(j)p FD(x)16 b Fs(\000)f FD(y)s Fs(j)23 b FD(<)g(l)k Fw(and)f(\011)2646 3556 y Fq(y)r(;l)2727 3586 y Fw(\()p FD(x)p Fw(\))e(=)f FD(g)2990 3598 y Fq(l)3015 3586 y Fw(\()p FD(t)p Fw(\))g(=)g FD(s)3259 3598 y Fq(l)3300 3586 y Fs(\000)15 b Fw(1:)456 3686 y(in)27 b(suc)n(h)h(circumstance,)f (since)g(\011)g(is)h FD(C)1706 3656 y Ft(1)1771 3686 y Fw(w)n(e)f(ha)n(v)n(e,)g(b)n(y)g(Lemma)g(4.7,)g(that)456 3863 y(\(5.15\))756 b(\001)1494 3875 y Fq(p)1533 3863 y Fw(\(\011)1630 3829 y Fq(y)r(;l)1711 3863 y Fw(\()p FD(x)p Fw(\)\))24 b Fs(\024)f Fw(0)f FD(<)h(h)2166 3829 y Fo(0)2166 3884 y Ft(0)2203 3863 y Fw(\()p FD(s)2274 3875 y Fq(l)2318 3863 y Fs(\000)18 b Fw(1\))456 4041 y(in)28 b(the)h(viscosit)n(y)e(sense.)39 b(This)28 b(completes)g(the)h (pro)r(of)f(of)g(the)h(desired)f(sup)r(ersolution)f(prop-)456 4141 y(ert)n(y)f(of)i(\011)786 4111 y Fq(y)r(;l)894 4141 y Fw(at)g(an)n(y)f(p)r(oin)n(ts)g FD(x)h Fw(so)f(that)h Fs(j)p FD(x)19 b Fs(\000)f FD(y)s Fs(j)23 b FD(<)f(l)r Fw(.)555 4416 y(Let)29 b(us)f(no)n(w)g(pro)n(v)n(e)e(\(5.4\))i(for)g FD(e)1576 4386 y Fo(\000)p 1638 4331 27 3 v 1638 4355 a Fn(c)1665 4367 y Fd(1)1697 4355 y Fn(l)p 1638 4373 80 3 v 1663 4406 a Fd(2)1750 4416 y Fs(\000)18 b Fw(1)24 b FD(<)g(s)g Fs(\024)g Fw(0.)39 b(Observ)n(e)26 b(that,)j(b)n(y)f (de\014nition)h(of)f FD(h)3396 4428 y Fq(l)3421 4416 y Fw(,)456 4516 y(recalling)e(\(1.2\),)1029 4753 y FD(h)1077 4765 y Ft(0)1114 4753 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f FD(h)1367 4765 y Fq(l)1393 4753 y Fw(\()p FD(s)p Fw(\))83 b Fs(\024)g FD(h)1775 4765 y Ft(0)1812 4753 y Fw(\()p FD(s)1883 4765 y Fq(l)1927 4753 y Fs(\000)18 b Fw(1\))g(+)p 2195 4630 66 4 v 2195 4696 a FD(C)2261 4708 y Ft(2)p 2195 4733 103 4 v 2233 4809 a FD(l)2308 4660 y Fp(\020)2357 4753 y Fw(\(1)h(+)f FD(s)p Fw(\))2604 4718 y Fq(p)2661 4753 y Fs(\000)g FD(s)2783 4713 y Fq(p)2783 4778 y(l)2821 4660 y Fp(\021)1579 4972 y Fs(\024)83 b FD(C)6 b(s)1831 4932 y Fq(p)1831 4997 y(l)1888 4972 y Fw(+)p 1981 4849 66 4 v 1981 4916 a FD(C)2046 4928 y Ft(2)p 1981 4953 103 4 v 2019 5029 a FD(l)2093 4880 y Fp(\020)2143 4972 y Fw(\(1)18 b(+)g FD(s)p Fw(\))2389 4938 y Fq(p)2446 4972 y Fs(\000)g FD(s)2568 4932 y Fq(p)2568 4997 y(l)2607 4880 y Fp(\021)1579 5192 y Fs(\024)p 1737 5069 66 4 v 1737 5136 a FD(C)1802 5148 y Ft(2)p 1737 5173 103 4 v 1775 5249 a FD(l)1849 5192 y Fw(\(1)h(+)f FD(s)p Fw(\))2096 5158 y Fq(p)2148 5192 y FD(;)-1715 b Fw(\(5.16\))p eop %%Page: 18 18 18 17 bop 456 251 a Ft(18)648 b(BERARDINO)23 b(SCIUNZI)g(AND)f(ENRICO)h (V)-7 b(ALDINOCI)456 450 y Fw(pro)n(vided)26 b FD(l)j Fw(is)f(su\016cien)n(tly)f(large.)456 550 y(Also,)g(using)g(Lemma)h (4.1,)638 761 y FD(H)707 773 y Ft(0)745 761 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f FD(H)1019 773 y Fq(l)1044 761 y Fw(\()p FD(s)p Fw(\))84 b(=)1378 648 y Fp(Z)1461 668 y Ft(0)1424 836 y Fq(s)1721 705 y Fw(1)p 1522 742 439 4 v 1522 838 a(\()1607 801 y Fq(p)p 1564 819 120 4 v 1564 866 a(p)p Fo(\000)p Ft(1)1694 838 y FD(h)1742 850 y Fq(l)1767 838 y Fw(\()p FD(\020)6 b Fw(\)\))1916 778 y Fd(1)p 1916 787 31 3 v 1916 821 a Fn(p)1989 761 y Fs(\000)2287 705 y Fw(1)p 2082 742 451 4 v 2082 838 a(\()2167 801 y Fq(p)p 2125 819 120 4 v 2125 866 a(p)p Fo(\000)p Ft(1)2254 838 y FD(h)2302 850 y Ft(0)2339 838 y Fw(\()p FD(\020)g Fw(\)\))2488 778 y Fd(1)p 2488 787 31 3 v 2488 821 a Fn(p)2557 761 y FD(d\020)1231 1041 y Fw(=)82 b(const)1595 928 y Fp(Z)1678 948 y Ft(0)1641 1116 y Fq(s)1739 985 y Fw(\()p 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FD(\020)6 b Fw(\)\))2529 1470 y Fn(p)p Fm(\000)p Fd(1)p 2530 1484 V 2567 1517 a Fn(p)2648 1442 y Fp(\021)14 b(\020)2761 1534 y FD(h)2809 1546 y Ft(0)2846 1534 y Fw(\()p FD(\020)6 b Fw(\))14 b FD(h)3014 1546 y Fq(l)3040 1534 y Fw(\()p FD(\020)6 b Fw(\))3146 1442 y Fp(\021)3208 1437 y Fd(1)p 3207 1446 31 3 v 3207 1479 a Fn(p)1231 1751 y Fs(\024)82 b Fw(const)1595 1638 y Fp(Z)1678 1659 y Ft(0)1641 1827 y Fq(s)1755 1695 y FD(h)1803 1707 y Ft(0)1840 1695 y Fw(\()p FD(\020)6 b Fw(\))20 b Fs(\000)e FD(h)2097 1707 y Fq(l)2122 1695 y Fw(\()p FD(\020)6 b Fw(\))p 1739 1732 506 4 v 1739 1828 a FD(h)1787 1840 y Ft(0)1825 1828 y Fw(\()p FD(\020)g Fw(\))14 b(\()p FD(h)2025 1840 y Fq(l)2051 1828 y Fw(\()p FD(\020)6 b Fw(\)\))2200 1769 y Fd(1)p 2200 1778 31 3 v 2200 1811 a Fn(p)2269 1751 y FD(d\020)20 b(:)456 1976 y Fw(Consequen)n(tly)-7 b(,)26 b(from)i(\(5.7\),)f(\(5.9\),)h(\(5.8\))f (and)g(\(1.2\),)487 2207 y FD(H)556 2219 y Ft(0)594 2207 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f FD(H)868 2219 y Fq(l)893 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b(\(5.16\),)f(w)n(e)i(deduce)f(that)892 3181 y FD(H)961 3193 y Ft(0)998 3181 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f FD(H)1272 3193 y Fq(l)1297 3181 y Fw(\()p FD(s)p Fw(\))84 b Fs(\024)1642 3124 y Fw(const)p 1642 3161 190 4 v 1723 3237 a FD(l)1855 3088 y Fp(\020)1904 3181 y Fw(1)18 b(+)2047 3068 y Fp(Z)2130 3088 y Fq(\022)2164 3063 y Fm(\003)2199 3088 y Fo(\000)p Ft(1)2094 3256 y Fq(s)2587 3124 y FD(d\020)p 2312 3161 638 4 v 2312 3259 a Fw(\(\(1)g(+)g FD(\020)6 b Fw(\))2593 3235 y Fq(p)2651 3259 y Fs(\000)18 b FD(s)2773 3219 y Fq(p)2773 3284 y(l)2811 3259 y Fw(\))2843 3215 y Ft(1)p Fq(=p)2959 3088 y Fp(\021)1484 3448 y Fs(\024)1642 3391 y Fw(const)p 1642 3428 190 4 v 1723 3505 a FD(l)1855 3355 y Fp(\020)1904 3448 y Fw(1)g(+)2047 3335 y Fp(Z)2130 3355 y Ft(0)2094 3523 y Fq(s)2467 3391 y FD(d\020)p 2192 3428 638 4 v 2192 3526 a Fw(\(\(1)g(+)g FD(\020)6 b Fw(\))2473 3502 y Fq(p)2531 3526 y Fs(\000)18 b FD(s)2653 3487 y Fq(p)2653 3551 y(l)2691 3526 y Fw(\))2723 3482 y Ft(1)p Fq(=p)2839 3355 y Fp(\021)2902 3448 y FD(:)456 3674 y Fw(By)27 b(Lemma)g(4.3,)g(w)n(e)g(can)g(estimate)h(the)g(last)f (in)n(tegral)g(and)g(get)1378 3847 y FD(H)1447 3859 y Ft(0)1484 3847 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f FD(H)1758 3859 y Fq(l)1783 3847 y Fw(\()p FD(s)p Fw(\))24 b Fs(\024)f(\000)2073 3791 y FD(c)p 2073 3828 36 4 v 2078 3904 a(l)2132 3847 y Fw(log\(1)18 b(+)g FD(s)p Fw(\))c FD(;)456 4059 y Fw(pro)n(ving)27 b(\(5.4\))h(for)g FD(e)1123 4029 y Fo(\000)p 1184 3974 27 3 v 1184 3998 a Fn(c)1211 4010 y Fd(1)1244 3998 y Fn(l)p 1184 4016 80 3 v 1210 4049 a Fd(2)1297 4059 y Fs(\000)18 b Fw(1)24 b FD(<)g(s)h Fs(\024)f Fw(0.)39 b(This)29 b(completes)f(the)h(pro)r(of)e(in)i(the)g (case)f FD(s)3186 4071 y Fq(l)3230 4059 y Fs(\000)19 b Fw(1)24 b FD(<)456 4159 y(s)f(<)f Fw(0.)555 4372 y(Let)36 b(us)f(no)n(w)g(consider)g(the)h(case)e(0)i Fs(\024)g FD(s)g(<)g Fw(1.)60 b(In)36 b(this)g(case,)g FD(h)2691 4384 y Fq(l)2753 4372 y FD(>)g Fw(0)f(b)n(y)g(insp)r(ection,)456 4472 y(th)n(us)f FD(H)714 4484 y Fq(l)774 4472 y Fw(is)g(w)n(ell)g (de\014ned)h(and)f(strictly)g(increasing)f(in)i([0,1\).)56 b(Denoting)35 b FD(t)f Fw(=)g Fs(j)p FD(x)23 b Fs(\000)g FD(y)s Fs(j)f(\000)h FD(l)456 4571 y Fw(and)k FD(s)d Fw(=)f FD(g)808 4583 y Fq(l)833 4571 y Fw(\()p FD(t)p Fw(\))h(=)f(\011)1104 4541 y Fq(y)r(;l)1184 4571 y Fw(\()p FD(x)p Fw(\),)29 b(w)n(e)f(notice)g(that)g FD(s)23 b(>)g Fw(0)28 b(corresp)r(onds)e(to)h Fs(j)p FD(x)19 b Fs(\000)g FD(y)s Fs(j)k(\025)g FD(l)r Fw(,)k(therefore,)456 4671 y(arguing)f(as)h(in)g(\(5.11\),)g(w)n(e)h(ha)n(v)n(e:)456 4875 y(\(5.17\))497 b(\001)1235 4887 y Fq(p)1274 4875 y Fw(\(\011)1371 4841 y Fq(y)r(;l)1452 4875 y Fw(\()p FD(x)p Fw(\)\))24 b FD(<)f(h)1755 4841 y Fo(0)1755 4896 y Fq(l)1780 4875 y Fw(\()p FD(s)p Fw(\))c(+)1995 4819 y FD(K)6 b Fw(\()p FD(N)27 b Fs(\000)18 b Fw(1\))p 1995 4856 360 4 v 2161 4932 a FD(l)2364 4875 y Fw(\()p FD(h)2444 4887 y Fq(l)2470 4875 y Fw(\()p FD(s)p Fw(\)\))2615 4811 y Fn(p)p Fm(\000)p Fd(1)p 2616 4825 104 3 v 2652 4858 a Fn(p)456 5059 y Fw(if)28 b Fs(j)p FD(x)19 b Fs(\000)f FD(y)s Fs(j)k(\025)h FD(l)r Fw(,)k(pro)n(vided)g FD(K)33 b Fw(is)27 b(large)g(enough.)36 b(Since,)28 b(b)n(y)f(de\014nition)h (of)g FD(h)2876 5071 y Fq(l)2929 5059 y Fw(and)f(\(1.2\),)1323 5216 y FD(h)1371 5228 y Fq(l)1397 5216 y Fw(\()p FD(s)p Fw(\))c Fs(\024)37 b Fw(const)13 b(\()p FD(h)1908 5228 y Ft(0)1945 5216 y Fw(\()p FD(s)p Fw(\))19 b(+)f FD(h)2198 5228 y Ft(0)2236 5216 y Fw(\(1)g Fs(\000)g FD(s)2450 5228 y Fq(l)2475 5216 y Fw(\)\))c FD(;)p eop %%Page: 19 19 19 18 bop 743 251 a Ft(MEAN)29 b(CUR)-7 b(V)g(A)i(TURE)29 b(PR)n(OPER)-5 b(TIES)28 b(F)n(OR)h Fq(p)p Ft(-LAPLA)n(CE)h(PHASE)e (TRANSITIONS)221 b(19)456 450 y Fw(for)p 583 383 66 4 v 27 w FD(C)648 462 y Ft(2)713 450 y Fw(large)26 b(enough,)h(it)h (follo)n(ws)f(that)978 667 y(\001)1047 679 y Fq(p)1085 667 y Fw(\(\011)1182 632 y Fq(y)r(;l)1263 667 y Fw(\()p FD(x)p Fw(\)\))85 b FD(<)d(h)1686 632 y Fo(0)1686 687 y Ft(0)1723 667 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)1938 610 y FD(p)p 1980 544 V(C)2045 622 y Ft(2)p 1938 648 145 4 v 1997 724 a FD(l)2092 574 y Fp(\020)2142 667 y Fw(\(1)f Fs(\000)g FD(s)p Fw(\))2388 632 y Fq(p)p Fo(\000)p Ft(1)2530 667 y Fw(+)g FD(s)2652 627 y Fq(p)p Fo(\000)p Ft(1)2652 692 y Fq(l)2776 574 y Fp(\021)1638 875 y Fw(+)1713 819 y FD(K)6 b Fw(\()p FD(N)27 b Fs(\000)18 b Fw(1\))p 1713 856 360 4 v 1879 932 a FD(l)2082 875 y Fw(\()p FD(h)2162 887 y Ft(0)2200 875 y Fw(\()p FD(s)p Fw(\))h(+)f FD(h)2453 887 y Ft(0)2490 875 y Fw(\(1)g Fs(\000)g FD(s)2704 887 y Fq(l)2730 875 y Fw(\)\))2804 811 y Fn(p)p Fm(\000)p Fd(1)p 2804 825 104 3 v 2841 858 a Fn(p)1491 1032 y Fs(\024)82 b FD(h)1686 998 y Fo(0)1686 1052 y Ft(0)1723 1032 y Fw(\()p FD(s)p Fw(\))14 b FD(;)456 1194 y Fw(where,)24 b(in)g(the)h(last)f (estimate,)h(\(1.2\))f(has)f(b)r(een)i(used)f(once)g(more)f(together)g (with)i(the)f(simple)456 1294 y(inequalit)n(y)34 b(\()p FD(a)24 b Fw(+)f FD(b)p Fw(\))1104 1264 y Fq(q)1176 1294 y Fs(\024)36 b Fw(2)1319 1264 y Fq(q)1355 1294 y Fw(\()p FD(a)1431 1264 y Fq(q)1491 1294 y Fw(+)23 b FD(b)1615 1264 y Fq(q)1652 1294 y Fw(\))35 b(with)h FD(q)j Fw(:=)c(\()p FD(p)24 b Fs(\000)f Fw(1\))p FD(=p)34 b Fw(.)60 b(Th)n(us)35 b(\011)2859 1264 y Fq(y)r(;l)2939 1294 y Fw(\()p FD(x)p Fw(\))i(is)e(a)g(strict)456 1393 y(viscosit)n(y)26 b(sup)r(ersolution)h (of)g(\(1.5\))h(for)f Fs(j)p FD(x)19 b Fs(\000)f FD(y)s Fs(j)k FD(>)h(l)r Fw(,)k(pro)n(vided)g(\011)2555 1363 y Fq(y)r(;l)2635 1393 y Fw(\()p FD(x)p Fw(\))i(is)f(w)n(ell)f (de\014ned.)555 1638 y(W)-7 b(e)34 b(need)g(no)n(w)f(to)h(pro)n(v)n(e)e (\(5.4\))h(in)h(the)g(case)f(0)f Fs(\024)h FD(s)g(<)g Fw(1)22 b Fs(\000)g FD(e)2601 1576 y Fm(\000)p 2646 1552 27 3 v Fn(c)2672 1588 y Fd(1)2705 1576 y Fn(l)p 2601 1595 125 3 v 2649 1628 a Fd(2)2739 1638 y Fw(.)55 b(T)-7 b(o)33 b(this)h(end,)i(\014rst)456 1757 y(notice)27 b(that,)h(if)g(0)23 b Fs(\024)f FD(s)h(<)g Fw(1)18 b Fs(\000)g FD(e)1473 1696 y Fm(\000)p 1518 1672 27 3 v Fn(c)1544 1708 y Fd(1)1577 1696 y Fn(l)p 1473 1714 125 3 v 1521 1747 a Fd(2)1611 1757 y Fw(,)28 b(w)n(e)f(ha)n(v)n(e)f(1)18 b Fs(\000)g FD(s)23 b(>)2268 1703 y Fs(p)p 2337 1703 65 4 v 54 x FD(s)2376 1769 y Fq(l)2429 1757 y Fw(and)28 b(therefore)456 1964 y(\(5.18\))719 b FD(s)1427 1924 y Fq(p)1427 1989 y(l)1488 1964 y Fs(\024)23 b FD(s)1615 1921 y Fq(p=)p Ft(2)1615 1989 y Fq(l)1720 1964 y Fw(\(1)18 b Fs(\000)g FD(s)p Fw(\))1966 1929 y Fq(p)2028 1964 y Fs(\024)2125 1908 y Fw(1)p 2125 1945 42 4 v 2133 2021 a FD(l)2191 1964 y Fw(\(1)g Fs(\000)g FD(s)p Fw(\))2437 1929 y Fq(p)2489 1964 y FD(;)456 2154 y Fw(if)28 b FD(l)h Fw(is)e(large)f(enough.)37 b(The)27 b(de\014nition)h(of)g FD(h)1873 2166 y Fq(l)1898 2154 y Fw(,)g(\(1.2\))f(and)h(\(5.18\))f(imply)g(that)456 2353 y(\(5.19\))321 b FD(h)1038 2365 y Fq(l)1063 2353 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f FD(h)1316 2365 y Ft(0)1353 2353 y Fw(\()p FD(s)p Fw(\))24 b Fs(\024)e FD(s)1606 2313 y Fq(p)1606 2378 y(l)1663 2353 y Fw(+)1756 2296 y(const)p 1756 2334 190 4 v 1838 2410 a FD(l)1970 2353 y Fw(\(1)c Fs(\000)g FD(s)p Fw(\))2216 2318 y Fq(p)2277 2353 y Fs(\024)2375 2296 y Fw(const)p 2375 2334 V 2457 2410 a FD(l)2588 2353 y Fw(\(1)h Fs(\000)f FD(s)p Fw(\))2835 2318 y Fq(p)2887 2353 y FD(;)456 2571 y Fw(for)27 b(0)22 b Fs(\024)h FD(s)g(<)f Fw(1)c Fs(\000)g FD(e)1076 2509 y Fm(\000)p 1121 2485 27 3 v Fn(c)1148 2521 y Fd(1)1180 2509 y Fn(l)p 1076 2527 125 3 v 1124 2560 a Fd(2)1214 2571 y Fw(.)37 b(On)28 b(the)g(other)f(hand,)g(the)h(de\014nition)g(of) g FD(h)2658 2583 y Fq(l)2711 2571 y Fw(and)f(\(1.2\))h(lead)f(to)456 2733 y(\(5.20\))386 b FD(h)1103 2745 y Fq(l)1128 2733 y Fw(\()p FD(s)p Fw(\))24 b Fs(\025)e FD(h)1390 2745 y Ft(0)1427 2733 y Fw(\()p FD(s)p Fw(\))d(+)f FD(h)1680 2745 y Ft(0)1717 2733 y Fw(\(1)h Fs(\000)f FD(s)1932 2745 y Fq(l)1957 2733 y Fw(\))23 b Fs(\025)37 b Fw(const)13 b(\(\(1)19 b Fs(\000)f FD(s)p Fw(\))2596 2699 y Fq(p)2653 2733 y Fw(+)g FD(s)2775 2693 y Fq(p)2775 2758 y(l)2813 2733 y Fw(\))456 2919 y(for)27 b(0)22 b Fs(\024)h FD(s)g(<)f Fw(1)c Fs(\000)g FD(e)1076 2858 y Fm(\000)p 1121 2834 27 3 v Fn(c)1148 2870 y Fd(1)1180 2858 y Fn(l)p 1076 2876 125 3 v 1124 2909 a Fd(2)1214 2919 y Fw(.)37 b(Also,)28 b(from)f(Lemma)g(4.1,)755 3123 y FD(H)824 3135 y Ft(0)862 3123 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f FD(H)1136 3135 y Fq(l)1161 3123 y Fw(\()p FD(s)p Fw(\))24 b(=)1375 3010 y Fp(Z)1458 3031 y Fq(s)1422 3199 y Ft(0)1722 3067 y Fw(1)p 1518 3104 451 4 v 1518 3200 a(\()1602 3163 y Fq(p)p 1560 3181 120 4 v 1560 3229 a(p)p Fo(\000)p Ft(1)1689 3200 y FD(h)1737 3212 y Ft(0)1774 3200 y Fw(\()p FD(\020)6 b Fw(\)\))1923 3141 y Fd(1)p 1923 3150 31 3 v 1923 3183 a Fn(p)1997 3123 y Fs(\000)2288 3067 y Fw(1)p 2090 3104 439 4 v 2090 3200 a(\()2174 3163 y Fq(p)p 2132 3181 120 4 v 2132 3229 a(p)p Fo(\000)p Ft(1)2261 3200 y FD(h)2309 3212 y Fq(l)2334 3200 y Fw(\()p FD(\020)g Fw(\)\))2483 3141 y Fd(1)p 2484 3150 31 3 v 2484 3183 a Fn(p)2552 3123 y FD(d\020)1288 3392 y Fs(\024)22 b Fw(const)1593 3279 y Fp(Z)1676 3299 y Fq(s)1639 3468 y Ft(0)2254 3336 y FD(h)2302 3348 y Fq(l)2327 3336 y Fw(\()p FD(\020)6 b Fw(\))20 b Fs(\000)e FD(h)2584 3348 y Ft(0)2621 3336 y Fw(\()p FD(\020)6 b Fw(\))p 1735 3373 1513 4 v 1735 3415 a Fp(\020)1785 3507 y Fw(\()p FD(h)1865 3519 y Fq(l)1890 3507 y Fw(\()p FD(\020)g Fw(\)\))2038 3443 y Fn(p)p Fm(\000)p Fd(1)p 2039 3457 104 3 v 2076 3490 a Fn(p)2176 3507 y Fw(+)18 b(\()p FD(h)2339 3519 y Ft(0)2376 3507 y Fw(\()p FD(\020)6 b Fw(\)\))2524 3443 y Fn(p)p Fm(\000)p Fd(1)p 2526 3457 V 2562 3490 a Fn(p)2644 3415 y Fp(\021)13 b(\020)2757 3507 y FD(h)2805 3519 y Fq(l)2830 3507 y Fw(\()p FD(\020)6 b Fw(\))14 b FD(h)2998 3519 y Ft(0)3036 3507 y Fw(\()p FD(\020)6 b Fw(\))3142 3415 y Fp(\021)3204 3410 y Fd(1)p 3202 3419 31 3 v 3202 3452 a Fn(p)3271 3392 y FD(d\020)1288 3712 y Fs(\024)22 b Fw(const)1593 3599 y Fp(Z)1676 3620 y Fq(s)1639 3788 y Ft(0)1751 3656 y FD(h)1799 3668 y Fq(l)1824 3656 y Fw(\()p FD(\020)6 b Fw(\))20 b Fs(\000)e FD(h)2081 3668 y Ft(0)2118 3656 y Fw(\()p FD(\020)6 b Fw(\))p 1735 3693 506 4 v 1735 3789 a FD(h)1783 3801 y Ft(0)1820 3789 y Fw(\()p FD(\020)g Fw(\))14 b(\()p FD(h)2020 3801 y Fq(l)2046 3789 y Fw(\()p FD(\020)6 b Fw(\)\))2195 3730 y Fd(1)p 2196 3739 31 3 v 2196 3772 a Fn(p)2264 3712 y FD(d\020)21 b(:)456 3429 y Fw(\(5.21\))456 3943 y(Th)n(us,)27 b(from)g(\(5.21\),)g(\(5.20\),)g(\(5.19\))g(and)g (\(1.2\),)1025 4152 y FD(H)1094 4164 y Ft(0)1131 4152 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f FD(H)1405 4164 y Fq(l)1430 4152 y Fw(\()p FD(s)p Fw(\))84 b Fs(\024)1774 4096 y Fw(const)p 1774 4133 190 4 v 1856 4209 a FD(l)2002 4039 y Fp(Z)2085 4060 y Fq(s)2048 4228 y Ft(0)2437 4096 y FD(d\020)p 2144 4133 672 4 v 2144 4170 a Fp(\020)2194 4262 y Fw(\(1)18 b Fs(\000)g FD(\020)6 b Fw(\))2443 4238 y Fq(p)2500 4262 y Fw(+)18 b FD(s)2622 4222 y Fq(p)2622 4287 y(l)2661 4170 y Fp(\021)2710 4187 y Ft(1)p Fq(=p)1617 4479 y Fw(=)1774 4423 y(const)p 1774 4460 190 4 v 1856 4536 a FD(l)2002 4366 y Fp(Z)2085 4386 y Ft(0)2048 4555 y Fo(\000)p Fq(s)2452 4423 y FD(d\030)p 2159 4460 670 4 v 2159 4496 a Fp(\020)2209 4588 y Fw(\(1)g(+)g FD(\030)t Fw(\))2456 4564 y Fq(p)2513 4588 y Fw(+)g FD(s)2635 4548 y Fq(p)2635 4613 y(l)2673 4496 y Fp(\021)2723 4513 y Ft(1)p Fq(=p)2852 4479 y FD(;)456 4803 y Fw(if)28 b(0)22 b Fs(\024)h FD(s)g(<)g Fw(1)17 b Fs(\000)i FD(e)1026 4742 y Fm(\000)p 1071 4718 27 3 v Fn(c)1097 4754 y Fd(1)1129 4742 y Fn(l)p 1025 4760 125 3 v 1073 4793 a Fd(2)1191 4803 y Fw(and,)28 b(therefore,)e(b)n(y)i(Lemma)f(4.3,)1294 4998 y FD(H)1363 5010 y Ft(0)1400 4998 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f FD(H)1674 5010 y Fq(l)1700 4998 y Fw(\()p FD(s)p Fw(\))23 b Fs(\024)g(\000)1989 4941 y Fw(const)p 1989 4978 190 4 v 2070 5055 a FD(l)2216 4998 y Fw(log\(1)18 b Fs(\000)g FD(s)p Fw(\))c FD(:)456 5216 y Fw(This)27 b(indeed)h(pro)n(v)n(es)e(\(5.4\))h(in)h(the)g(case)f (0)22 b Fs(\024)h FD(s)g(<)f Fw(1)c Fs(\000)g FD(e)2272 5154 y Fm(\000)p 2317 5130 27 3 v Fn(c)2344 5166 y Fd(1)2376 5154 y Fn(l)p 2272 5172 125 3 v 2320 5205 a Fd(2)2410 5216 y Fw(.)p eop %%Page: 20 20 20 19 bop 456 251 a Ft(20)648 b(BERARDINO)23 b(SCIUNZI)g(AND)f(ENRICO)h (V)-7 b(ALDINOCI)555 450 y Fw(Let)28 b(us)g(no)n(w)e(pro)n(v)n(e)g (\(5.2\).)37 b(Using)27 b(the)h(de\014nitions)g(of)g FD(H)2380 462 y Fq(l)2405 450 y Fw(,)g FD(h)2504 462 y Fq(l)2557 450 y Fw(and)f(\(1.2\),)1121 666 y FD(H)1190 678 y Fq(l)1216 666 y Fw(\(1\))83 b Fs(\024)f Fw(const)1770 553 y Fp(Z)1853 573 y Ft(1)1816 741 y(0)2298 609 y FD(d\020)p 1914 646 856 4 v 1914 683 a Fp(\020)1963 775 y FD(h)2011 787 y Ft(0)2048 775 y Fw(\()p FD(\020)6 b Fw(\))20 b(+)e FD(h)2305 787 y Ft(0)2342 775 y Fw(\(1)g Fs(\000)g FD(s)2556 787 y Fq(l)2582 775 y Fw(\))2614 683 y Fp(\021)2663 700 y Ft(1)p Fq(=p)1405 992 y Fs(\024)82 b Fw(const)1770 879 y Fp(Z)1853 900 y Ft(1)1816 1068 y(0)2207 936 y FD(d\020)p 1914 973 672 4 v 1914 1009 a Fp(\020)1963 1101 y Fw(\(1)18 b Fs(\000)g FD(\020)6 b Fw(\))2212 1078 y Fq(p)2270 1101 y Fw(+)18 b FD(s)2392 1062 y Fq(p)2392 1127 y(l)2430 1009 y Fp(\021)2480 1027 y Ft(1)p Fq(=p)1405 1319 y Fs(\024)82 b Fw(const)1770 1206 y Fp(Z)1853 1226 y Ft(1)1816 1394 y(0)2046 1262 y FD(d\020)p 1914 1300 352 4 v 1914 1376 a Fw(1)18 b Fs(\000)g FD(\020)25 b Fw(+)18 b FD(s)2240 1388 y Fq(l)1405 1537 y Fs(\024)82 b Fw(const)28 b(log)1912 1481 y(1)p 1901 1518 65 4 v 1901 1594 a FD(s)1940 1606 y Fq(l)1989 1537 y FD(:)456 1734 y Fw(This)e(pro)n(v)n(es)e(\(5.2\),)j (pro)n(vided)p 1460 1688 36 4 v 25 w FD(c)1496 1746 y Ft(1)1559 1734 y Fw(is)f(c)n(hosen)g(to)g(b)r(e)h(suitably)f(small,)g (and)g(ends)g(the)h(pro)r(of)f(of)456 1833 y(Lemma)h(5.1.)2498 b Fc(\003)555 2014 y Fw(W)-7 b(e)31 b(no)n(w)f(in)n(tro)r(duce)g(an)g (appropriate)e(mo)r(di\014cation)j(of)f(the)h(barrier)d(in)j(Lemma)f (5.1,)g(in)456 2113 y(order)24 b(to)h(deal)h(with)g(the)g(distance)f (function.)37 b(Giv)n(en)25 b(a)g(smo)r(oth)h(h)n(yp)r(ersurface)e Fs(S)32 b Fw(w)n(e)25 b(de\014ne)456 2213 y FD(d)499 2225 y Fo(S)548 2213 y Fw(\()p FD(x)p Fw(\))e(to)e(b)r(e)i(the)f (signed)f(distance)h(from)f FD(x)h Fw(to)g Fs(S)6 b Fw(,)23 b(with)g(the)f(assumption)f(that)h FD(d)3016 2225 y Fo(S)3087 2213 y Fw(is)g(p)r(ositiv)n(e)456 2312 y Fg(ab)l(ove)k Fs(S)6 b Fw(.)36 b(F)-7 b(or)25 b(some)f(prop)r(erties)g(of)h(the)h (distance)e(function,)j(see)d([9].)36 b(With)26 b(this)f(de\014nition,) 456 2412 y(w)n(e)i(are)f(no)n(w)h(in)h(p)r(osition)g(to)f(in)n(tro)r (duce)g(another)g(barrier:)456 2540 y FE(Lemma)33 b(5.2.)42 b Fg(L)l(et)33 b Fw(0)28 b FD(<)h(")g Fs(\024)f FD(\033)33 b Fs(\024)28 b FD(\016)k(<)d Fw(1)p Fg(,)34 b FD(\030)f Fs(2)c Fr(R)2106 2510 y Fq(N)6 b Fo(\000)p Ft(1)2260 2540 y Fg(,)34 b FD(M)k Fs(2)29 b Fw(Mat\(\()p FD(N)h Fs(\000)21 b Fw(1\))f Fs(\002)h Fw(\()p FD(N)29 b Fs(\000)21 b Fw(1\)\))p Fg(.)456 2640 y(L)l(et)29 b Fw(\000)g Fg(b)l(e)h(the)g (hyp)l(ersurfac)l(e)i(de\014ne)l(d)e(as)1112 2812 y Fw(\000)23 b(:=)g Fs(f)14 b FD(x)1401 2824 y Fq(n)1469 2812 y Fw(=)1568 2756 y FD(")p 1566 2793 42 4 v 1566 2869 a Fw(2)1618 2812 y FD(x)1665 2778 y Fo(0)1707 2812 y Fs(\001)19 b FD(M)9 b(x)1886 2778 y Fo(0)1927 2812 y Fw(+)18 b FD(\033)s(\030)23 b Fs(\001)c FD(x)2208 2778 y Fo(0)2246 2812 y Fs(g)e(\\)i(f)14 b(j)p FD(x)2505 2778 y Fo(0)2528 2812 y Fs(j)24 b FD(<)2672 2756 y(\033)p 2672 2793 51 4 v 2678 2869 a(")2746 2812 y Fs(g)456 2992 y Fg(and)30 b(assume)g(that)1250 3131 y Fw(tr)14 b FD(M)31 b Fs(\025)23 b FD(\016)17 b(;)183 b Fs(k)p FD(M)9 b Fs(k)22 b(\024)2083 3075 y Fw(2)p 2083 3112 42 4 v 2084 3188 a FD(\016)2148 3131 y(;)184 b Fs(j)p FD(\030)t Fs(j)23 b(\024)2562 3075 y Fw(1)p 2562 3112 V 2563 3188 a FD(\016)2627 3131 y(:)456 3298 y Fg(Then,)35 b(ther)l(e)e(exist)f(functions)h FD(\033)1519 3310 y Ft(0)1586 3298 y Fw(:)c(\(0)p FD(;)14 b Fw(+)p Fs(1)p Fw(\))29 b Fs(\000)-14 b(!)29 b Fw(\(0)p FD(;)14 b Fw(1\))32 b Fg(and)i FD(C)2562 3310 y Ft(0)2628 3298 y Fw(:)c(\(0)p FD(;)14 b Fw(+)p Fs(1)p Fw(\))28 b Fs(\000)-14 b(!)29 b Fw([1)p FD(;)14 b Fw(+)p Fs(1)p Fw(\))456 3422 y Fg(and)25 b FD(T)661 3434 y Fq(";\016)771 3422 y Fs(2)849 3330 y Fp(h)888 3422 y Fw(0)p FD(;)14 b(C)1026 3434 y Ft(0)1063 3422 y Fw(\()p FD(\016)s Fw(\))g(log)1313 3390 y Ft(1)p 1313 3404 34 4 v 1314 3451 a Fq(")1356 3330 y Fp(i)1420 3422 y Fg(such)24 b(that,)i(if)g FD(")c Fs(\024)h FD(\033)j Fs(\024)d FD(\033)2227 3434 y Ft(0)2265 3422 y Fw(\()p FD(\016)s Fw(\))p Fg(,)j(we)f(c)l(an)g(\014nd)f(a)h(non-de)l(cr)l(e)l (asing)456 3580 y(function)k FD(g)822 3592 y Ft(\000)889 3580 y Fs(2)24 b FD(C)1033 3550 y Ft(1)p Fq(;)p Ft(1)1123 3580 y Fw(\()p Fs(\0001)p FD(;)14 b(T)1389 3592 y Fq(";\016)1476 3580 y Fw(\))p Fg(,)30 b(c)l(onstant)e(in)1994 3488 y Fp(\020)2060 3580 y Fs(\000)17 b(1)p FD(;)d Fs(\000)p FD(C)2386 3592 y Ft(0)2423 3580 y Fw(\()p FD(\016)s Fw(\))g(log)2672 3548 y Ft(1)p 2672 3562 V 2673 3609 a Fq(")2715 3488 y Fp(i)2754 3580 y Fg(,)30 b(such)f(that)g FD(g)3206 3592 y Ft(\000)3251 3580 y Fw(\(0\))23 b(=)456 3705 y(0)p Fg(,)29 b FD(g)592 3717 y Ft(\000)637 3705 y Fw(\()p FD(T)718 3717 y Fq(";\016)805 3705 y Fw(\))23 b(=)g(1)p Fg(,)29 b(and)h(for)g(which)h FD(g)1610 3717 y Ft(\000)1654 3705 y Fw(\()p FD(d)1729 3717 y Ft(\000)1775 3705 y Fw(\()p FD(x)p Fw(\)\))f Fg(is)g(a)g(strict)e(visc)l(osity)j(sup)l(ersolution)e (of)48 b Fw(\(1.5\))456 3804 y Fg(pr)l(ovide)l(d)31 b FD(d)826 3816 y Ft(\000)872 3804 y Fw(\()p FD(x)p Fw(\))24 b FD(<)f(T)1144 3816 y Fq(";\016)1231 3804 y Fg(.)555 3904 y(Mor)l(e)30 b(pr)l(e)l(cisely,)g FD(g)1168 3916 y Ft(\000)1242 3904 y Fg(is)f(c)l(onstructe)l(d)e(as)i(fol)t(lows.)41 b(L)l(et)28 b FD(c)2357 3916 y Ft(1)2417 3904 y FD(>)23 b Fw(0)28 b Fg(b)l(e)h(suitably)g(smal)t(l)g(and)g(let)456 4004 y FD(\032)c Fs(2)g FD(C)669 3973 y Ft(1)707 4004 y Fw(\()p Fr(R)p Fw(\))37 b Fg(b)l(e)31 b(a)h(non-de)l(cr)l(e)l(asing)f (function)g(so)g(that)g FD(\032)p Fw(\(0\))25 b(=)g(0)p Fg(,)31 b FD(\032)p Fw(\()p FD(s)p Fw(\))26 b(=)f Fs(\000)p Fw(1)30 b Fg(for)h FD(s)26 b Fs(\024)e(\000)p Fw(1)p FD(=)p Fw(2)456 4103 y Fg(and)30 b FD(\032)p Fw(\()p FD(s)p Fw(\))23 b(=)g(1)29 b Fg(for)i FD(s)23 b Fs(\025)f Fw(1)p FD(=)p Fw(2)p Fg(.)38 b(F)-6 b(or)30 b(any)g FD(s)23 b Fs(2)g Fw(\(0)p FD(;)14 b Fw(1\))p Fg(,)30 b(de\014ne)456 4259 y Fw(\(5.22\))638 b FD(h)1355 4271 y Ft(\000)1400 4259 y Fw(\()p FD(s)p Fw(\))24 b(:=)f(max)o Fs(f)14 b Fw(0)p FD(;)g(h)1975 4271 y Ft(0)2011 4259 y Fw(\()p FD(s)p Fw(\))19 b(+)f FD(c)2252 4271 y Ft(1)2289 4259 y FD(\016)s("\032)p Fw(\()p FD(s)p Fw(\))p Fs(g)c FD(:)456 4415 y Fg(L)l(et)29 b FD(s)638 4427 y Fq(\016)o(;")751 4415 y Fg(b)l(e)h(the)g(p)l(oint)g(ne)l(ar)g Fs(\000)p Fw(1)f Fg(with)h(for)h(which)g FD(h)2119 4427 y Ft(0)2156 4415 y Fw(\()p FD(s)2227 4427 y Fq(\016)o(;")2311 4415 y Fw(\))24 b(=)e FD(c)2490 4427 y Ft(1)2528 4415 y FD(\016)s(")p Fg(.)38 b(De\014ne)29 b(also)1439 4634 y FD(H)1508 4646 y Ft(\000)1553 4634 y Fw(\()p FD(s)p Fw(\))37 b(:=)1818 4521 y Fp(Z)1901 4541 y Fq(s)1864 4710 y Ft(0)1960 4578 y Fw(\()p FD(p)19 b Fs(\000)f Fw(1\))2210 4548 y Ft(1)p Fq(=p)2329 4578 y FD(d\020)p 1960 4615 455 4 v 1974 4692 a Fw(\()p FD(p)c(h)2110 4704 y Ft(\000)2155 4692 y Fw(\()p FD(\020)6 b Fw(\)\))2293 4668 y Ft(1)p Fq(=p)2438 4634 y FD(:)456 4836 y Fg(Then,)30 b(we)h(have:)615 4963 y Fw(\(i\))42 b Fg(Ther)l(e)31 b(exist)e(a)i(c)l(onstant)e FD(c)1616 4933 y Fq(])1670 4963 y Fs(2)23 b Fw(\(0)p FD(;)14 b Fw(1\))29 b Fg(so)h(that)456 5159 y Fw(\(5.23\))730 b FD(c)1435 5124 y Fq(])1480 5159 y Fw(\()p FD(\016)s(")p Fw(\))1634 5099 y Fd(1)p 1633 5108 31 3 v 1633 5142 a Fn(p)1701 5159 y Fs(\024)23 b Fw(1)18 b(+)g FD(s)1971 5171 y Fq(\016)o(;")2078 5159 y Fs(\024)2188 5102 y Fw(1)p 2175 5140 67 4 v 2175 5216 a FD(c)2211 5192 y Fq(])2266 5159 y Fw(\()p FD(\016)s(")p Fw(\))2420 5099 y Fd(1)p 2419 5108 31 3 v 2419 5142 a Fn(p)2478 5159 y Fw(;)p eop %%Page: 21 21 21 20 bop 743 251 a Ft(MEAN)29 b(CUR)-7 b(V)g(A)i(TURE)29 b(PR)n(OPER)-5 b(TIES)28 b(F)n(OR)h Fq(p)p Ft(-LAPLA)n(CE)h(PHASE)e (TRANSITIONS)221 b(21)592 450 y Fw(\(ii\))42 b Fg(for)31 b(any)f(for)h FD(s)1208 462 y Fq(\016)o(;")1315 450 y FD(<)22 b(s)h Fs(\024)g Fw(1)p Fg(,)456 589 y Fw(\(5.24\))1095 b FD(h)1812 601 y Ft(\000)1857 589 y Fw(\()p FD(s)p Fw(\))24 b FD(>)e Fw(0)744 729 y Fg(in)29 b(p)l(articular,)i FD(H)1319 741 y Ft(\000)1393 729 y Fg(is)e(wel)t(l)h(de\014ne)l(d)f(and)h (strictly)f(incr)l(e)l(asing)g(in)g Fw([)p FD(s)2921 741 y Fq(\016)o(;")3005 729 y FD(;)14 b Fw(1])29 b Fg(and)g(thus)744 834 y(we)35 b(may)h(de\014ne)e FD(g)1344 846 y Ft(\000)1389 834 y Fw(\()p FD(t)p Fw(\))f(:=)e FD(H)1711 798 y Fo(\000)p Ft(1)1704 858 y(\000)1800 834 y Fw(\()p FD(t)p Fw(\))p Fg(,)37 b(for)f(any)f FD(t)d Fs(2)g Fw([)p FD(H)2499 846 y Ft(\000)2544 834 y Fw(\()p FD(s)2615 846 y Fq(\016)o(;")2699 834 y Fw(\))p FD(;)14 b(H)2837 846 y Ft(\000)2883 834 y Fw(\(1\)])35 b Fg(and)g(extend)744 933 y FD(g)784 945 y Ft(\000)829 933 y Fw(\()p FD(t)p Fw(\))d Fg(to)g(b)l(e)f(c)l (onstantly)h(e)l(qual)g FD(s)1806 945 y Fq(\016)o(;")1921 933 y Fg(for)h FD(t)26 b Fs(\024)g FD(H)2272 945 y Ft(\000)2317 933 y Fw(\()p FD(s)2388 945 y Fq(\016)o(;")2472 933 y Fw(\))p Fg(.)45 b(In)31 b(p)l(articular,)j(if)e FD(g)3214 945 y Ft(\000)3259 933 y Fw(\()p FD(t)p Fw(\))27 b FD(>)744 1033 y(s)783 1045 y Fq(\016)o(;")867 1033 y Fg(,)j(then)g FD(g)1150 1003 y Fo(0)1147 1056 y Ft(\000)1192 1033 y Fw(\()p FD(t)p Fw(\))23 b FD(>)g Fw(0)p Fg(.)456 1189 y(Pr)l(o)l(of.)43 b Fw(First)35 b(observ)n(e)f(that)i(\(5.23\))f(follo) n(ws)f(from)h(\(1.2\):)53 b(indeed,)38 b(if)e FD(c)f Fw(and)h FD(C)42 b Fw(are)34 b(as)h(in)456 1289 y(\(1.2\),)1086 1349 y Fp(\020)1146 1385 y FD(c)1182 1397 y Ft(1)p 1146 1422 74 4 v 1150 1498 a FD(C)1229 1349 y Fp(\021)1278 1366 y Ft(1)p Fq(=p)1411 1441 y Fw(\()p FD(\016)s(")p Fw(\))1554 1407 y Ft(1)p Fq(=p)1697 1441 y Fs(\024)i Fw(1)18 b(+)g FD(s)1981 1453 y Fq(\016)o(;")2101 1441 y Fs(\024)2203 1349 y Fp(\020)2262 1385 y FD(c)2298 1397 y Ft(1)p 2262 1422 V 2281 1498 a FD(c)2346 1349 y Fp(\021)2395 1366 y Ft(1)p Fq(=p)2528 1441 y Fw(\()p FD(\016)s(")p Fw(\))2671 1407 y Ft(1)p Fq(=p)2791 1441 y FD(:)456 1591 y Fw(Also,)23 b(with)g(no)f(loss)f(of)i(generalit)n(y)-7 b(,)22 b(w)n(e)g(ma)n(y)f(assume)h FD(s)2205 1603 y Fq(\016)o(;")2312 1591 y FD(<)h Fs(\000)p Fw(1)8 b(+)g FD(\022)2629 1561 y Fo(\003)2666 1591 y Fw(,)23 b(in)g(order)e(to)h(use)h(\(1.3\).)456 1691 y(Note)35 b(that,)i(since)e(b)n(y)f(\(1.3\),)j FD(h)1488 1703 y Ft(0)1560 1691 y Fw(is)e(increasing)e(in)j([)p FD(s)2213 1703 y Fq(\016)o(;")2297 1691 y FD(;)14 b(\022)2375 1660 y Fo(\003)2413 1691 y Fw(\),)37 b(w)n(e)e(get)f FD(h)2828 1703 y Ft(0)2866 1691 y Fw(\()p FD(s)2937 1703 y Fq(\016)o(;")3021 1691 y Fw(\))h FD(>)g(c)3224 1703 y Ft(1)3261 1691 y FD(\016)s(")g Fw(in)456 1790 y(\()p FD(s)527 1802 y Fq(\016)o(;")611 1790 y FD(;)14 b(\022)689 1760 y Fo(\003)727 1790 y Fw(\).)36 b(Moreo)n(v)n(er,)23 b(from)h(\(1.2\),)h(if)g FD(c)1725 1802 y Ft(1)1787 1790 y Fw(is)f(small)g(enough,)h(w)n(e)f(ma)n(y)g(supp)r(ose)g FD(h)3041 1802 y Ft(0)3078 1790 y Fw(\()p FD(s)p Fw(\))g FD(>)e(c)3328 1802 y Ft(1)3366 1790 y FD(\016)s(")456 1890 y Fw(for)27 b FD(s)622 1902 y Fq(\016)o(;")729 1890 y FD(<)22 b(s)h(<)g Fw(0.)36 b(F)-7 b(rom)27 b(the)h(ab)r(o)n(v)n(e)f (discussions,)f(\(5.24\))h(follo)n(ws.)456 1989 y(Notice)19 b(that)h(the)h(constan)n(t)e(extension)g(of)h FD(g)1829 2001 y Ft(\000)1893 1989 y Fw(is)g FD(C)2034 1959 y Ft(1)p Fq(;)p Ft(1)2144 1989 y Fw(since,)h(b)n(y)f(Lemma)f(4.6,)i(if)f FD(t)j Fw(=)g FD(H)3189 2001 y Ft(\000)3234 1989 y Fw(\()p FD(s)3305 2001 y Fq(\016)o(;")3389 1989 y Fw(\),)1365 2180 y FD(g)1408 2145 y Fo(0)1405 2200 y Ft(\000)1450 2180 y Fw(\()p FD(t)p Fw(\))83 b(=)1775 2088 y Fp(\020)1906 2123 y FD(p)p 1834 2161 185 4 v 1834 2237 a(p)19 b Fs(\000)f Fw(1)2043 2180 y FD(h)2091 2192 y Ft(\000)2136 2180 y Fw(\()p FD(g)2208 2192 y Ft(\000)2253 2180 y Fw(\()p FD(t)p Fw(\)\))2379 2088 y Fp(\021)2430 2105 y Ft(1)p Fq(=p)1627 2404 y Fw(=)1775 2312 y Fp(\020)1906 2348 y FD(p)p 1834 2385 V 1834 2461 a(p)h Fs(\000)f Fw(1)2043 2404 y FD(h)2091 2416 y Ft(\000)2136 2404 y Fw(\()p FD(s)2207 2416 y Fq(\016)o(;")2291 2404 y Fw(\))2323 2312 y Fp(\021)2373 2330 y Ft(1)p Fq(=p)1627 2564 y Fw(=)83 b(0)14 b FD(:)456 2704 y Fw(T)-7 b(o)37 b(estimate)i(the)f(domain)g(on)g(whic)n(h)g FD(g)1805 2716 y Ft(\000)1888 2704 y Fw(is)h(strictly)f(increasing)e(w) n(e)i(ha)n(v)n(e)f(therefore)h(to)456 2803 y(estimate)27 b FD(H)857 2815 y Ft(\000)902 2803 y Fw(\()p FD(s)973 2815 y Fq(\016)o(;")1058 2803 y Fw(\))g(and)h FD(H)1348 2815 y Ft(\000)1393 2803 y Fw(\(1\).)37 b(Using)28 b(Lemma)f(4.2,)g (one)g(obtains)1155 2942 y FD(h)1203 2954 y Ft(\000)1248 2942 y Fw(\()p FD(s)p Fw(\))83 b Fs(\025)g FD(h)1630 2954 y Ft(0)1667 2942 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f FD(c)1908 2954 y Ft(1)1945 2942 y FD(\016)s(")1434 3067 y Fw(=)83 b FD(h)1630 3079 y Ft(0)1667 3067 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f FD(h)1920 3079 y Ft(0)1957 3067 y Fw(\()p FD(s)2028 3079 y Fq(\016)o(;")2112 3067 y Fw(\))1434 3242 y Fs(\025)83 b Fw(const)1785 3150 y Fp(\020)1835 3242 y Fw(\(1)18 b(+)g FD(s)p Fw(\))2081 3208 y Fq(p)2138 3242 y Fs(\000)g Fw(\(1)h(+)f FD(s)2436 3254 y Fq(\016)o(;")2519 3242 y Fw(\))2551 3208 y Fq(p)2590 3150 y Fp(\021)2640 3168 y Ft(1)p Fq(=p)456 3242 y Fw(\(5.25\))456 3407 y(for)24 b(an)n(y)f FD(s)g Fs(2)h Fw([)p FD(s)936 3419 y Fq(\016)o(;")1020 3407 y FD(;)14 b Fs(\000)p Fw(1)e(+)g FD(\022)1294 3377 y Fo(\003)1331 3407 y Fw(].)36 b(On)24 b(the)h(other)f(hand,)h(for)f(an)n(y)f FD(s)g Fs(2)h Fw([)p Fs(\000)p Fw(1)12 b(+)g FD(\022)2808 3377 y Fo(\003)2845 3407 y FD(;)i Fw(0],)25 b(\(1.4\))f(implies)456 3506 y(that)456 3673 y(\(5.26\))480 b FD(h)1197 3685 y Ft(\000)1242 3673 y Fw(\()p FD(s)p Fw(\))24 b Fs(\025)e FD(h)1504 3685 y Ft(0)1542 3673 y Fw(\()p Fs(\000)p Fw(1)17 b(+)h FD(\022)1822 3638 y Fo(\003)1861 3673 y Fw(\))h Fs(\000)f FD(c)2031 3685 y Ft(1)2068 3673 y FD(\016)s(")23 b Fs(\025)2267 3617 y FD(h)2315 3629 y Ft(0)2352 3617 y Fw(\()p Fs(\000)p Fw(1)18 b(+)g FD(\022)2633 3586 y Fo(\003)2672 3617 y Fw(\))p 2267 3654 437 4 v 2465 3730 a(2)2728 3673 y FD(:)456 3840 y Fw(Therefore,)29 b(using)h(the)g(de\014nition)g(of)g FD(H)1760 3852 y Ft(\000)1805 3840 y Fw(,)h(\(5.23\),)f(\(5.25\))f(and) h(\(5.26\))f(and)h(making)f(use)h(of)456 3939 y(Lemma)d(4.3,)g(w)n(e)g (get)772 4133 y Fs(\000)p FD(H)906 4145 y Ft(\000)951 4133 y Fw(\()p FD(s)1022 4145 y Fq(\016)o(;")1106 4133 y Fw(\))83 b(=)1369 4020 y Fp(Z)1452 4041 y Ft(0)1415 4209 y Fq(s)1446 4218 y Fn(\016)n(;")1551 4077 y Fw(\()p FD(p)19 b Fs(\000)f Fw(1\))1801 4047 y Ft(1)p Fq(=p)1920 4077 y FD(d\020)p 1551 4114 455 4 v 1565 4192 a Fw(\()p FD(p)c(h)1701 4204 y Ft(\000)1746 4192 y Fw(\()p FD(\020)6 b Fw(\)\))1884 4168 y Ft(1)p Fq(=p)1221 4396 y Fw(=)1369 4283 y Fp(Z)1452 4304 y Ft(0)1415 4472 y Fo(\000)p Ft(1+)p Fq(\022)1585 4455 y Fm(\003)1647 4340 y Fw(\()p FD(p)18 b Fs(\000)g Fw(1\))1896 4310 y Ft(1)p Fq(=p)2016 4340 y FD(d\020)p 1647 4377 V 1661 4455 a Fw(\()p FD(p)c(h)1797 4467 y Ft(\000)1842 4455 y Fw(\()p FD(\020)6 b Fw(\)\))1980 4431 y Ft(1)p Fq(=p)2130 4396 y Fw(+)2213 4283 y Fp(Z)2296 4304 y Fo(\000)p Ft(1+)p Fq(\022)2466 4279 y Fm(\003)2259 4472 y Fq(s)2290 4481 y Fn(\016)n(;")2528 4340 y Fw(\()p FD(p)18 b Fs(\000)g Fw(1\))2777 4310 y Ft(1)p Fq(=p)2896 4340 y FD(d\020)p 2528 4377 V 2542 4455 a Fw(\()p FD(p)c(h)2678 4467 y Ft(\000)2723 4455 y Fw(\()p FD(\020)6 b Fw(\)\))2861 4431 y Ft(1)p Fq(=p)1221 4667 y Fs(\024)83 b Fw(const)1586 4525 y Fp( )1652 4667 y Fw(1)18 b(+)1795 4554 y Fp(Z)1878 4575 y Fo(\000)p Ft(1+)p Fq(\022)2048 4550 y Fm(\003)1841 4743 y Fq(s)1872 4752 y Fn(\016)n(;")2531 4611 y FD(d\020)p 2110 4648 929 4 v 2110 4743 a Fw(\(\(1)g(+)g FD(\020)6 b Fw(\))2391 4719 y Fq(p)2449 4743 y Fs(\000)18 b Fw(\(1)g(+)g FD(s)2746 4755 y Fq(\016)o(;")2830 4743 y Fw(\))2862 4719 y Fq(p)2901 4743 y Fw(\))2933 4702 y Ft(1)p Fq(=p)3048 4525 y Fp(!)1221 4914 y Fs(\024)83 b FD(C)1428 4926 y Ft(0)1465 4914 y Fw(\()p FD(\016)s Fw(\))14 b(log)1715 4858 y(1)p 1715 4895 42 4 v 1716 4971 a FD(")1780 4914 y(;)456 5076 y Fw(or,)26 b(equiv)-5 b(alen)n(tly)e(,)456 5216 y(\(5.27\))855 b FD(H)1593 5228 y Ft(\000)1638 5216 y Fw(\()p FD(s)1709 5228 y Fq(\016)o(;")1793 5216 y Fw(\))38 b Fs(\025)e FD(C)2023 5228 y Ft(0)2061 5216 y Fw(\()p FD(\016)s Fw(\))14 b(log)g FD(")g(:)p eop %%Page: 22 22 22 21 bop 456 251 a Ft(22)648 b(BERARDINO)23 b(SCIUNZI)g(AND)f(ENRICO)h (V)-7 b(ALDINOCI)456 450 y Fw(This)27 b(completes)g(the)h(desired)f (estimate)h(on)f FD(H)1970 462 y Ft(\000)2016 450 y Fw(\()p FD(s)2087 462 y Fq(\016)o(;")2171 450 y Fw(\).)555 550 y(Let)h(us)g(no)n(w)e(estimate)i FD(H)1385 562 y Ft(\000)1430 550 y Fw(\(1\):)37 b(from)28 b(the)g(de\014nition)g(of)f FD(h)2447 562 y Ft(\000)2520 550 y Fw(and)g(\(1.2\),)1021 771 y FD(H)1090 783 y Ft(\000)1135 771 y Fw(\(1\))c(=)1352 658 y Fp(Z)1435 679 y Ft(1)1398 847 y(0)1496 715 y Fw(\()p FD(p)18 b Fs(\000)g Fw(1\))1745 685 y Ft(1)p Fq(=p)1864 715 y FD(d\020)p 1496 752 455 4 v 1510 829 a Fw(\()p FD(p)c(h)1646 841 y Ft(\000)1691 829 y Fw(\()p FD(\020)6 b Fw(\)\))1829 805 y Ft(1)p Fq(=p)1983 771 y Fs(\024)1264 1019 y(\024)1352 906 y Fp(Z)1435 927 y Ft(1)p Fq(=)p Ft(2)1398 1095 y(0)1563 963 y Fw(const)13 b FD(d\020)p 1563 1000 290 4 v 1582 1076 a Fw(\(1)19 b Fs(\000)f FD(\020)6 b Fw(\))1880 1019 y(+)1963 906 y Fp(Z)2046 927 y Ft(1)2010 1095 y(1)p Fq(=)p Ft(2)2359 963 y Fw(const)13 b FD(d\020)p 2138 1000 732 4 v 2138 1042 a Fp(\020)2187 1134 y FD(c)p Fw(\(1)18 b Fs(\000)g FD(\020)6 b Fw(\))2472 1110 y Fq(p)2530 1134 y Fw(+)18 b FD(c)2649 1146 y Ft(1)2686 1134 y FD(\016)s(")2765 1042 y Fp(\021)2826 1037 y Fd(1)p 2824 1046 31 3 v 2824 1079 a Fn(p)1264 1358 y Fs(\024)1352 1245 y Fp(Z)1435 1265 y Ft(1)p Fq(=)p Ft(2)1398 1433 y(0)1563 1301 y Fw(const)13 b FD(d\020)p 1563 1338 290 4 v 1582 1415 a Fw(\(1)19 b Fs(\000)f FD(\020)6 b Fw(\))1880 1358 y(+)1963 1245 y Fp(Z)2046 1265 y Ft(1)2010 1433 y(1)p Fq(=)p Ft(2)2236 1301 y Fw(const)13 b FD(d\020)p 2138 1338 486 4 v 2138 1435 a Fw(1)18 b Fs(\000)g FD(\020)25 b Fw(+)18 b(\()p FD(\016)s(")p Fw(\))2579 1375 y Fd(1)p 2578 1384 31 3 v 2578 1418 a Fn(p)1264 1551 y Fs(\024)23 b Fw(const)13 b(\(1)18 b Fs(\000)g Fw(log)q(\()p FD(\016)s(")p Fw(\)\))37 b Fs(\024)g(\000)p FD(C)2276 1563 y Ft(0)2313 1551 y Fw(\()p FD(\016)s Fw(\))14 b(log)g FD(")g(:)456 1134 y Fw(\(5.28\))456 1718 y(The)27 b(claims)g(on)h(the)g(domain)f(of)g FD(g)1568 1730 y Ft(\000)1641 1718 y Fw(are)g(th)n(us)g(a)g (consequence)g(of)h(\(5.27\))e(and)i(\(5.28\).)555 1942 y(No)n(w)g(w)n(e)h(deal)f(with)h(the)g(pro)r(of)f(of)h(the)g(viscosit)n (y)f(sup)r(ersolution)f(prop)r(ert)n(y)h(of)h FD(g)3162 1954 y Ft(\000)3206 1942 y Fw(.)41 b(First)456 2041 y(of)27 b(all,)h(notice)f(that,)h(in)g(an)f(appropriate)f(system)h(of)h(co)r (ordinates)e(w)n(e)h(ha)n(v)n(e)823 2249 y FD(D)894 2214 y Ft(2)931 2249 y FD(d)974 2261 y Ft(\000)1043 2249 y Fw(=)c(diag)1298 2131 y Fp(\022)1452 2192 y Fs(\000)p FD(k)1560 2204 y Ft(1)p 1369 2229 312 4 v 1369 2305 a Fw(1)18 b Fs(\000)g FD(d)1555 2317 y Ft(\000)1600 2305 y FD(k)1643 2317 y Ft(1)1691 2249 y FD(;)c(:)g(:)g(:)f(;)1969 2192 y Fs(\000)p FD(k)2077 2204 y Fq(N)6 b Fo(\000)p Ft(1)p 1885 2229 423 4 v 1885 2305 a Fw(1)18 b Fs(\000)g FD(d)2071 2317 y Ft(\000)2117 2305 y FD(k)2160 2317 y Fq(N)6 b Fo(\000)p Ft(1)2318 2249 y FD(;)14 b Fw(0)2397 2131 y Fp(\023)2480 2249 y Fs(2)24 b Fw(Mat)13 b(\()p FD(N)28 b Fs(\002)18 b FD(N)9 b Fw(\))14 b FD(;)456 2456 y Fw(where)28 b(the)i FD(k)885 2468 y Fq(i)912 2456 y Fw('s)f(are)f(the)i(principal)e(curv)-5 b(atures)28 b(of)h(\000)g(at)g (the)h(p)r(oin)n(t)f(where)f(the)i(distance)f(is)456 2555 y(realized)j(\(see)h Fs(x)p Fw(14.6)e(in)i([9])g(for)g(further)g (details\).)53 b(W)-7 b(e)33 b(also)f(de\014ne)h FD(P)45 b Fw(as)33 b(the)g(parab)r(oloid)456 2655 y(describing)26 b(\000,)i(i.e.,)1427 2799 y FD(P)12 b Fw(\()p FD(x)1571 2764 y Fo(0)1595 2799 y Fw(\))23 b(:=)1772 2742 y FD(")p 1771 2780 42 4 v 1771 2856 a Fw(2)1822 2799 y FD(x)1869 2764 y Fo(0)1912 2799 y Fs(\001)18 b FD(M)9 b(x)2090 2764 y Fo(0)2132 2799 y Fw(+)18 b FD(\033)s(\030)23 b Fs(\001)18 b FD(x)2412 2764 y Fo(0)2450 2799 y FD(:)456 2971 y Fw(Notice)k(that,)i(b)n(y)e(construction,)h Fs(jr)p FD(P)12 b Fs(j)23 b(\024)f Fw(1)g(;)j(th)n(us,)e(b)n(y)f(the)h(mean)f (curv)-5 b(ature)22 b(equation)g(\(see,)456 3071 y(for)27 b(instance,)g(equation)g(\(14.103\))f(of)h([9]\),)h(it)g(follo)n(ws)f (that)1138 3210 y Fq(N)6 b Fo(\000)p Ft(1)1150 3235 y Fp(X)1156 3412 y Fq(i)p Ft(=1)1296 3314 y FD(k)1339 3326 y Fq(i)1450 3314 y Fw(=)1690 3235 y Fp(X)1597 3413 y Ft(1)p Fo(\024)p Fq(i)p Fo(\024)p Fq(N)g Fo(\000)p Ft(1)1915 3314 y FD(@)1959 3326 y Fq(i)2001 3172 y Fp( )2230 3258 y FD(@)2274 3270 y Fq(i)2302 3258 y FD(P)p 2077 3295 444 4 v 2077 3311 a Fp(p)p 2160 3311 361 4 v 71 x Fw(1)18 b(+)g Fs(jr)p FD(P)12 b Fs(j)2483 3358 y Ft(2)2530 3172 y Fp(!)1450 3589 y Fw(=)1762 3533 y(\001)p FD(P)p 1607 3570 444 4 v 1607 3587 a Fp(p)p 1690 3587 361 4 v 71 x Fw(1)18 b(+)g Fs(jr)p FD(P)12 b Fs(j)2013 3634 y Ft(2)2079 3589 y Fs(\000)2172 3533 y Fw(\()p FD(D)2275 3503 y Ft(2)2313 3533 y FD(P)25 b Fs(r)p FD(P)12 b Fw(\))19 b Fs(\001)g(r)p FD(P)p 2172 3570 580 4 v 2197 3648 a Fw(\(1)g(+)f Fs(jr)p FD(P)12 b Fs(j)2553 3624 y Ft(2)2590 3648 y Fw(\))2622 3624 y Ft(3)p Fq(=)p Ft(2)1450 3788 y Fs(\025)82 b Fw(\001)p FD(P)31 b Fs(\000)18 b Fw(const)13 b Fs(jr)p FD(P)f Fs(j)2216 3754 y Ft(2)2254 3788 y Fs(k)p FD(D)2367 3754 y Ft(2)2404 3788 y FD(P)g Fs(k)i FD(:)456 3950 y Fw(Consequen)n(tly)-7 b(,)26 b(if)j FD(x)f Fw(is)f(so)g(that)h Fs(j)p FD(d)1569 3962 y Ft(\000)1614 3950 y Fw(\()p FD(x)p Fw(\))p Fs(j)d(\024)d FD(C)1919 3962 y Ft(0)1957 3950 y Fw(\()p FD(\016)s Fw(\))14 b(log)2206 3917 y Ft(1)p 2206 3931 34 4 v 2207 3978 a Fq(")2249 3950 y Fw(,)28 b(since)g Fs(j)p FD(k)2570 3962 y Fq(i)2597 3950 y Fs(j)c(\024)e FD(C)2790 3962 y Ft(1)2828 3950 y Fw(\()p FD(\016)s Fw(\))p FD(")p Fw(,)28 b(w)n(e)f(ha)n(v)n(e) 1036 4201 y(\001)p FD(d)1148 4213 y Ft(\000)1216 4201 y Fs(\024)1304 4097 y Fq(N)6 b Fo(\000)p Ft(1)1316 4122 y Fp(X)1322 4299 y Fq(i)p Ft(=1)1555 4145 y Fs(\000)p FD(k)1663 4157 y Fq(i)p 1472 4182 303 4 v 1472 4258 a Fw(1)18 b Fs(\000)g FD(d)1658 4270 y Ft(\000)1703 4258 y FD(k)1746 4270 y Fq(i)1807 4201 y Fs(\024)23 b(\000)1974 4097 y Fq(N)6 b Fo(\000)p Ft(1)1985 4122 y Fp(X)1991 4299 y Fq(i)p Ft(=1)2131 4201 y FD(k)2174 4213 y Fq(i)2220 4201 y Fw(+)18 b FD(C)2362 4213 y Ft(1)2400 4201 y Fw(\()p FD(\016)s Fw(\))p FD(")2543 4167 y Ft(2)2594 4201 y Fw(log)2725 4145 y(1)p 2725 4182 42 4 v 2726 4258 a FD(")2800 4201 y Fs(\024)1216 4424 y(\024)23 b(\000)p Fw(\001)p FD(P)30 b Fw(+)18 b(const)c Fs(jr)p FD(P)e Fs(j)1988 4389 y Ft(2)2025 4424 y Fs(k)p FD(D)2138 4389 y Ft(2)2175 4424 y FD(P)g Fs(k)18 b Fw(+)g FD(C)2442 4436 y Ft(1)2479 4424 y Fw(\()p FD(\016)s Fw(\))p FD(")2632 4367 y Fd(3)p 2632 4376 29 3 v 2632 4409 a(2)2698 4424 y Fs(\024)1216 4574 y(\024)23 b(\000)p FD("\016)e Fw(+)d FD(C)1608 4586 y Ft(2)1646 4574 y Fw(\()p FD(\016)s Fw(\)\()p FD("\033)1871 4540 y Ft(2)1927 4574 y Fw(+)g FD(")2059 4518 y Fd(3)p 2059 4527 V 2059 4560 a(2)2101 4574 y Fw(\))24 b Fs(\024)e(\000)p FD("\016)f Fw(+)d FD(C)2548 4586 y Ft(3)2586 4574 y Fw(\()p FD(\016)s Fw(\))p FD("\033)2789 4518 y Fd(1)p 2789 4527 V 2789 4560 a(2)456 4350 y Fw(\(5.29\))456 4774 y(Therefore,)27 b(if)i FD(d)976 4786 y Ft(\000)1021 4774 y Fw(\()p FD(x)p Fw(\))d Fs(2)1238 4682 y Fp(\020)1287 4774 y FD(H)1356 4786 y Ft(\000)1402 4774 y Fw(\()p FD(s)1473 4786 y Fq(\016)o(;")1557 4774 y Fw(\))p FD(;)14 b(H)1695 4786 y Ft(\000)1740 4774 y Fw(\(1\))1846 4682 y Fp(\021)1924 4774 y Fw(\(and)29 b(th)n(us,)g(b)n(y)f(\(5.27\))g(and)g(\(5.28\),)g Fs(j)p FD(d)3175 4786 y Ft(\000)3220 4774 y Fw(\()p FD(x)p Fw(\))p Fs(j)e(\024)456 4899 y FD(C)515 4911 y Ft(0)552 4899 y Fw(\()p FD(\016)s Fw(\))14 b(log)q(\(1)p FD(=")p Fw(\))27 b(and)g FD(g)1196 4869 y Fo(0)1193 4922 y Ft(\000)1238 4899 y Fw(\()p FD(d)1313 4911 y Ft(\000)1359 4899 y Fw(\()p FD(x)p Fw(\)\))d FD(>)f Fw(0\),)k(w)n(e)h(ha)n(v)n(e)e(b)n(y)h(Lemma)h (4.5)e(that)983 5064 y(\001)1052 5076 y Fq(p)1090 5064 y Fw(\()p FD(g)1162 5076 y Ft(\000)1207 5064 y Fw(\()p FD(t)p Fw(\)\))e(=)f(\()p FD(p)18 b Fs(\000)g Fw(1\))p FD(g)1737 5030 y Fo(0)1734 5085 y Ft(\000)1779 5064 y Fw(\()p FD(t)p Fw(\))1873 5030 y Fq(p)p Fo(\000)p Ft(2)1997 5064 y FD(g)2040 5030 y Fo(00)2037 5085 y Ft(\000)2082 5064 y Fw(\()p FD(t)p Fw(\))h(+)f FD(g)2321 5030 y Fo(0)2318 5085 y Ft(\000)2363 5064 y Fw(\()p FD(t)p Fw(\))2457 5030 y Fq(p)p Fo(\000)p Ft(1)2581 5064 y Fw(\001)p FD(d)2693 5076 y Ft(\000)2738 5064 y Fw(\()p FD(t)p Fw(\))24 b Fs(\024)1357 5215 y(\024)f Fw(\()p FD(p)18 b Fs(\000)g Fw(1\))p FD(g)1737 5180 y Fo(0)1734 5235 y Ft(\000)1779 5215 y Fw(\()p FD(t)p Fw(\))1873 5180 y Fq(p)p Fo(\000)p Ft(2)1997 5215 y FD(g)2040 5180 y Fo(00)2037 5235 y Ft(\000)2082 5215 y Fw(\()p FD(t)p Fw(\))h Fs(\000)f FD(")p Fw(\()p FD(\016)j Fs(\000)d FD(C)2549 5227 y Ft(4)2587 5215 y Fw(\()p FD(\016)s Fw(\))p FD(\033)2751 5158 y Fd(1)p 2752 5167 V 2752 5200 a(2)2794 5215 y Fw(\))p FD(g)2869 5180 y Fo(0)2866 5235 y Ft(\000)2912 5215 y Fw(\()p FD(t)p Fw(\))3006 5180 y Fq(p)p Fo(\000)p Ft(1)456 5142 y Fw(\(5.30\))p eop %%Page: 23 23 23 22 bop 743 251 a Ft(MEAN)29 b(CUR)-7 b(V)g(A)i(TURE)29 b(PR)n(OPER)-5 b(TIES)28 b(F)n(OR)h Fq(p)p Ft(-LAPLA)n(CE)h(PHASE)e (TRANSITIONS)221 b(23)456 450 y Fw(where)32 b(w)n(e)h(are)g(using)g (the)h(notation)f FD(t)f Fw(=)h FD(d)1887 462 y Ft(\000)1932 450 y Fw(\()p FD(x)p Fw(\).)55 b(T)-7 b(aking)33 b(in)n(to)g(accoun)n (t)g(Lemma)g(4.6,)h(b)n(y)456 550 y(\(5.30\))26 b(w)n(e)i(get)913 774 y(\001)982 786 y Fq(p)1021 774 y Fw(\()p FD(g)1093 786 y Ft(\000)1138 774 y Fw(\()p FD(t)p Fw(\)\))c Fs(\024)f FD(h)1424 740 y Fo(0)1424 795 y Ft(\000)1469 774 y Fw(\()p FD(s)p Fw(\))18 b Fs(\000)h FD(")1727 682 y Fp(\020)1776 774 y FD(\016)i Fs(\000)d FD(C)1976 786 y Ft(4)2014 774 y Fw(\()p FD(\016)s Fw(\))p FD(\033)2178 718 y Fd(1)p 2179 727 29 3 v 2179 760 a(2)2221 682 y Fp(\021)2285 657 y(\022)2427 718 y FD(p)p 2356 755 185 4 v 2356 831 a(p)g Fs(\000)g Fw(1)2550 774 y FD(h)2598 786 y Ft(\000)2643 774 y Fw(\()p FD(s)p Fw(\))2746 657 y Fp(\023)2818 648 y Fn(p)p Fm(\000)p Fd(1)p 2818 662 104 3 v 2855 695 a Fn(p)2964 774 y FD(:)456 965 y Fw(where)27 b(w)n(e)g(are)g(using)g(the) h(notation)f FD(s)c Fw(=)f FD(g)1838 977 y Ft(\000)1883 965 y Fw(\()p FD(d)1958 977 y Ft(\000)2004 965 y Fw(\()p FD(x)p Fw(\)\).)456 1073 y(No)n(w)36 b(w)n(e)h(c)n(hose)f FD(\033)1062 1085 y Ft(0)1099 1073 y Fw(\()p FD(\016)s Fw(\))i(small)f(suc)n(h)g(that)g FD(\016)28 b Fs(\000)c FD(C)2067 1085 y Ft(4)2105 1073 y Fw(\()p FD(\016)s Fw(\))p FD(\033)2269 1021 y Fd(1)p 2270 1030 29 3 v 2270 1063 a(2)2351 1073 y Fs(\025)2464 1041 y Fq(\016)p 2464 1055 34 4 v 2464 1102 a Ft(2)2544 1073 y Fw(for)37 b FD(\033)42 b Fs(\024)c FD(\033)2920 1085 y Ft(0)2958 1073 y Fw(\()p FD(\016)s Fw(\).)66 b(Th)n(us,)39 b(if)456 1173 y Fs(j)p FD(d)522 1185 y Ft(\000)567 1173 y Fw(\()p FD(x)p Fw(\))p Fs(j)45 b(\024)d FD(C)912 1185 y Ft(0)950 1173 y Fw(\()p FD(\016)s Fw(\))14 b(log)q(\(1)p FD(=")p Fw(\))39 b(\(and)h(so)f FD(s)k Fw(=)g FD(g)1952 1185 y Ft(\000)1997 1173 y Fw(\()p FD(d)2072 1185 y Ft(\000)2118 1173 y Fw(\()p FD(x)p Fw(\)\))h FD(>)f(s)2452 1185 y Fq(\016)o(;")2536 1173 y Fw(\),)g(w)n(e)d(gather)e (\(recall)i(also)456 1273 y(\(5.24\)\))27 b(that)764 1439 y(\001)833 1451 y Fq(p)871 1439 y Fw(\()p FD(g)943 1451 y Ft(\000)989 1439 y Fw(\()p FD(t)p Fw(\)\))83 b Fs(\024)g FD(h)1394 1405 y Fo(0)1394 1460 y Ft(\000)1439 1439 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f Fw(const)13 b FD(\016)s(")h Fw(\()p FD(h)2020 1451 y Ft(\000)2065 1439 y Fw(\()p FD(s)p Fw(\)\))2211 1371 y Fn(p)p Fm(\000)p Fd(1)p 2211 1385 104 3 v 2248 1418 a Fn(p)1198 1593 y Fs(\024)83 b FD(h)1394 1558 y Fo(0)1394 1613 y Ft(0)1431 1593 y Fw(\()p FD(s)p Fw(\))19 b(+)f FD(c)1672 1605 y Ft(1)1709 1593 y FD(\016)s("\032)1831 1558 y Fo(0)1854 1593 y Fw(\()p FD(s)p Fw(\))h Fs(\000)f Fw(const)c FD(\016)s(")g Fw(\()o FD(h)2435 1605 y Ft(0)2473 1593 y Fw(\()p FD(s)p Fw(\))k(+)g FD(c)2713 1605 y Ft(1)2751 1593 y FD(\016)s("\032)p Fw(\()p FD(s)p Fw(\)\))3018 1524 y Fn(p)p Fm(\000)p Fd(1)p 3018 1538 V 3055 1571 a Fn(p)456 1593 y Fw(\(5.31\))456 1734 y(W)-7 b(e)28 b(no)n(w)f(claim)g(that)456 1879 y(\(5.32\))515 b FD(c)1220 1891 y Ft(1)1257 1879 y FD(\032)1300 1845 y Fo(0)1324 1879 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f Fw(const)27 b(\()p FD(h)1826 1891 y Ft(0)1863 1879 y Fw(\()p FD(s)p Fw(\))19 b(+)f FD(c)2104 1891 y Ft(1)2141 1879 y FD(\016)s("\032)p Fw(\()p FD(s)p Fw(\)\))2409 1811 y Fn(p)p Fm(\000)p Fd(1)p 2409 1825 V 2445 1858 a Fn(p)2550 1879 y FD(<)k Fw(0)14 b FD(;)456 2020 y Fw(for)29 b(an)n(y)g FD(s)d Fs(2)h Fw(\()p FD(s)962 2032 y Fq(\016)o(;")1046 2020 y FD(;)14 b Fw(1\),)31 b(pro)n(vided)d FD(c)1590 2032 y Ft(1)1657 2020 y Fw(is)i(small)f(enough.)43 b(Indeed,)30 b(if)h FD(s)26 b Fs(\024)g(\000)p Fw(1)p FD(=)p Fw(2)i(or)h FD(s)e Fs(\025)f Fw(1)p FD(=)p Fw(2,)456 2120 y FD(\032)499 2090 y Fo(0)522 2120 y Fw(\()p FD(s)p Fw(\))g(=)f(0)k(and)g(therefore)g (the)g(left)h(hand)f(side)g(of)g(\(5.32\))g(is)g(under)g(con)n(trol.)40 b(On)29 b(the)h(other)456 2220 y(hand,)k(if)f FD(s)f Fs(2)g Fw(\()p Fs(\000)p Fw(1)p FD(=)p Fw(2)p FD(;)14 b Fw(1)p FD(=)p Fw(2\),)32 b(b)n(y)g(denoting)h FD(c)1908 2190 y Fo(\003)1978 2220 y Fw(:=)f(inf)2199 2235 y Fq(s)p Fo(2)p Ft([)p Fo(\000)p Ft(1)p Fq(=)p Ft(2)p Fq(;)p Ft(1)p Fq(=)p Ft(2])2602 2220 y FD(h)2650 2232 y Ft(0)2687 2220 y Fw(\()p FD(s)p Fw(\))i(\(whic)n(h)f(is)g(strictly)456 2320 y(p)r(ositiv)n(e)27 b(on)g(accoun)n(t)g(of)g(\(1.2\)\),)h(w)n(e)f (b)r(ound)h(the)g(left)h(hand)e(side)h(of)f(\(5.32\))g(b)n(y)1486 2486 y FD(c)1522 2498 y Ft(1)1559 2486 y Fs(k)p FD(\032)1644 2452 y Fo(0)1666 2486 y Fs(k)1708 2498 y Fo(1)1797 2486 y Fs(\000)18 b Fw(const)27 b(\()p FD(c)2165 2452 y Fo(\003)2203 2486 y Fw(\))2246 2418 y Fn(p)p Fm(\000)p Fd(1)p 2246 2432 V 2282 2465 a Fn(p)2391 2486 y FD(;)456 2627 y Fw(whic)n(h)g(is)h (negativ)n(e)e(for)h(suitably)h(small)f FD(c)1801 2639 y Ft(1)1838 2627 y Fw(.)37 b(This)28 b(pro)n(v)n(es)d(\(5.32\).)456 2752 y(Therefore,)h(b)n(y)h(virtue)h(of)f(\(5.31\))g(and)h(\(5.32\),)e (if)j FD(d)2091 2764 y Ft(\000)2136 2752 y Fw(\()p FD(x)p Fw(\))24 b Fs(2)2349 2660 y Fp(\020)2399 2752 y FD(H)2468 2764 y Ft(\000)2513 2752 y Fw(\()p FD(s)2584 2764 y Fq(\016)o(;")2668 2752 y Fw(\))p FD(;)14 b(H)2806 2764 y Ft(\000)2852 2752 y Fw(\(1\))2958 2660 y Fp(\021)3007 2752 y Fw(,)28 b(w)n(e)f(get)1536 2922 y(\001)1605 2934 y Fq(p)1644 2922 y Fw(\()p FD(g)1716 2934 y Ft(\000)1761 2922 y Fw(\()p FD(t)p Fw(\)\))d FD(<)e(h)2046 2888 y Fo(0)2046 2943 y Ft(0)2083 2922 y Fw(\()p FD(g)2155 2934 y Ft(\000)2200 2922 y Fw(\()p FD(t)p Fw(\)\))14 b FD(:)456 3063 y Fw(If)28 b(else)f FD(d)739 3075 y Ft(\000)784 3063 y Fw(\()p FD(x)p Fw(\))d FD(<)f(H)1076 3075 y Ft(\000)1121 3063 y Fw(\()p FD(s)1192 3075 y Fq(\016)o(;")1276 3063 y Fw(\),)28 b(w)n(e)g(ha)n(v)n(e)1268 3204 y(\001)1337 3216 y Fq(p)1376 3204 y Fw(\()p FD(g)1448 3216 y Ft(\000)1493 3204 y Fw(\()p FD(t)p Fw(\)\))c(=)e(0)h FD(<)g(h)1931 3170 y Fo(0)1931 3224 y Ft(0)1968 3204 y Fw(\()p FD(s)2039 3216 y Fq(\016)o(;")2123 3204 y Fw(\))g(=)g FD(h)2314 3170 y Fo(0)2314 3224 y Ft(0)2351 3204 y Fw(\()p FD(g)2423 3216 y Ft(\000)2468 3204 y Fw(\()p FD(t)p Fw(\)\))14 b FD(:)456 3345 y Fw(Finally)-7 b(,)21 b(in)f(the)h(case)e FD(d)1184 3357 y Ft(\000)1229 3345 y Fw(\()p FD(x)p Fw(\))24 b(=)f FD(H)1521 3357 y Ft(\000)1566 3345 y Fw(\()p FD(s)1637 3357 y Fq(\016)o(;")1721 3345 y Fw(\),)f(since)e FD(g)2034 3357 y Ft(\000)2099 3345 y Fw(is)g FD(C)2240 3315 y Ft(1)p Fq(;)p Ft(1)2330 3345 y Fw(,)i(w)n(e)d(get)h(the)g(thesis)g(b)n(y)g (exploiting)456 3444 y(Lemma)27 b(4.7)g(as)g(w)n(e)g(did)h(in)g(Lemma)f (5.1)g(ab)r(o)n(v)n(e.)1368 b Fc(\003)1556 3632 y Fw(6.)41 b Fv(Sliding)31 b(methods)555 3782 y Fw(W)-7 b(e)28 b(no)n(w)f(use)g (the)g(barriers)f(in)n(tro)r(duced)h(in)g Fs(x)p Fw(5)g(and)g(an)g (appropriate)f(sliding)h(tec)n(hniques)456 3881 y(to)36 b(deduce)i(an)e(estimate)h(on)g(the)g(curv)-5 b(ature)36 b(of)h(touc)n(hing)g(parab)r(oloids)e(for)h(solutions)h(of)456 3981 y(\(1.5\):)456 4101 y FE(Lemma)32 b(6.1.)42 b Fg(L)l(et)32 b FD(l)r(;)14 b(\022)r(;)g(\016)31 b(>)c Fw(0)32 b Fg(and)h FD(M)1756 4113 y Ft(1)1821 4101 y Fs(2)28 b Fw(Mat\(\()p FD(N)h Fs(\000)20 b Fw(1\))g Fs(\002)g Fw(\()p FD(N)30 b Fs(\000)20 b Fw(1\)\))p Fg(.)47 b(L)l(et)31 b FD(u)h Fg(b)l(e)h(a)g(we)l(ak)456 4201 y(Sob)l(olev)d(solution)f(of)h(\(1.5\)) g(in)f(the)g(whole)i Fr(R)1886 4171 y Fq(N)1955 4201 y Fg(,)e(satisfying)i(\(2.1\).)39 b(Assume)28 b(that)h Fs(j)p FD(u)p Fs(j)23 b FD(<)g Fw(1)28 b Fg(in)456 4301 y Fw([)p Fs(\000)p FD(l)r(;)14 b(l)r Fw(])658 4270 y Fq(N)719 4301 y Fg(,)30 b FD(u)p Fw(\(0\))23 b(=)f(0)30 b Fg(and)g FD(u)p Fw(\()p FD(x)p Fw(\))24 b FD(<)e Fw(0)29 b Fg(for)i(any)f FD(x)24 b Fw(=)e(\()p FD(x)2141 4270 y Fo(0)2165 4301 y FD(;)14 b(x)2249 4313 y Fq(n)2295 4301 y Fw(\))23 b Fs(2)h Fw([)p Fs(\000)p FD(l)r(;)14 b(l)r Fw(])2631 4270 y Fq(N)2722 4301 y Fg(so)30 b(that)1441 4485 y FD(x)1488 4497 y Fq(n)1557 4485 y FD(<)1686 4429 y(\022)p 1654 4466 106 4 v 1654 4542 a Fw(2)p FD(l)1723 4518 y Ft(2)1769 4485 y FD(x)1816 4450 y Fo(0)1859 4485 y Fs(\001)18 b FD(M)1981 4497 y Ft(1)2018 4485 y FD(x)2065 4450 y Fo(0)2107 4485 y Fw(+)2200 4429 y FD(\022)p 2200 4466 42 4 v 2207 4542 a(l)2251 4485 y(\030)23 b Fs(\001)18 b FD(x)2398 4450 y Fo(0)2436 4485 y FD(:)456 4654 y Fg(Then,)35 b(ther)l(e)f(exist)f(a)h(universal)g(c)l(onstant)e FD(\016)1916 4666 y Ft(0)1983 4654 y FD(>)e Fw(0)i Fg(and)i(a)g(function)g FD(\033)f Fw(:)d(\(0)p FD(;)14 b Fw(1\))29 b Fs(\000)-14 b(!)29 b Fw(\(0)p FD(;)14 b Fw(1\))p Fg(,)456 4753 y(so)30 b(that,)g(if)653 4933 y FD(\016)c Fs(2)d Fw(\(0)p FD(;)14 b(\016)942 4945 y Ft(0)979 4933 y Fw(])g FD(;)99 b(\016)26 b Fs(\024)c FD(\022)k Fs(\024)c Fw(2)p FD(\016)17 b(;)1667 4877 y(\022)p 1667 4914 V 1674 4990 a(l)1742 4933 y Fs(2)1820 4841 y Fp(\020)1870 4933 y Fw(0)p FD(;)d(\033)s Fw(\()p FD(\016)s Fw(\))2103 4841 y Fp(i)2156 4933 y FD(;)99 b Fs(k)p FD(M)2401 4945 y Ft(1)2437 4933 y Fs(k)23 b(\024)2599 4877 y Fw(1)p 2599 4914 V 2600 4990 a FD(\016)2736 4933 y Fg(and)85 b Fs(j)p FD(\030)t Fs(j)23 b(\024)3159 4877 y Fw(1)p 3159 4914 V 3160 4990 a FD(\016)3224 4933 y(;)456 5097 y Fg(then)1765 5203 y Fw(tr)p FD(M)1911 5215 y Ft(1)1971 5203 y Fs(\024)f FD(\016)17 b(:)p eop %%Page: 24 24 24 23 bop 456 251 a Ft(24)648 b(BERARDINO)23 b(SCIUNZI)g(AND)f(ENRICO)h (V)-7 b(ALDINOCI)456 450 y Fg(Pr)l(o)l(of.)43 b Fw(Let)27 b FD(g)904 462 y Fq(l)957 450 y Fw(and)h(\011)1184 420 y Fq(y)r(;l)1292 450 y Fw(b)r(e)g(the)g(functions)g(de\014ned)g(in)g (Lemma)f(5.1.)36 b(De\014ne)28 b(also)680 648 y(\000)732 660 y Ft(1)806 648 y Fw(:=)931 531 y Fp(\032)993 648 y FD(x)23 b Fw(=)g(\()p FD(x)1230 614 y Fo(0)1254 648 y FD(;)14 b(x)1338 660 y Fq(N)1401 648 y Fw(\))24 b Fs(2)f Fr(R)1589 614 y Fq(N)6 b Fo(\000)p Ft(1)1762 648 y Fs(\002)18 b Fr(R)52 b Fw(s.t.)46 b FD(x)2155 660 y Fq(N)2241 648 y Fw(=)2371 592 y FD(\022)p 2339 629 106 4 v 2339 705 a Fw(2)p FD(l)2408 681 y Ft(2)2454 648 y FD(x)2501 614 y Fo(0)2543 648 y Fs(\001)19 b FD(M)2666 660 y Ft(1)2703 648 y FD(x)2750 614 y Fo(0)2792 648 y Fw(+)2885 592 y FD(\022)p 2885 629 42 4 v 2892 705 a(l)2936 648 y(\030)k Fs(\001)18 b FD(x)3083 614 y Fo(0)3107 531 y Fp(\033)3197 648 y FD(:)456 858 y Fw(Notice)31 b(that,)i(b)n(y)f(construction,)g FD(u)f Fw(is)h(negativ)n(e)e(b)r(elo)n(w)i(\000)2342 870 y Ft(1)2410 858 y Fw(in)g([)p Fs(\000)p FD(l)r(;)14 b(l)r Fw(])2713 828 y Fq(N)2775 858 y Fw(.)49 b(F)-7 b(urthermore,)32 b(b)n(y)456 958 y(our)26 b(assumptions,)1551 1062 y FD(\022)r Fs(k)p FD(M)1715 1074 y Ft(1)1751 1062 y Fs(k)p 1551 1099 243 4 v 1640 1175 a FD(l)1667 1151 y Ft(2)1826 1119 y Fs(\024)1954 1062 y FD(\022)p 1923 1099 104 4 v 1923 1175 a(l)1950 1151 y Ft(2)1987 1175 y FD(\016)2060 1119 y Fs(\024)2158 1062 y FD(\033)s Fw(\()p FD(\016)s Fw(\))p 2158 1099 156 4 v 2202 1175 a FD(l)r(\016)2336 1119 y(:)456 1282 y Fw(Fixed)32 b FD(\024)740 1294 y Ft(1)808 1282 y Fs(2)g Fw(\(0)p FD(;)14 b Fw(1)p FD(=)p Fw(2\),)32 b(to)g(b)r(e)h(b)r(etter)g(sp)r(eci\014ed)g(in)f(the)h (sequel,)g(the)g(ab)r(o)n(v)n(e)e(b)r(ound)i(on)f(the)456 1381 y(curv)-5 b(ature)28 b(of)h(\000)977 1393 y Ft(1)1043 1381 y Fw(implies)g(that,)h(if)f FD(\033)s Fw(\()p FD(\016)s Fw(\))q FD(=\016)j Fw(is)d(su\016cien)n(tly)g(small,)g(then,)h(giv)n (en)e(an)n(y)g FD(x)i Fw(suc)n(h)456 1481 y(that)k(there)h(is)f FD(X)41 b Fs(2)34 b Fw(\000)1203 1493 y Ft(1)1275 1481 y Fw(with)h Fs(j)p FD(X)29 b Fs(\000)23 b FD(x)p Fs(j)35 b Fw(=)f FD(d)1927 1493 y Ft(\000)1968 1501 y Fd(1)2005 1481 y Fw(\()p FD(x)p Fw(\))h Fs(\024)f FD(\024)2298 1493 y Ft(1)2335 1481 y FD(l)r Fw(,)i(there)e(is)h(a)f(ball)g(of)g (radius)g FD(\024)3381 1493 y Ft(1)3418 1481 y FD(l)456 1580 y Fw(whic)n(h)29 b(touc)n(hes)f(\000)1046 1592 y Ft(1)1113 1580 y Fw(b)n(y)g(b)r(elo)n(w)h(at)g FD(X)36 b Fw(and)29 b(whose)f(cen)n(ter)h FD(X)2408 1550 y Fo(0)2459 1580 y Fw(lies)g(in)h(the)f(direction)g FD(X)d Fs(\000)19 b FD(x)p Fw(.)456 1680 y(Therefore,)26 b(since)h(w)n(e)h(c)n(hose)e (the)i(signed)f(distance)h FD(d)2164 1692 y Ft(\000)2205 1700 y Fd(1)2269 1680 y Fw(to)g(b)r(e)g(p)r(ositiv)n(e)f(ab)r(o)n(v)n (e)f(\000)3078 1692 y Ft(1)3115 1680 y Fw(,)456 1833 y(\(6.1\))866 b Fs(j)p FD(x)19 b Fs(\000)f FD(X)1741 1798 y Fo(0)1763 1833 y Fs(j)h(\000)f FD(d)1931 1845 y Ft(\000)1972 1853 y Fd(1)2009 1833 y Fw(\()p FD(x)p Fw(\))37 b(=)g FD(\024)2307 1845 y Ft(1)2344 1833 y FD(l)15 b(:)555 1985 y Fw(W)-7 b(e)28 b(claim)g(that,)g(for)f(suitable)g(univ)n (ersal)f(p)r(ositiv)n(e)i(constan)n(ts)e FD(\024)2633 1997 y Ft(1)2698 1985 y Fw(and)h FD(\024)2907 1997 y Ft(2)2944 1985 y Fw(,)h(w)n(e)f(ha)n(v)n(e)456 2138 y(\(6.2\))971 b FD(u)p Fw(\()p FD(x)p Fw(\))24 b Fs(\024)e FD(g)1908 2150 y Fq(\024)1947 2158 y Fd(1)1979 2150 y Fq(l)2005 2138 y Fw(\()p FD(d)2080 2150 y Ft(\000)2121 2158 y Fd(1)2158 2138 y Fw(\()p FD(x)p Fw(\)\))456 2291 y(for)27 b(an)n(y)f FD(x)e Fs(2)f Fw([)p Fs(\000)p FD(\024)1024 2303 y Ft(2)1061 2291 y FD(l)r(;)14 b(\024)1173 2303 y Ft(2)1210 2291 y FD(l)r Fw(])1260 2260 y Fq(N)1322 2291 y Fw(.)456 2390 y(T)-7 b(o)43 b(pro)n(v)n(e)f(\(6.2\),)47 b(\014rst)c(recall)g(that)g (\011)1760 2360 y Fq(y)r(;\024)1855 2368 y Fd(1)1887 2360 y Fq(l)1956 2390 y Fw(is)g(de\014ned)h(in)g FD(B)2533 2402 y Fq(\024)2572 2410 y Fd(1)2604 2402 y Fq(l)p Ft(+)p Fq(T)2715 2411 y Fn(\024)2749 2423 y Fd(1)2782 2411 y Fn(l)2810 2390 y Fw(\()p FD(y)s Fw(\),)k(with)c FD(T)3243 2402 y Fq(\024)3282 2410 y Fd(1)3314 2402 y Fq(l)3389 2390 y Fs(2)456 2503 y Fw([)r(\026)-44 b FD(cl)r(;)14 b(\024)627 2515 y Ft(1)663 2503 y FD(l)r(=)p Fw(2],)21 b(and)h(that)f(\011)1235 2473 y Fq(y)r(;\024)1330 2481 y Fd(1)1362 2473 y Fq(l)1410 2503 y Fs(\025)i(\000)p Fw(1)6 b(+)g FD(e)1721 2473 y Fo(\000)r Ft(~)-35 b Fq(c)n(l)1848 2503 y Fw(in)22 b(its)g(domain)f(of)g(de\014nition)h(\(see)g(the)f (statemen)n(t)456 2602 y(of)27 b(Lemma)h(5.1\).)36 b(Therefore,)26 b(since)862 2755 y FD(B)925 2770 y Ft(\()p Fq(\024)990 2778 y Fd(1)1022 2770 y Ft(+)r(\026)-35 b Fq(c)p Ft(\))p Fq(l)1154 2755 y Fw(\(0)p FD(;)14 b(:)g(:)g(:)g(;)g Fw(0)p FD(;)g Fs(\000)p FD(l)r(=)p Fw(2\))80 b Fs(\022)j FD(B)1991 2767 y Fq(\024)2030 2775 y Fd(1)2062 2767 y Fq(l)p Ft(+)p Fq(T)2173 2776 y Fn(\024)2207 2788 y Fd(1)2240 2776 y Fn(l)2269 2755 y Fw(\(0)p FD(;)14 b(:)g(:)g(:)f(;)h Fw(0)p FD(;)g Fs(\000)p FD(l)r(=)p Fw(2\))1780 2896 y Fs(\022)83 b FD(B)1991 2908 y Ft(2)p Fq(\024)2063 2916 y Fd(1)2095 2908 y Fq(l)2121 2896 y Fw(\(0)p FD(;)14 b(:)g(:)g(:)f(;)h Fw(0)p FD(;)g Fs(\000)p FD(l)r(=)p Fw(2\))21 b Fs(\032)h Fw([)p Fs(\000)p FD(l)r(;)14 b(l)r Fw(])2976 2862 y Fq(N)456 3048 y Fw(if)28 b FD(\024)580 3060 y Ft(1)644 3048 y Fw(is)g(con)n(v)n(enien)n(tly)e(small,)i(w)n(e)f(deduce)h(from)f (\(2.1\))g(that)841 3208 y FD(u)p Fw(\()p FD(x)p Fw(\))38 b FD(<)e Fw(\011)1204 3174 y Ft(\(0)p Fq(;:::)o(;)p Ft(0)p Fq(;)p Fo(\000)p Fq(l=)p Ft(2\))p Fq(;\024)1640 3182 y Fd(1)1672 3174 y Fq(l)1697 3208 y Fw(\()p FD(x)p Fw(\))14 b FD(;)98 b Fs(8)p FD(x)23 b Fs(2)g FD(B)2201 3220 y Fq(\024)2240 3228 y Fd(1)2272 3220 y Fq(l)p Ft(+)p Fq(T)2383 3229 y Fn(\024)2417 3241 y Fd(1)2450 3229 y Fn(l)2479 3208 y Fw(\(0)p FD(;)14 b(:)g(:)g(:)f(;)h Fw(0)p FD(;)g Fs(\000)p FD(l)r(=)p Fw(2\))g FD(:)456 3361 y Fw(Giv)n(en)28 b FD(X)j Fs(2)25 b Fw(\000)929 3373 y Ft(1)994 3361 y Fw(let)k FD(\027)1156 3373 y Fq(X)1248 3361 y Fw(b)r(e)g(the)f(normal)g (direction)g(of)g(\000)2282 3373 y Ft(1)2348 3361 y Fw(at)g FD(X)35 b Fw(p)r(oin)n(ting)28 b(do)n(wn)n(w)n(ards)f(and)456 3461 y(de\014ne)1619 3578 y FD(X)1695 3544 y Fo(0)1741 3578 y Fw(:=)c FD(X)h Fw(+)18 b FD(\024)2076 3590 y Ft(1)2113 3578 y FD(l)r(\027)2181 3590 y Fq(X)2258 3578 y Fw(;)456 3713 y(w)n(e)27 b(no)n(w)g(slide)g(the)h(surface)f(\011)1428 3683 y Ft(\(0)p Fq(;:::)o(;)p Ft(0)p Fq(;)p Fo(\000)p Fq(l=)p Ft(2\))p Fq(;\024)1864 3691 y Fd(1)1895 3683 y Fq(l)1949 3713 y Fw(in)g(the)h(direction)g(of)f(the)h(v)n(ector)1472 3866 y FD(v)e Fw(:=)d FD(X)1725 3832 y Fo(0)1766 3866 y Fs(\000)18 b Fw(\(0)p FD(;)c(:)g(:)g(:)f(;)h Fw(0)p FD(;)g Fs(\000)p FD(l)r(=)p Fw(2\))d FD(;)456 4022 y Fw(that)27 b(is,)h(w)n(e)f(will)h(consider)f(the)h(surface)e(\011)1832 3992 y Fq(t)1884 4022 y Fw(:=)d(\011)2060 3992 y Ft(\(0)p Fq(;:::)n(;)p Ft(0)p Fq(;)p Fo(\000)p Fq(l=)p Ft(2\)+)p Fq(tv)r(;\024)2606 4000 y Fd(1)2639 3992 y Fq(l)2692 4022 y Fw(for)k FD(t)c(>)f Fw(0.)456 4121 y(W)-7 b(e)28 b(will)f(sho)n(w)g(that)456 4278 y(\(6.3\))46 b(\011)738 4248 y Fq(t)767 4278 y Fw(\()p Ff(x)p Fw(\))24 b FD(>)f(u)p Fw(\()p Ff(x)p Fw(\))28 b(for)f(an)n(y)f FD(t)d Fs(2)h Fw([0)p FD(;)14 b Fw(1\))27 b(and)g(an)n(y)g Ff(x)c Fs(2)h FD(B)2280 4290 y Fq(\024)2319 4298 y Fd(1)2351 4290 y Fq(l)p Ft(+)p Fq(T)2462 4299 y Fn(\024)2496 4311 y Fd(1)2529 4299 y Fn(l)2558 4186 y Fp(\020)2607 4278 y Fw(\(0)p FD(;)14 b(:)g(:)g(:)g(;)g Fw(0)p FD(;)g Fs(\000)p FD(l)r(=)p Fw(2\))i(+)i FD(tv)3325 4186 y Fp(\021)3374 4278 y Fw(.)456 4459 y(Indeed,)37 b(let)e FD(t)g Fs(2)h Fw([0)p FD(;)14 b Fw(1\))34 b(b)r(e)i(the)f(\014rst)g(time)g(on)g(whic)n(h)g(\011)2330 4429 y Fq(t)2393 4459 y Fw(touc)n(hes)g FD(u)p Fw(.)58 b(First)35 b(of)g(all,)i(note)456 4584 y(that,)g(since)d FD(t)i(<)f Fw(1,)h(w)n(e)e(ha)n(v)n(e)g FD(u)h(<)g Fw(0)f(on)h FD(@)5 b(B)1967 4596 y Fq(\024)2006 4604 y Fd(1)2038 4596 y Fq(l)2063 4492 y Fp(\020)2113 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y(no)n(w)k(going)f(to)h(sho)n(w.)p eop %%Page: 25 25 25 24 bop 743 251 a Ft(MEAN)29 b(CUR)-7 b(V)g(A)i(TURE)29 b(PR)n(OPER)-5 b(TIES)28 b(F)n(OR)h Fq(p)p Ft(-LAPLA)n(CE)h(PHASE)e (TRANSITIONS)221 b(25)555 463 y Fw(Assume,)23 b(indeed,)h(b)n(y)d(con)n (tradiction)f(that)i FD(u)h Fs(\024)g Fw(\011)2169 432 y Fq(t)2219 463 y Fw(in)f FD(B)2373 475 y Fq(\024)2412 483 y Fd(1)2445 475 y Fq(l)p Ft(+)p Fq(T)2556 484 y Fn(\024)2590 496 y Fd(1)2623 484 y Fn(l)2651 370 y Fp(\020)2701 463 y Fw(\(0)p FD(;)14 b(:)g(:)g(:)f(;)h Fw(0)p FD(;)g Fs(\000)p FD(l)r(=)p Fw(2\))7 b(+)g FD(tv)3398 370 y Fp(\021)456 620 y Fw(and)29 b FD(u)p Fw(\()p FD(z)t Fw(\))d(=)h(\011)957 590 y Fq(t)986 620 y Fw(\()p FD(z)t Fw(\))i(for)h(some)f FD(z)h Fs(2)d FD(@)5 b(B)1725 632 y Fq(\024)1764 640 y Fd(1)1796 632 y Fq(l)p Ft(+)p Fq(T)1907 641 y Fn(\024)1941 653 y Fd(1)1974 641 y Fn(l)2002 528 y Fp(\020)2052 620 y Fw(\(0)p FD(;)14 b(:)g(:)g(:)f(;)h Fw(0)p FD(;)g Fs(\000)p FD(l)r(=)p Fw(2\))k(+)h FD(tv)2772 528 y Fp(\021)2822 620 y Fw(.)44 b(Let)30 b(us)g(consider)456 740 y(the)e(\\radial)e (direction")1302 1033 y FD(w)40 b Fw(:=)1563 947 y FD(z)22 b Fs(\000)1707 855 y Fp(\020)1756 947 y Fw(\(0)p FD(;)14 b(:)g(:)g(:)g(;)g Fw(0)p FD(;)g Fs(\000)p FD(l)r(=)p Fw(2\))i(+)i FD(tv)2474 855 y Fp(\021)p 1535 1014 1016 4 v 1535 1024 a(\014)1535 1074 y(\014)1535 1124 y(\014)1563 1120 y FD(z)k Fs(\000)1707 1028 y Fp(\020)1756 1120 y Fw(\(0)p FD(;)14 b(:)g(:)g(:)g(;)g Fw(0)p FD(;)g Fs(\000)p FD(l)r(=)p Fw(2\))i(+)i FD(tv)2474 1028 y Fp(\021)2523 1024 y(\014)2523 1074 y(\014)2523 1124 y(\014)2575 1033 y FD(:)456 1351 y Fw(Then,)31 b(b)n(y)g(the)g(construction)e(in)i (Lemma)g(5.1,)f(\011)2067 1321 y Fq(t)2096 1351 y Fw(\()p FD(z)t Fw(\))e(=)g(1)i(and)h Fs(r)p Fw(\011)2695 1321 y Fq(t)2724 1351 y Fw(\()p FD(z)t Fw(\))20 b Fs(\001)g FD(w)31 b(>)d Fw(0.)46 b(On)30 b(the)456 1451 y(other)j(hand,)j FD(u)e Fs(\024)g Fw(1)g(and,)i(since)e FD(u)p Fw(\()p FD(z)t Fw(\))f(=)h(1,)i(w)n(e)e(ha)n(v)n(e)f(that)h Fs(r)p FD(u)p Fw(\()p 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col0 sh gr /Symbol ff 195.00 scf sf 11520 4455 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 225.00 scf sf 8820 6795 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 330.00 scf sf 3375 5400 m gs 1 -1 sc (T) col0 sh gr /Symbol ff 225.00 scf sf 3555 5535 m gs 1 -1 sc (k) col0 sh gr /Times-Roman ff 150.00 scf sf 3690 5625 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 225.00 scf sf 3780 5535 m gs 1 -1 sc (l) col0 sh gr % here ends figure; $F2psEnd rs showpage %%EndDocument @endspecial 1440 2217 a Fh(T)-7 b(ouc)n(hing)26 b(b)r(et)n(w)n(een)i FC(\011)2163 2194 y Fz(1)2224 2217 y Fh(and)f FC(\000)2425 2226 y Fz(1)555 2469 y Fw(Indeed,)e(w)n(e)f(deduce)g(from)g(\(6.3\))g (that)g(\011)1866 2439 y Ft(1)1903 2469 y Fw(\()p Ff(x)p Fw(\))g Fs(\025)f FD(u)p Fw(\()p Ff(x)p Fw(\))h(for)g(an)n(y)f Ff(x)h Fs(2)f FD(B)2753 2484 y Ft(\()p Fq(\024)2818 2492 y Fd(1)2850 2484 y Ft(+)r(\026)-35 b Fq(c)p Ft(\))p Fq(l)2982 2469 y Fw(\()p FD(X)3090 2439 y Fo(0)3113 2469 y Fw(\).)36 b(There-)456 2569 y(fore,)27 b(using)g(\(6.1\),)665 2746 y FD(g)705 2758 y Fq(\024)744 2766 y Fd(1)776 2758 y Fq(l)802 2654 y Fp(\020)851 2746 y FD(d)894 2758 y Ft(\000)935 2766 y Fd(1)972 2746 y Fw(\()p FD(x)p Fw(\))1083 2654 y Fp(\021)1157 2746 y Fw(=)22 b FD(g)1284 2758 y Fq(\024)1323 2766 y Fd(1)1355 2758 y Fq(l)1381 2746 y Fw(\()p Fs(j)p FD(x)d Fs(\000)f FD(X)1661 2712 y Fo(0)1684 2746 y Fs(j)g(\000)g FD(\024)1856 2758 y Ft(1)1893 2746 y FD(l)r Fw(\))23 b(=)g(\011)2128 2712 y Fq(X)2187 2687 y Fm(0)2209 2712 y Fq(;\024)2268 2720 y Fd(1)2300 2712 y Fq(l)2325 2746 y Fw(\()p FD(x)p Fw(\))h(=)f(\011)2613 2712 y Ft(1)2650 2746 y Fw(\()p FD(x)p Fw(\))h Fs(\025)e FD(u)p Fw(\()p FD(x)p Fw(\))14 b FD(:)-2612 b Fw(\(6.5\))456 2919 y(By)26 b(our)g(assumptions)f(and)i(an)f(easy)g(co)n(v)n(ering)e(argumen)n(t)i (o)n(v)n(er)e(\000)2582 2931 y Ft(1)2619 2919 y Fw(,)j(the)g(reader)e (ma)n(y)h(easily)456 3018 y(con)n(vince)g(herself)i(that)1189 3170 y([)p Fs(\000)p FD(\024)1325 3182 y Ft(2)1362 3170 y FD(l)r(;)14 b(\024)1474 3182 y Ft(2)1510 3170 y FD(l)r Fw(])1560 3136 y Fq(N)1646 3170 y Fs(\022)1776 3091 y Fp([)1734 3270 y Fq(X)5 b Fo(2)p Ft(\000)1879 3278 y Fd(1)1925 3170 y FD(B)1988 3185 y Ft(\()p Fq(\024)2053 3193 y Fd(1)2085 3185 y Ft(+)r(\026)-35 b Fq(c)o Ft(\))p Fq(l)2217 3170 y Fw(\()p FD(X)25 b Fw(+)18 b FD(\024)2474 3182 y Ft(1)2511 3170 y FD(l)r(\027)2579 3182 y Fq(X)2641 3170 y Fw(\))c FD(;)456 3409 y Fw(pro)n(vided)21 b FD(\024)840 3421 y Ft(2)900 3409 y Fw(is)h(suitably)h(small)f(with)h(resp)r(ect)f (to)g FD(\024)2109 3421 y Ft(1)2169 3409 y Fw(and)i(\026)-44 b FD(c)p Fw(.)36 b(Therefore,)22 b(\(6.2\))g(follo)n(ws)g(from)456 3509 y(\(6.5\).)555 3713 y(W)-7 b(e)33 b(no)n(w)f(complete)g(the)h(pro) r(of)f(of)g(the)h(desired)f(result)g(arguing)e(b)n(y)j(con)n (tradiction)e(and)456 3813 y(supp)r(osing)18 b(that)h(tr)p FD(M)1150 3825 y Ft(1)1210 3813 y FD(>)k(\016)s Fw(:)32 b(with)20 b(this)f(assumption,)i(b)n(y)d(Lemma)h(5.2,)h(w)n(e)f(get)f (that)i FD(g)3182 3825 y Ft(\000)3223 3833 y Fd(2)3259 3813 y Fw(\()p FD(d)3334 3825 y Ft(\000)3375 3833 y Fd(2)3412 3813 y Fw(\))456 3913 y(is)27 b(a)g(strict)h(sup)r(ersolution)f(of)g (\(1.5\),)h(where)1146 4090 y(\000)1198 4102 y Ft(2)1318 4090 y Fw(:=)1488 3998 y Fp(n)1544 4090 y FD(x)23 b Fw(=)g(\()p FD(x)1781 4055 y Fo(0)1805 4090 y FD(;)14 b(x)1889 4102 y Fq(n)1935 4090 y Fw(\))23 b Fs(2)g Fr(R)2122 4055 y Fq(N)6 b Fo(\000)p Ft(1)2295 4090 y Fs(\002)18 b Fr(R)52 b Fw(s.t.)1488 4291 y FD(x)1535 4303 y Fq(n)1604 4291 y Fw(=)1734 4235 y FD(\022)p 1702 4272 106 4 v 1702 4348 a Fw(2)p FD(l)1771 4324 y Ft(2)1817 4291 y FD(x)1864 4256 y Fo(0)1906 4291 y Fs(\001)19 b FD(M)2029 4303 y Ft(1)2065 4291 y FD(x)2112 4256 y Fo(0)2155 4291 y Fw(+)2248 4235 y FD(\022)p 2248 4272 42 4 v 2255 4348 a(l)2299 4291 y(\030)j Fs(\001)d FD(x)2446 4256 y Fo(0)2488 4291 y Fs(\000)2581 4235 y FD("\016)p 2581 4272 79 4 v 2600 4348 a Fw(4)2670 4291 y Fs(j)p FD(x)2740 4256 y Fo(0)2764 4291 y Fs(j)2787 4256 y Ft(2)2824 4199 y Fp(o)1021 4495 y Fw(and)41 b FD(")83 b Fw(:=)1498 4439 y(2)p FD(\022)p 1498 4476 83 4 v 1508 4552 a(l)1535 4528 y Ft(2)1605 4495 y FD(:)456 4675 y Fw(Moreo)n(v)n(er,)25 b(if)j FD(\022)r(=l)g Fw(and)g FD(\016)j Fw(are)26 b(su\016cien)n(tly)i(small,)f(w)n(e)g (claim)h(that)835 4830 y(if)g FD(s)950 4842 y Fq(\016)o(;")1057 4830 y FD(<)23 b(s)g Fs(\024)f(\000)p Fw(1)c(+)g(\()p FD(\016)s(\022)r(l)1642 4800 y Fo(\000)p Ft(2)1731 4830 y Fw(\))1763 4800 y Ft(1)p Fq(=p)1869 4830 y Fw(,)166 b FD(h)2106 4842 y Ft(\000)2147 4850 y Fd(2)2183 4830 y Fw(\()p FD(s)p Fw(\))84 b Fs(\024)e FD(h)2565 4842 y Fq(\024)2604 4850 y Fd(1)2636 4842 y Fq(l)2662 4830 y Fw(\()p FD(s)p Fw(\))166 b(and)-2609 b(\(6.6\))981 4961 y(if)28 b(1)18 b Fs(\000)g Fw(\()p FD(\016)s(\022)r(l)1340 4931 y Fo(\000)p Ft(2)1429 4961 y Fw(\))1461 4931 y Ft(1)p Fq(=p)1590 4961 y Fs(\024)k FD(s)h Fs(\024)g Fw(1,)166 b FD(h)2106 4973 y Ft(\000)2147 4981 y Fd(2)2183 4961 y Fw(\()p FD(s)p Fw(\))84 b Fs(\025)e FD(h)2565 4973 y Fq(\024)2604 4981 y Fd(1)2636 4973 y Fq(l)2662 4961 y Fw(\()p FD(s)p Fw(\))14 b FD(:)-2346 b Fw(\(6.7\))456 5113 y(Indeed,)26 b(if)g FD(h)869 5125 y Ft(\000)910 5133 y Fd(2)946 5113 y Fw(\()p FD(s)p Fw(\))e(=)e(0,)k(\(6.6\))f(follo) n(ws)f(from)h(item)h(\(i\))g(in)g(Lemma)f(5.1.)35 b(If,)27 b(on)e(the)g(con)n(trary)-7 b(,)456 5216 y FD(h)504 5228 y Ft(\000)545 5236 y Fd(2)581 5216 y Fw(\()p FD(s)p Fw(\))34 b FD(>)f Fw(0)h(and)f FD(s)h Fs(2)g Fw([)p FD(s)1283 5228 y Fq(\016)o(;")1367 5216 y FD(;)14 b Fs(\000)p Fw(1)21 b(+)h(\()p FD(\016)s(")p Fw(\))1762 5185 y Ft(1)p Fq(=p)1868 5216 y Fw(],)36 b(then,)g(b)n(y)e(the)g(de\014nitions)g(of)g FD(h)2998 5228 y Ft(\000)3039 5236 y Fd(2)3109 5216 y Fw(and)g FD(h)3325 5228 y Fq(\024)3364 5236 y Fd(1)3396 5228 y Fq(l)3421 5216 y Fw(,)p eop %%Page: 28 28 28 27 bop 456 251 a Ft(28)648 b(BERARDINO)23 b(SCIUNZI)g(AND)f(ENRICO)h (V)-7 b(ALDINOCI)456 450 y Fw(\(1.2\))27 b(and)g(\(5.23\),)547 629 y FD(h)595 641 y Ft(\000)636 649 y Fd(2)672 629 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f FD(h)925 641 y Fq(\024)964 649 y Fd(1)996 641 y Fq(l)1022 629 y Fw(\()p FD(s)p Fw(\))83 b Fs(\024)g(\000)p Fw(const)13 b FD(\016)s(")18 b Fw(+)1814 573 y(const)p 1814 610 190 4 v 1896 686 a FD(l)2027 629 y Fw(\(1)h(+)f FD(s)p Fw(\))2274 595 y Fq(p)2331 629 y Fs(\000)2424 573 y Fw(const)p 2424 610 V 2505 686 a FD(l)2623 629 y(s)2662 590 y Fq(p)2662 654 y(\024)2701 662 y Fd(1)2733 654 y Fq(l)2777 629 y Fw(+)g FD(h)2908 641 y Ft(0)2945 629 y Fw(\()p Fs(\000)p Fw(1)g(+)g FD(s)3224 641 y Fq(\024)3263 649 y Fd(1)3295 641 y Fq(l)3321 629 y Fw(\))1208 833 y Fs(\024)83 b(\000)p Fw(const)1634 777 y FD(\016)s(\022)p 1634 814 82 4 v 1643 890 a(l)1670 866 y Ft(2)1744 833 y Fw(+)1837 777 y(const)p 1837 814 190 4 v 1918 890 a FD(l)2050 833 y Fw(\(1)18 b(+)g FD(s)p Fw(\))2296 799 y Fq(p)2353 833 y Fw(+)g(const)c FD(s)2679 793 y Fq(p)2679 858 y(\024)2718 866 y Fd(1)2750 858 y Fq(l)1208 1037 y Fs(\024)83 b(\000)p Fw(const)1634 981 y FD(\016)s(\022)p 1634 1018 82 4 v 1643 1094 a(l)1670 1070 y Ft(2)1744 1037 y Fw(+)18 b(const)2040 981 y FD(\016)s(")p 2040 1018 79 4 v 2066 1094 a(l)2147 1037 y Fw(+)g(const)c FD(s)2473 997 y Fq(p)2473 1062 y(\024)2512 1070 y Fd(1)2544 1062 y Fq(l)1208 1241 y Fw(=)83 b Fs(\000)p Fw(const)1634 1185 y FD(\016)s(\022)p 1634 1222 82 4 v 1643 1298 a(l)1670 1274 y Ft(2)1744 1241 y Fw(+)18 b(const)2040 1185 y FD(\016)s(\022)p 2040 1222 V 2049 1298 a(l)2076 1274 y Ft(3)2150 1241 y Fw(+)g(const)13 b FD(e)2475 1207 y Fo(\000)p Ft(const)d Fq(l)2729 1241 y FD(;)456 1412 y Fw(whic)n(h)29 b(is)g(negativ)n(e)g (for)f(con)n(v)n(enien)n(tly)h(large)f FD(l)r Fw(,)h(completing)g(the)h (pro)r(of)f(of)g(\(6.6\).)42 b(T)-7 b(o)29 b(pro)n(v)n(e)456 1512 y(\(6.7\),)34 b(use)f(\(1.2\),)i(\(5.23\))d(and)h(the)h (de\014nitions)g(of)f FD(h)2179 1524 y Ft(\000)2220 1532 y Fd(2)2290 1512 y Fw(and)g FD(h)2505 1524 y Fq(\024)2544 1532 y Fd(1)2576 1524 y Fq(l)2635 1512 y Fw(to)g(deduce)g(that,)i(if)f (1)22 b Fs(\000)456 1614 y Fw(\()p FD(\016)s(\022)r(l)596 1584 y Fo(\000)p Ft(2)685 1614 y Fw(\))717 1584 y Ft(1)p Fq(=p)846 1614 y Fs(\024)g FD(s)h Fs(\024)g Fw(1,)673 1757 y FD(h)721 1769 y Ft(\000)762 1777 y Fd(2)799 1757 y Fw(\()p FD(s)p Fw(\))c Fs(\000)f FD(h)1052 1769 y Fq(\024)1091 1777 y Fd(1)1123 1769 y Fq(l)1148 1757 y Fw(\()p FD(s)p Fw(\))84 b Fs(\025)f FD(h)1531 1769 y Ft(0)1568 1757 y Fw(\()p FD(s)p Fw(\))19 b(+)f(const)13 b FD(\016)s(")18 b Fs(\000)g FD(h)2204 1769 y Fq(\024)2243 1777 y Fd(1)2275 1769 y Fq(l)2301 1757 y Fw(\()p FD(s)p Fw(\))1335 1918 y Fs(\025)83 b Fw(const)13 b FD(\016)s(")18 b Fs(\000)g FD(h)1914 1930 y Ft(0)1951 1918 y Fw(\(1)g Fs(\000)g FD(s)2165 1930 y Fq(\024)2204 1938 y Fd(1)2237 1930 y Fq(l)2262 1918 y Fw(\))h Fs(\000)2406 1862 y Fw(const)p 2406 1899 190 4 v 2487 1975 a FD(l)2605 1826 y Fp(\020)2655 1918 y Fw(\(1)f Fs(\000)g FD(s)p Fw(\))2901 1884 y Fq(p)2958 1918 y Fw(+)g FD(s)3080 1878 y Fq(p)3080 1943 y(\024)3119 1951 y Fd(1)3152 1943 y Fq(l)3177 1826 y Fp(\021)1335 2116 y Fs(\025)83 b Fw(const)13 b FD(\016)s(")18 b Fs(\000)g Fw(const)c FD(s)2109 2076 y Fq(p)2109 2141 y(\024)2148 2149 y Fd(1)2180 2141 y Fq(l)2224 2116 y Fs(\000)2317 2059 y Fw(const)p 2317 2096 V 2398 2172 a FD(l)2516 2023 y Fp(\020)2566 2116 y Fw(\(1)k Fs(\000)g FD(s)p Fw(\))2812 2081 y Fq(p)2869 2116 y Fw(+)g FD(s)2991 2076 y Fq(p)2991 2141 y(\024)3030 2149 y Fd(1)3062 2141 y Fq(l)3088 2023 y Fp(\021)1335 2320 y Fs(\025)83 b Fw(const)13 b FD(\016)s(")18 b Fs(\000)g Fw(const)2080 2263 y FD(\016)s(\022)p 2080 2300 82 4 v 2089 2376 a(l)2116 2352 y Ft(3)2189 2320 y Fs(\000)g Fw(const)c FD(e)2515 2285 y Fo(\000)p Ft(const)c Fq(l)1335 2523 y Fw(=)83 b(const)1696 2467 y FD(\016)s(\022)p 1696 2504 V 1705 2580 a(l)1732 2556 y Ft(2)1806 2523 y Fs(\000)18 b Fw(const)2102 2467 y FD(\016)s(\022)p 2102 2504 V 2111 2580 a(l)2138 2556 y Ft(3)2212 2523 y Fs(\000)g Fw(const)13 b FD(e)2537 2489 y Fo(\000)p Ft(const)e Fq(l)2791 2523 y FD(:)456 2694 y Fw(This,)27 b(taking)g FD(l)i Fw(big)f(enough,)f(pro)n(vide)f(the)i(pro)r(of)f (\(6.7\).)555 2880 y(According)d(to)g(\(6.6\))g(and)g(\(6.7\),)h(the)g (function)g FD(s)e Fs(7!)g FD(H)2316 2892 y Ft(\000)2357 2900 y Fd(2)2394 2880 y Fw(\()p FD(s)p Fw(\))12 b Fs(\000)g FD(H)2655 2892 y Fq(\024)2694 2900 y Fd(1)2726 2892 y Fq(l)2752 2880 y Fw(\()p FD(s)p Fw(\))25 b(is)f(increasing)f(for)456 2982 y FD(s)h Fs(\024)g(\000)p Fw(1)18 b(+)g(\()p FD(\016)s(\022)r(l) 956 2952 y Fo(\000)p Ft(2)1045 2982 y Fw(\))1077 2952 y Ft(1)p Fq(=p)1211 2982 y Fw(and)29 b(decreasing)d(for)i FD(s)c Fs(\025)g Fw(1)19 b Fs(\000)f Fw(\()p FD(\016)s(\022)r(l)2340 2952 y Fo(\000)p Ft(2)2429 2982 y Fw(\))2461 2952 y Ft(1)p Fq(=p)2567 2982 y Fw(,)29 b(therefore)e(its)i(maxim)n(um)456 3085 y(o)r(ccurs)d(in)i([)p Fs(\000)p Fw(1)18 b(+)g(\()p FD(\016)s(\022)r(l)1180 3055 y Fo(\000)p Ft(2)1269 3085 y Fw(\))1301 3055 y Ft(1)p Fq(=p)1407 3085 y FD(;)c Fw(1)k Fs(\000)g Fw(\()p FD(\016)s(\022)r(l)1727 3055 y Fo(\000)p Ft(2)1816 3085 y Fw(\))1848 3055 y Ft(1)p Fq(=p)1953 3085 y Fw(],)28 b(i.e.,)456 3228 y(\(6.8\))154 b(max)721 3285 y Fq(s)p Fo(2)p Ft([)p Fq(s)847 3294 y Fn(\016)n(;")924 3285 y Fq(;)p Ft(1])1009 3228 y FD(H)1078 3240 y Ft(\000)1119 3248 y Fd(2)1156 3228 y Fw(\()p FD(s)p Fw(\))19 b Fs(\000)f FD(H)1430 3240 y Fq(\024)1469 3248 y Fd(1)1501 3240 y Fq(l)1527 3228 y Fw(\()p FD(s)p Fw(\))23 b(=)433 b(max)1741 3290 y Fq(s)p Fo(2)p Ft([)p Fo(\000)p Ft(1+\()p Fq(\016)r(\022)r(l)2085 3273 y Fm(\000)p Fd(2)2162 3290 y Ft(\))2188 3273 y Fd(1)p Fn(=p)2282 3290 y Fq(;)p Ft(1)p Fo(\000)p Ft(\()p Fq(\016)r(\022)r(l) 2500 3273 y Fm(\000)p Fd(2)2577 3290 y Ft(\))2603 3273 y Fd(1)p Fn(=p)2696 3290 y Ft(])2729 3228 y FD(H)2798 3240 y Ft(\000)2839 3248 y Fd(2)2876 3228 y Fw(\()p FD(s)p Fw(\))18 b Fs(\000)h FD(H)3150 3240 y Fq(\024)3189 3248 y Fd(1)3221 3240 y Fq(l)3246 3228 y Fw(\()p FD(s)p Fw(\))456 3437 y(Also,)27 b(recalling)f(the)i(de\014nition)g(of)g FD(H)1674 3449 y Ft(0)1739 3437 y Fw(in)g(Lemma)f(5.1,)g(if)h FD(s)23 b Fs(2)g Fw([0)p FD(;)14 b Fw(1)k Fs(\000)g Fw(\()p FD(\016)s(\022)r(l)2890 3407 y Fo(\000)p Ft(2)2979 3437 y Fw(\))3011 3407 y Ft(1)p Fq(=p)3117 3437 y Fw(],)456 3639 y(\(6.9\))319 b FD(H)1015 3651 y Ft(\000)1056 3659 y Fd(2)1092 3639 y Fw(\()p FD(s)p Fw(\))24 b(=)1306 3526 y Fp(Z)1389 3547 y Fq(s)1352 3715 y Ft(0)1451 3583 y Fw(\()p FD(p)18 b Fs(\000)g Fw(1\))1700 3553 y Ft(1)p Fq(=p)1819 3583 y FD(d\020)p 1449 3620 459 4 v 1449 3698 a Fw(\()p FD(p)c(h)1585 3710 y Ft(\000)1626 3718 y Fd(2)1662 3698 y Fw(\()p FD(\020)6 b Fw(\)\))1800 3674 y Ft(1)p Fq(=p)1940 3639 y Fs(\024)2027 3526 y Fp(Z)2110 3547 y Fq(s)2074 3715 y Ft(0)2170 3583 y Fw(\()p FD(p)18 b Fs(\000)g Fw(1\))2419 3553 y Ft(1)p Fq(=p)2538 3583 y FD(d\020)p 2170 3620 455 4 v 2188 3698 a Fw(\()p FD(p)c(h)2324 3710 y Ft(0)2361 3698 y Fw(\()p FD(\020)6 b Fw(\)\))2499 3674 y Ft(1)p Fq(=p)2657 3639 y Fw(=)23 b FD(H)2814 3651 y Ft(0)2851 3639 y Fw(\()p FD(s)p Fw(\))456 3844 y(and)k(analogously)-7 b(,)25 b(if)j FD(s)23 b Fs(2)h Fw([)p Fs(\000)p Fw(1)17 b(+)h(\()p FD(\016)s(\022)r(l)1670 3813 y Fo(\000)p Ft(2)1760 3844 y Fw(\))1792 3813 y Ft(1)p Fq(=p)1897 3844 y FD(;)c Fw(0],)456 4046 y(\(6.10\))299 b Fs(\000)p FD(H)1102 4058 y Ft(\000)1143 4066 y Fd(2)1179 4046 y Fw(\()p FD(s)p Fw(\))23 b(=)1393 3933 y Fp(Z)1476 3953 y Ft(0)1439 4121 y Fq(s)1539 3989 y Fw(\()p FD(p)c Fs(\000)f Fw(1\))1789 3959 y Ft(1)p Fq(=p)1908 3989 y FD(d\020)p 1537 4027 459 4 v 1537 4104 a Fw(\()p FD(p)c(h)1673 4116 y Ft(\000)1714 4124 y Fd(2)1750 4104 y Fw(\()p FD(\020)6 b Fw(\)\))1888 4080 y Ft(1)p Fq(=p)2028 4046 y Fs(\025)2116 3933 y Fp(Z)2199 3953 y Fq(s)2162 4121 y Ft(0)2258 3989 y Fw(\()p FD(p)19 b Fs(\000)f Fw(1\))2508 3959 y Ft(1)p Fq(=p)2627 3989 y FD(d\020)p 2258 4027 455 4 v 2277 4104 a Fw(\()p FD(p)c(h)2413 4116 y Ft(0)2449 4104 y Fw(\()p FD(\020)6 b Fw(\)\))2587 4080 y Ft(1)p Fq(=p)2746 4046 y Fw(=)22 b Fs(\000)p FD(H)2967 4058 y Ft(0)3004 4046 y Fw(\()p FD(s)p Fw(\))14 b FD(:)456 4238 y Fw(Hence,)27 b(from)h(\(6.9\))f(and)g(\(6.10\),)1665 4381 y FD(H)1734 4393 y Ft(\000)1775 4401 y Fd(2)1811 4381 y Fw(\()p FD(s)p Fw(\))d Fs(\024)f FD(H)2095 4393 y Ft(0)2132 4381 y Fw(\()p FD(s)p Fw(\))456 4527 y(for)38 b(an)n(y)g FD(s)k Fs(2)h Fw([)p Fs(\000)p Fw(1)25 b(+)g(\()p FD(\016)s(\022)r(l)1326 4497 y Fo(\000)p Ft(2)1415 4527 y Fw(\))1447 4497 y Ft(1)p Fq(=p)1553 4527 y FD(;)14 b Fw(1)26 b Fs(\000)f Fw(\()p FD(\016)s(\022)r(l)1888 4497 y Fo(\000)p Ft(2)1977 4527 y Fw(\))2009 4497 y Ft(1)p Fq(=p)2115 4527 y Fw(].)71 b(Consequen)n(tly)-7 b(,)41 b(from)e(\(5.4\),)j(if)d FD(s)j Fs(2)456 4630 y Fw([)p Fs(\000)p Fw(1)17 b(+)h(\()p FD(\016)s(\022)r(l)826 4599 y Fo(\000)p Ft(2)915 4630 y Fw(\))947 4599 y Ft(1)p Fq(=p)1053 4630 y FD(;)c Fw(1)k Fs(\000)g Fw(\()p FD(\016)s(\022)r(l)1373 4599 y Fo(\000)p Ft(2)1462 4630 y Fw(\))1494 4599 y Ft(1)p Fq(=p)1600 4630 y Fw(],)28 b(then)1343 4825 y FD(H)1412 4837 y Ft(\000)1453 4845 y Fd(2)1490 4825 y Fw(\()p FD(s)p Fw(\))23 b Fs(\024)g FD(H)1773 4837 y Fq(\024)1812 4845 y Fd(1)1844 4837 y Fq(l)1870 4825 y Fw(\()p FD(s)p Fw(\))18 b(+)2084 4769 y(const)p 2084 4806 190 4 v 2166 4882 a FD(l)2298 4825 y Fw(log)2438 4769 y FD(l)2465 4739 y Ft(2)p 2429 4806 82 4 v 2429 4882 a FD(\016)s(\022)2534 4825 y(:)456 4996 y Fw(Therefore,)26 b(b)n(y)h(\(6.8\),)456 5192 y(\(6.11\))692 b FD(H)1430 5204 y Ft(\000)1471 5212 y Fd(2)1508 5192 y Fw(\()p FD(s)p Fw(\))24 b Fs(\024)e FD(H)1791 5204 y Fq(\024)1830 5212 y Fd(1)1862 5204 y Fq(l)1888 5192 y Fw(\()p FD(s)p Fw(\))d(+)2103 5136 y(const)p 2103 5173 190 4 v 2184 5249 a FD(l)2316 5192 y Fw(log)2456 5136 y FD(l)2483 5106 y Ft(2)p 2447 5173 82 4 v 2447 5249 a FD(\016)s(\022)p eop %%Page: 29 29 29 28 bop 743 251 a Ft(MEAN)29 b(CUR)-7 b(V)g(A)i(TURE)29 b(PR)n(OPER)-5 b(TIES)28 b(F)n(OR)h Fq(p)p Ft(-LAPLA)n(CE)h(PHASE)e (TRANSITIONS)221 b(29)456 450 y Fw(for)27 b(an)n(y)f FD(s)d Fs(2)h Fw([)p FD(s)942 462 y Fq(\016)o(;")1026 450 y FD(;)14 b Fw(1].)456 550 y(F)-7 b(urthermore,)26 b(b)n(y)i(de\014nition)f(of)h(\000)1591 562 y Ft(1)1656 550 y Fw(and)f(\000)1869 562 y Ft(2)1906 550 y Fw(,)h(if)g Fs(j)p FD(x)2103 520 y Fo(0)2127 550 y Fs(j)2150 562 y Fo(1)2244 550 y Fw(=)22 b FD(\024)2379 562 y Ft(2)2416 550 y FD(l)r Fw(,)27 b(then)1541 686 y FD(d)1584 698 y Ft(\000)1625 706 y Fd(2)1662 686 y Fw(\()p FD(x)p Fw(\))d Fs(\025)e FD(d)1927 698 y Ft(\000)1968 706 y Fd(1)2005 686 y Fw(\()p FD(x)p Fw(\))e(+)e FD(c)p Fw(\()p FD(\016)s Fw(\))456 823 y(for)32 b(a)g(suitable)h FD(c)p Fw(\()p FD(\016)s Fw(\))f Fs(2)g Fw(\(0)p FD(;)14 b Fw(1\).)52 b(Hence,)34 b(using)e(\(6.11\))g(and)h(taking)f FD(l)i Fw(appropriately)d(large,)456 922 y(with)d FD(s)23 b Fw(=)f FD(g)834 934 y Fq(\024)873 942 y Fd(1)905 934 y Fq(l)931 922 y Fw(\()p FD(d)1006 934 y Ft(\000)1047 942 y Fd(1)1084 922 y Fw(\()p FD(x)p Fw(\)\),)872 1112 y FD(H)941 1124 y Ft(\000)982 1132 y Fd(2)1019 1020 y Fp(\020)1068 1112 y FD(g)1108 1124 y Fq(\024)1147 1132 y Fd(1)1179 1124 y Fq(l)1205 1112 y Fw(\()p FD(d)1280 1124 y Ft(\000)1321 1132 y Fd(1)1358 1112 y Fw(\()p FD(x)p Fw(\)\))1501 1020 y Fp(\021)1635 1112 y FD(<)82 b(H)1851 1124 y Fq(\024)1890 1132 y Fd(1)1922 1124 y Fq(l)1948 1020 y Fp(\020)1998 1112 y FD(g)2038 1124 y Fq(\024)2077 1132 y Fd(1)2108 1124 y Fq(l)2134 1112 y Fw(\()p FD(d)2209 1124 y Ft(\000)2250 1132 y Fd(1)2287 1112 y Fw(\()p FD(x)p Fw(\)\))2430 1020 y Fp(\021)2499 1112 y Fw(+)2592 1056 y(const)p 2592 1093 190 4 v 2674 1169 a FD(l)2806 1112 y Fw(log)2946 1056 y FD(l)2973 1025 y Ft(2)p 2937 1093 82 4 v 2937 1169 a FD(\016)s(\022)1635 1326 y Fw(=)g FD(d)1825 1338 y Ft(\000)1866 1346 y Fd(1)1903 1326 y Fw(\()p FD(x)p Fw(\))19 b(+)2126 1269 y(const)p 2126 1307 190 4 v 2208 1383 a FD(l)2340 1326 y Fw(log)2480 1269 y FD(l)2507 1239 y Ft(2)p 2471 1307 82 4 v 2471 1383 a FD(\016)s(\022)2585 1326 y Fs(\024)k FD(d)2716 1338 y Ft(\000)2757 1346 y Fd(2)2794 1326 y Fw(\()p FD(x)p Fw(\))14 b FD(;)456 1490 y Fw(pro)n(vided)33 b FD(g)844 1502 y Fq(\024)883 1510 y Fd(1)915 1502 y Fq(l)941 1490 y Fw(\()p FD(d)1016 1502 y Ft(\000)1057 1510 y Fd(1)1094 1490 y Fw(\()p FD(x)p Fw(\)\))j Fs(\025)e FD(s)1411 1502 y Fq(\016)o(;")1530 1490 y Fw(and)h Fs(j)p FD(x)1769 1460 y Fo(0)1792 1490 y Fs(j)1815 1502 y Fo(1)1921 1490 y Fw(=)f FD(\024)2068 1502 y Ft(2)2105 1490 y FD(l)r Fw(;)k(therefore,)e(since)e FD(H)2853 1502 y Ft(\000)2894 1510 y Fd(2)2966 1490 y Fw(is)g(strictly)h(in-)456 1590 y(creasing)26 b(in)h([)p FD(s)933 1602 y Fq(\016)o(;")1017 1590 y FD(;)14 b Fw(1],)456 1726 y(\(6.12\))783 b FD(g)1492 1738 y Fq(\024)1531 1746 y Fd(1)1563 1738 y Fq(l)1588 1726 y Fw(\()p FD(d)1663 1738 y Ft(\000)1704 1746 y Fd(1)1741 1726 y Fw(\()p FD(x)p Fw(\)\))25 b FD(<)d(g)2036 1738 y Ft(\000)2077 1746 y Fd(2)2114 1726 y Fw(\()p FD(d)2189 1738 y Ft(\000)2230 1746 y Fd(2)2267 1726 y Fw(\()p FD(x)p Fw(\)\))14 b FD(;)456 1862 y Fw(for)28 b(an)n(y)g FD(x)i Fw(so)e(that)h FD(g)1143 1874 y Fq(\024)1182 1882 y Fd(1)1214 1874 y Fq(l)1240 1862 y Fw(\()p FD(d)1315 1874 y Ft(\000)1356 1882 y Fd(1)1393 1862 y Fw(\()p FD(x)p Fw(\)\))d Fs(\025)f FD(s)1691 1874 y Fq(\016)o(;")1804 1862 y Fw(and)k Fs(j)p FD(x)2037 1832 y Fo(0)2061 1862 y Fs(j)2084 1874 y Fo(1)2180 1862 y Fw(=)c FD(\024)2318 1874 y Ft(2)2355 1862 y FD(l)r Fw(.)40 b(Of)29 b(course,)f(if)i FD(g)2961 1874 y Fq(\024)3000 1882 y Fd(1)3032 1874 y Fq(l)3057 1862 y Fw(\()p FD(d)3132 1874 y Ft(\000)3173 1882 y Fd(1)3210 1862 y Fw(\()p FD(x)p Fw(\)\))d FD(<)456 1962 y(s)495 1974 y Fq(\016)o(;")578 1962 y Fw(,)32 b(\(6.12\))e(holds)h(since)g FD(g)1344 1974 y Ft(\000)1385 1982 y Fd(2)1450 1962 y Fs(\025)d FD(s)1582 1974 y Fq(\016)o(;")1696 1962 y Fw(b)n(y)j(construction)f (\(recall)g(item)i(\(ii\))f(of)g(Lemma)g(5.2\).)456 2062 y(Th)n(us,)456 2198 y(\(6.13\))447 b FD(g)1156 2210 y Fq(\024)1195 2218 y Fd(1)1227 2210 y Fq(l)1252 2198 y Fw(\()p FD(d)1327 2210 y Ft(\000)1368 2218 y Fd(1)1405 2198 y Fw(\()p FD(x)p Fw(\)\))25 b FD(<)d(g)1700 2210 y Ft(\000)1741 2218 y Fd(2)1778 2198 y Fw(\()p FD(d)1853 2210 y Ft(\000)1894 2218 y Fd(2)1931 2198 y Fw(\()p FD(x)p Fw(\)\))14 b FD(;)181 b Fs(8j)p FD(x)2409 2164 y Fo(0)2432 2198 y Fs(j)2455 2210 y Fo(1)2548 2198 y Fw(=)23 b FD(\024)2684 2210 y Ft(2)2721 2198 y FD(l)15 b(;)456 2335 y Fw(pro)n(vided)35 b(that)i FD(d)1038 2347 y Ft(\000)1079 2355 y Fd(1)1116 2335 y Fw(\()p FD(x)p Fw(\))h(is)f(in)g(the)g(domain)f(of)h FD(g)2064 2347 y Fq(\024)2103 2355 y Fd(1)2135 2347 y Fq(l)2197 2335 y Fw(and)g FD(d)2411 2347 y Ft(\000)2452 2355 y Fd(2)2489 2335 y Fw(\()p FD(x)p Fw(\))h(in)f(the)g(domain)f(of)h FD(g)3344 2347 y Ft(\000)3385 2355 y Fd(2)3421 2335 y Fw(.)456 2434 y(Notice,)24 b(ho)n(w)n(ev)n(er,)e(that)h(the)h(\014rst)f (of)g(these)g(conditions)g(is)g(ful\014lled)h(for)e(free)h(in)h([)p Fs(\000)p FD(\024)3124 2446 y Ft(2)3161 2434 y FD(l)r(;)14 b(\024)3273 2446 y Ft(2)3309 2434 y FD(l)r Fw(])3359 2404 y Fq(N)3421 2434 y Fw(,)456 2534 y(pro)n(vided)27 b FD(\024)846 2546 y Ft(2)911 2534 y Fw(is)h(c)n(hosen)f(to)h(b)r(e)h (suitably)e(small,)h(since,)g(b)n(y)g(Lemma)g(5.1,)g FD(T)2881 2546 y Fq(l)2929 2534 y Fs(\025)e Fw(\026)-44 b FD(cl)29 b Fw(and)f(so,)g(if)456 2633 y FD(x)23 b Fs(2)h Fw([)p Fs(\000)p FD(\024)741 2645 y Ft(2)777 2633 y FD(l)r(;)14 b(\024)889 2645 y Ft(2)926 2633 y FD(l)r Fw(])976 2603 y Fq(N)1038 2633 y Fw(,)1591 2735 y FD(d)1634 2747 y Ft(\000)1675 2755 y Fd(1)1712 2735 y Fw(\()p FD(x)p Fw(\))24 b Fs(\024)e Fw(2)p FD(\024)2024 2747 y Ft(2)2061 2735 y FD(l)j Fs(\024)d FD(T)2247 2747 y Fq(l)2286 2735 y FD(:)456 2854 y Fw(Then,)27 b(\(6.13\))g(and)h(\(6.2\))f(imply)h(that) 456 2990 y(\(6.14\))920 b FD(u)p Fw(\()p FD(x)p Fw(\))24 b FD(<)e(g)1899 3002 y Ft(\000)1940 3010 y Fd(2)1977 2990 y Fw(\()p FD(d)2052 3002 y Ft(\000)2093 3010 y Fd(2)2130 2990 y Fw(\()p FD(x)p Fw(\)\))14 b FD(;)456 3127 y Fw(for)26 b(an)n(y)f(for)h(an)n(y)g FD(x)e Fs(2)f Fw([)p Fs(\000)p FD(\024)1304 3139 y Ft(2)1341 3127 y FD(l)r(;)14 b(\024)1453 3139 y Ft(2)1489 3127 y FD(l)r Fw(])1539 3097 y Fq(N)1628 3127 y Fw(so)26 b(that)h Fs(j)p FD(x)1978 3097 y Fo(0)2001 3127 y Fs(j)2024 3139 y Fo(1)2118 3127 y Fw(=)c FD(\024)2254 3139 y Ft(2)2291 3127 y FD(l)28 b Fw(and)e FD(d)2547 3139 y Ft(\000)2588 3147 y Fd(2)2625 3127 y Fw(\()p FD(x)p Fw(\))h(is)g(in)g(the)f(domain)h(of)456 3226 y FD(g)496 3238 y Ft(\000)537 3246 y Fd(2)573 3226 y Fw(.)555 3399 y(With)35 b(these)f(estimates,)h(w)n(e)e(are)g(no)n(w)g(ready)g(to)h (deduce)g(the)g(con)n(tradiction)e(that)i(will)456 3499 y(\014nish)j(the)h(pro)r(of)e(of)i(the)f(desired)g(result.)65 b(T)-7 b(o)37 b(this)h(end,)i(w)n(e)d(slide)g FD(g)2774 3511 y Ft(\000)2815 3519 y Fd(2)2851 3499 y Fw(\()p FD(d)2926 3511 y Ft(\000)2967 3519 y Fd(2)3004 3499 y Fw(\))h(in)f(the)h FD(e)3372 3511 y Fq(n)3417 3499 y Fw(-)456 3599 y(direction)27 b(till)h(w)n(e)f(touc)n(h)g FD(u)p Fw(,)h(i.e.,)g(w)n(e)f(consider,)f (for)h FD(t)c Fs(2)h Fr(R)p Fw(,)1441 3755 y FD(g)1484 3721 y Fq(t)1512 3755 y Fw(\()p FD(x)p Fw(\))g(:=)f FD(g)1798 3767 y Ft(\000)1839 3775 y Fd(2)1875 3663 y Fp(\020)1925 3755 y FD(d)1968 3767 y Ft(\000)2009 3775 y Fd(2)2046 3755 y Fw(\()p FD(x)c Fs(\000)f FD(te)2296 3767 y Fq(n)2341 3755 y Fw(\))2373 3663 y Fp(\021)2436 3755 y FD(:)456 3917 y Fw(If)28 b(w)n(e)f(denote)g(b)n(y)h Fs(D)1108 3929 y Ft(0)1173 3917 y Fw(the)g(domain)f(of)g FD(g)1745 3929 y Ft(\000)1786 3937 y Fd(2)1823 3917 y Fw(\()p FD(d)1898 3929 y Ft(\000)1939 3937 y Fd(2)1976 3917 y Fw(\),)h(w)n(e)f(ha)n(v)n (e)f(b)n(y)i(Lemma)f(5.2)g(that)1445 4106 y Fs(D)1509 4118 y Ft(0)1569 4106 y Fs(\022)1657 3989 y Fp(\032)1719 4106 y FD(x)1766 4118 y Fq(n)1835 4106 y Fs(\024)22 b FD(C)1981 4118 y Ft(0)2019 4106 y Fw(\()p FD(\016)s Fw(\))14 b(log)2269 4050 y FD(l)2296 4020 y Ft(2)p 2269 4087 64 4 v 2280 4163 a FD(\022)2342 3989 y Fp(\033)2432 4106 y FD(;)456 4288 y Fw(and)27 b(so,)g(in)h(particular,)1589 4435 y Fs(D)1653 4447 y Ft(0)1713 4435 y Fs(\022)1801 4318 y Fp(\032)1863 4435 y FD(x)1910 4447 y Fq(n)1979 4435 y Fs(\024)2077 4378 y FD(\024)2125 4390 y Ft(2)2162 4378 y FD(l)p 2077 4415 112 4 v 2112 4492 a Fw(8)2198 4318 y Fp(\033)2288 4435 y FD(;)456 4607 y Fw(if)g FD(l)h Fw(is)e(su\016cien)n(tly)h(large.)35 b(Notice)28 b(that,)g(with)g(this) g(notation,)f FD(g)2533 4577 y Fq(t)2589 4607 y Fw(is)h(de\014ned)g(in) 456 4789 y(\(6.15\))633 b Fs(D)1366 4801 y Fq(t)1418 4789 y Fw(:=)23 b Fs(D)1593 4801 y Ft(0)1649 4789 y Fw(+)18 b FD(te)1801 4801 y Fq(n)1869 4789 y Fs(\022)1956 4672 y Fp(\032)2019 4789 y FD(x)2066 4801 y Fq(n)2134 4789 y Fs(\024)2232 4732 y FD(\024)2280 4744 y Ft(2)2317 4732 y FD(l)p 2232 4769 V 2267 4846 a Fw(8)2372 4789 y(+)g FD(t)2485 4672 y Fp(\033)2575 4789 y FD(:)456 4978 y Fw(Th)n(us,)35 b(if)f FD(t)f Fs(\024)f(\000)p FD(\024)1052 4990 y Ft(2)1089 4978 y FD(l)r(=)p Fw(4,)i(then)g Fs(D)1516 4990 y Fq(t)1578 4978 y Fs(\022)f(f)p FD(x)1765 4990 y Fq(n)1843 4978 y Fs(\024)g(\000)p Fw(\()p FD(\024)2086 4990 y Ft(2)2123 4978 y FD(l)r Fw(\))p FD(=)p Fw(8)p Fs(g)f Fw(and)h(therefore,)i FD(g)2930 4948 y Fq(t)2991 4978 y FD(>)e(u)g Fw(in)h Fs(D)3337 4990 y Fq(t)3389 4978 y Fs(\\)456 5078 y Fw([)p Fs(\000)p FD(\024)592 5090 y Ft(2)628 5078 y FD(l)r(;)14 b(\024)740 5090 y Ft(2)777 5078 y FD(l)r Fw(])27 b(b)r(ecause)g(of)h(\(2.1\),)f(pro)n (vided)g FD(l)i Fw(is)e(large)f(enough.)37 b(On)27 b(the)h(other)f (hand,)1204 5216 y FD(g)1247 5181 y Ft(0)1283 5216 y Fw(\(0\))d(=)e FD(g)1540 5228 y Ft(\000)1581 5236 y Fd(2)1617 5216 y Fw(\()p FD(d)1692 5228 y Ft(\000)1733 5236 y Fd(2)1771 5216 y Fw(\(0\)\))h(=)g FD(g)2060 5228 y Ft(\000)2101 5236 y Fd(2)2137 5216 y Fw(\(0\))g(=)g(0)f(=)h FD(u)p Fw(\(0\))14 b FD(;)p eop %%Page: 30 30 30 29 bop 456 251 a Ft(30)648 b(BERARDINO)23 b(SCIUNZI)g(AND)f(ENRICO)h (V)-7 b(ALDINOCI)456 450 y Fw(therefore,)41 b(there)d(is)h(a)g(time)g FD(t)k Fs(2)f Fw([)p Fs(\000)p FD(\024)1747 462 y Ft(2)1784 450 y FD(l)r(=)p Fw(4)p FD(;)14 b Fw(0])37 b(of)i(\014rst)f(touc)n (hing)h(b)r(et)n(w)n(een)g FD(g)3048 420 y Fq(t)3116 450 y Fw(and)g FD(u)f Fw(in)456 550 y Fs(D)520 562 y Fq(t)569 550 y Fs(\\)21 b Fw([)p Fs(\000)p FD(\024)781 562 y Ft(2)818 550 y FD(l)r(;)14 b(\024)930 562 y Ft(2)966 550 y FD(l)r Fw(].)46 b(Since)31 b FD(g)1345 562 y Ft(\000)1386 570 y Fd(2)1422 550 y Fw(\()p FD(d)1497 562 y Ft(\000)1538 570 y Fd(2)1575 550 y Fw(\))g(is)f(a)h(strict)f(sup)r(ersolution)g(of)h (\(1.5\))f(exploiting)g(Corol-)456 649 y(lary)i(3.3)h(and)g(Lemma)h (4.7)e(as)h(ab)r(o)n(v)n(e)f(w)n(e)h(pro)n(v)n(e)f(that)i(con)n(tact)f (p)r(oin)n(ts)g(in)h(the)g(in)n(terior)f(of)456 749 y Fs(D)520 761 y Fq(t)557 749 y Fs(\\)8 b Fw([)p Fs(\000)p FD(\024)756 761 y Ft(2)792 749 y FD(l)r(;)14 b(\024)904 761 y Ft(2)941 749 y FD(l)r Fw(])21 b(ma)n(y)h(only)g(happ)r(en)g(at)g (critical)g(p)r(oin)n(ts)g(of)g FD(u)g Fw(in)g(the)h(region)e(where)g FD(g)3172 719 y Fq(t)3223 749 y Fw(is)h(\015at.)456 849 y(Arguing)27 b(exactly)g(as)h(ab)r(o)n(v)n(e)f(w)n(e)g(can)h(exploit)g (Theorem)f(3.4)h(and)f(rule)h(out)g(this)h(p)r(ossibilit)n(y)456 948 y(\(see)e(the)h(argumen)n(ts)e(on)i(page)e(26\).)555 1147 y(Hence,)31 b(the)f(\014rst)g(touc)n(hing)f(p)r(oin)n(t)h(m)n(ust) g(o)r(ccur)f(at)g(the)i(b)r(oundary)e(of)g Fs(D)2926 1159 y Fq(t)2976 1147 y Fs(\\)20 b Fw([)p Fs(\000)p FD(\024)3187 1159 y Ft(2)3223 1147 y FD(l)r(;)14 b(\024)3335 1159 y Ft(2)3372 1147 y FD(l)r Fw(].)456 1247 y(W)-7 b(e)32 b(\014rst)g(observ)n(e)e(that)j(no)f(con)n(tact)f(p)r(oin)n(t)h(ma)n(y) g(lie)g(in)g FD(@)5 b Fs(D)2416 1259 y Fq(t)2445 1247 y Fw(.)51 b(Indeed,)33 b(if)g FD(x)e Fs(2)g FD(@)5 b Fs(D)3175 1259 y Fq(t)3204 1247 y Fw(,)33 b(then,)456 1347 y(from)27 b(Lemma)h(5.2,)f(w)n(e)g(ha)n(v)n(e)g(that)h Fs(r)p FD(g)1712 1317 y Fq(t)1741 1347 y Fw(\()p FD(x)p Fw(\))c Fs(6)p Fw(=)g(0,)j FD(g)2100 1317 y Fq(t)2129 1347 y Fw(\()p FD(x)p Fw(\))d(=)f(1,)28 b(while)g FD(u)23 b Fs(\024)g Fw(1.)37 b(Hence,)28 b(since)g FD(u)456 1446 y Fw(is)f FD(C)604 1416 y Ft(1)669 1446 y Fw(\(see)h([8)o(])g(or)f([19) o(]\),)h(if)g(a)f(con)n(tact)g(p)r(oin)n(t)33 b(\026)-47 b FD(x)28 b Fw(o)r(ccur)f(on)g FD(@)5 b Fs(D)2447 1458 y Fq(t)2504 1446 y Fw(w)n(e)27 b(w)n(ould)g(ha)n(v)n(e)1483 1582 y(0)c(=)f Fs(r)p FD(u)p Fw(\()5 b(\026)-47 b FD(x)q Fw(\))23 b(=)g Fs(r)p FD(g)2087 1548 y Fq(t)2116 1582 y Fw(\()5 b(\026)-47 b FD(x)p Fw(\))24 b Fs(6)p Fw(=)e(0)14 b FD(:)555 1717 y Fw(This)24 b(con)n(tradiction)e(sho)n(ws)g(that)h (con)n(tact)g(p)r(oin)n(ts)g(ma)n(y)g(only)g(happ)r(en)g(either)h(on)f Fs(j)p FD(x)3240 1687 y Fo(0)3263 1717 y Fs(j)3286 1729 y Fo(1)3380 1717 y Fw(=)456 1817 y FD(\024)504 1829 y Ft(2)541 1817 y FD(l)29 b Fw(or)d(on)h FD(x)858 1829 y Fq(N)945 1817 y Fw(=)22 b Fs(\000)p FD(\024)1145 1829 y Ft(2)1182 1817 y FD(l)29 b Fw(\(the)f(case)e FD(x)1633 1829 y Fq(N)1720 1817 y Fw(=)c FD(\024)1855 1829 y Ft(2)1892 1817 y FD(l)29 b Fw(cannot)e(hold)g(for)g FD(t)c Fs(\024)g Fw(0)k(b)r(ecause)g(of)g(\(6.15\)\).)555 1988 y(W)-7 b(e)34 b(no)n(w)f(exclude)h(the)g(p)r(ossibilit)n(y)f(of)h(touc)n(hing) f(at)g FD(x)2337 2000 y Fq(N)2434 1988 y Fw(=)g Fs(\000)p FD(\024)2645 2000 y Ft(2)2681 1988 y FD(l)r Fw(.)55 b(Indeed,)35 b(if)f FD(x)3215 1958 y Fq(])3281 1988 y Fw(is)f(so)456 2088 y(that)27 b FD(x)682 2058 y Fq(])682 2108 y(n)751 2088 y Fw(=)c Fs(\000)p FD(\024)952 2100 y Ft(2)988 2088 y FD(l)30 b Fw(and)d FD(u)p Fw(\()p FD(x)1331 2058 y Fq(])1362 2088 y Fw(\))d(=)e FD(g)1548 2058 y Fq(t)1577 2088 y Fw(\()p FD(x)1656 2058 y Fq(])1688 2088 y Fw(\))28 b(for)f(some)g FD(t)c Fs(2)g Fw([)p Fs(\000)p FD(\024)2350 2100 y Ft(2)2387 2088 y FD(l)r(=)p Fw(4)p FD(;)14 b Fw(0],)26 b(then)456 2267 y(\(6.16\))717 b FD(x)1433 2232 y Fq(])1433 2287 y(n)1497 2267 y Fs(\000)18 b FD(t)23 b Fs(\024)f(\000)p FD(\024)1833 2279 y Ft(2)1870 2267 y FD(l)e Fw(+)2008 2210 y FD(\024)2056 2222 y Ft(2)2093 2210 y FD(l)p 2008 2247 112 4 v 2043 2324 a Fw(4)2152 2267 y(=)j Fs(\000)2315 2210 y Fw(3)p 2315 2247 42 4 v 2315 2324 a(4)2366 2267 y FD(\024)2414 2279 y Ft(2)2451 2267 y FD(l)15 b Fw(;)456 2430 y(on)27 b(the)h(other)f(hand,)h(b)n(y)f(construction,)g(\000)1828 2442 y Ft(2)1888 2430 y Fs(\022)22 b(fj)p FD(x)2087 2442 y Fq(n)2133 2430 y Fs(j)h(\024)f Fw(const)p Fs(g)p Fw(,)27 b(therefore,)g(\(6.16\))g(leads)g(to)1231 2604 y FD(d)1274 2616 y Ft(\000)1315 2624 y Fd(2)1352 2604 y Fw(\()p FD(x)1431 2570 y Fq(])1481 2604 y Fs(\000)19 b FD(te)1634 2616 y Fq(n)1678 2604 y Fw(\))24 b Fs(\024)e(\000)1896 2548 y Fw(1)p 1896 2585 V 1896 2661 a(2)1947 2604 y FD(\024)1995 2616 y Ft(2)2032 2604 y FD(l)j Fs(\024)e(\000)p FD(C)2294 2616 y Ft(0)2331 2604 y Fw(\()p FD(\016)s Fw(\))14 b(log)2580 2548 y(1)p 2580 2585 V 2581 2661 a FD(")2645 2604 y(;)456 2763 y Fw(if)43 b FD(l)h Fw(is)e(large)g(enough,)k(for)c(our)g(c)n (hoices)f(of)i(the)g(parameters)e(and)h(therefore,)k(recalling)456 2863 y(Lemma)39 b(5.2,)k FD(x)984 2833 y Fq(])1055 2863 y Fw(is)d(a)g(critical)f(p)r(oin)n(t)h(for)g FD(g)1934 2833 y Fq(t)1962 2863 y Fw(.)75 b(Since)40 b FD(u)j Fs(2)h FD(C)2544 2833 y Ft(1)2622 2863 y Fw(\(see)c([8)o(])g(or)f([19)o(]\),) 44 b FD(x)3317 2833 y Fq(])3389 2863 y Fw(is)456 2962 y(also)35 b(critical)i(for)f FD(u)p Fw(:)55 b(but)37 b(this)g(p)r(ossibilit)n(y)g(ma)n(y)f(b)r(e)h(excluded)g(using)g (Theorem)f(3.4)g(and)456 3062 y(arguing)26 b(b)n(y)h(con)n(tradiction)f (exactly)h(as)g(ab)r(o)n(v)n(e.)555 3233 y(Therefore,)33 b(a)f(con)n(tact)g(p)r(oin)n(t)g FD(x)1601 3203 y Fq(?)1671 3233 y Fs(2)g(D)1822 3245 y Fq(t)1873 3233 y Fs(\\)22 b Fw([)p Fs(\000)p FD(\024)2086 3245 y Ft(2)2123 3233 y FD(l)r(;)14 b(\024)2235 3245 y Ft(2)2271 3233 y FD(l)r Fw(])32 b(do)r(es)g(o)r(ccur)g(when)h Fs(j)p FD(x)3066 3203 y Fo(0)3089 3233 y Fs(j)3112 3245 y Fo(1)3214 3233 y Fw(=)e FD(\024)3358 3245 y Ft(2)3395 3233 y FD(l)r Fw(.)456 3333 y(Notice)c(no)n(w)g(that)h(since)f FD(t)c Fs(\024)g Fw(0)k(and)h FD(d)1687 3345 y Ft(\000)1728 3353 y Fd(2)1792 3333 y Fw(is)g(de\014ned)g(to)f(b)r(e)h(p)r(ositiv)n (e)f(ab)r(o)n(v)n(e)g(\000)2971 3345 y Ft(2)3008 3333 y Fw(,)1498 3469 y FD(d)1541 3481 y Ft(\000)1582 3489 y Fd(2)1618 3469 y Fw(\()p FD(x)1697 3434 y Fq(?)1755 3469 y Fs(\000)18 b FD(te)1907 3481 y Fq(n)1952 3469 y Fw(\))23 b Fs(\025)g FD(d)2138 3481 y Ft(\000)2179 3489 y Fd(2)2215 3469 y Fw(\()p FD(x)2294 3434 y Fq(?)2333 3469 y Fw(\))14 b FD(:)456 3604 y Fw(But)27 b(then,)i(since)e FD(g)1076 3616 y Ft(\000)1117 3624 y Fd(2)1181 3604 y Fw(is)g(non-decreasing,)f(w)n(e)h(deduce)h(from)f(\(6.14\))g(that)523 3739 y FD(g)563 3751 y Ft(\000)604 3759 y Fd(2)640 3739 y Fw(\()p FD(d)715 3751 y Ft(\000)756 3759 y Fd(2)793 3739 y Fw(\()p FD(x)872 3705 y Fq(?)929 3739 y Fs(\000)18 b FD(te)1081 3751 y Fq(n)1126 3739 y Fw(\)\))24 b(=)f FD(g)1345 3705 y Fq(t)1373 3739 y Fw(\()p FD(x)1452 3705 y Fq(?)1491 3739 y Fw(\))h(=)e FD(u)p Fw(\()p FD(x)1761 3705 y Fq(?)1800 3739 y Fw(\))h FD(<)g(g)1983 3751 y Ft(\000)2024 3759 y Fd(2)2060 3739 y Fw(\()p FD(d)2135 3751 y Ft(\000)2176 3759 y Fd(2)2213 3739 y Fw(\()p FD(x)2292 3705 y Fq(?)2331 3739 y Fw(\)\))h Fs(\024)e FD(g)2546 3751 y Ft(\000)2587 3759 y Fd(2)2623 3739 y Fw(\()p FD(d)2698 3751 y Ft(\000)2739 3759 y Fd(2)2776 3739 y Fw(\()p FD(x)2855 3705 y Fq(?)2913 3739 y Fs(\000)c FD(te)3065 3751 y Fq(n)3110 3739 y Fw(\)\))c FD(:)456 3875 y Fw(This)27 b(con)n(tradiction)f (concludes)i(the)f(pro)r(of)h(of)f(Lemma)g(6.1.)975 b Fc(\003)461 4047 y Fw(7.)41 b Fv(A)31 b(viscosity)h(solution)e(pr)n (oper)-6 b(ty)33 b(f)n(or)f(the)f(limiting)h(equa)-6 b(tion)31 b(and)g(pr)n(oof)1677 4147 y(of)h(Theorem)g(2.2)555 4296 y Fw(W)-7 b(e)28 b(no)n(w)f(sho)n(w)g(that)h(rescaled)e(solutions) h(of)h(\(1.5\))f(satisfy)h(a)f(mean)g(curv)-5 b(ature)27 b(equation)456 4396 y(in)g(a)h(w)n(eak)e(viscosit)n(y)h(sense.)456 4513 y FE(Prop)s(osition)44 b(7.1.)k Fg(L)l(et)40 b FD(u)h Fg(b)l(e)g(a)h(Sob)l(olev)g(we)l(ak)g(solution)f(of)h(\(1.5\))h(in)e (the)g(whole)i Fr(R)3350 4483 y Fq(N)3419 4513 y Fg(,)456 4613 y(satisfying)d(\(2.1\),)i(so)d(that)g Fs(j)p FD(u)p Fs(j)g FD(<)g Fw(1)f Fg(and)h FD(u)p Fw(\(0\))g(=)f(0)p Fg(.)65 b(L)l(et)39 b Ff(d)g Fs(2)h Fw(\(0)p FD(;)14 b Fw(1\))p Fg(,)41 b FD(V)58 b Fs(2)39 b Fr(R)3120 4583 y Fq(N)6 b Fo(\000)p Ft(1)3313 4613 y Fg(and)456 4713 y FD(M)31 b Fs(2)24 b Fw(Mat\(\()p FD(N)j Fs(\000)18 b Fw(1\))h Fs(\002)f Fw(\()p FD(N)27 b Fs(\000)18 b Fw(1\)\))30 b Fg(with)456 4849 y Fw(\(7.1\))309 b Fs(j)p FD(V)19 b Fs(j)24 b(\024)e Fw(tr)14 b FD(M)22 b(;)184 b Fw(tr)13 b FD(M)32 b(>)23 b Ff(d)p Fs(k)p FD(M)9 b Fs(k)169 b Fg(and)h Fs(k)p FD(M)9 b Fs(k)21 b(\024)i Ff(d)2837 4815 y Fo(\000)p Ft(1)2940 4849 y FD(:)456 4985 y Fg(L)l(et)29 b FD(u)647 4997 y Fq(")682 4985 y Fw(\()p FD(x)p Fw(\))24 b(:=)f FD(u)p Fw(\()p FD(x=")p Fw(\))29 b Fg(and)784 5170 y Fw(\000)23 b(:=)970 5053 y Fp(\032)1032 5170 y FD(x)g Fw(=)g(\()p FD(x)1269 5136 y Fo(0)1293 5170 y FD(;)14 b(x)1377 5182 y Fq(n)1423 5170 y Fw(\))23 b Fs(2)g Fr(R)1610 5136 y Fq(N)6 b Fo(\000)p Ft(1)1783 5170 y Fs(\002)18 b Fr(R)52 b Fg(s.t.)46 b FD(x)2177 5182 y Fq(N)2264 5170 y Fw(=)2361 5114 y(1)p 2361 5151 V 2361 5227 a(2)2413 5170 y FD(x)2460 5136 y Fo(0)2502 5170 y Fs(\001)19 b FD(M)9 b(x)2681 5136 y Fo(0)2722 5170 y Fw(+)18 b FD(V)38 b Fs(\001)18 b FD(x)2979 5136 y Fo(0)3003 5053 y Fp(\033)3093 5170 y FD(:)p eop %%Page: 31 31 31 30 bop 743 251 a Ft(MEAN)29 b(CUR)-7 b(V)g(A)i(TURE)29 b(PR)n(OPER)-5 b(TIES)28 b(F)n(OR)h Fq(p)p Ft(-LAPLA)n(CE)h(PHASE)e (TRANSITIONS)221 b(31)456 450 y Fg(Then,)27 b(ther)l(e)g(exist)e(a)i (universal)f Ff(d)1548 420 y Fq(?)1610 450 y FD(>)c Fw(0)k Fg(and)g(a)h(function)f FD(\033)2361 462 y Ft(0)2421 450 y Fw(:)e(\(0)p FD(;)14 b Fw(1\))22 b Fs(\000)-14 b(!)23 b Fw(\(0)p FD(;)14 b Fw(1\))25 b Fg(such)i(that)e(if)456 550 y FD(")f Fs(2)g Fw(\(0)p FD(;)14 b(\033)756 562 y Ft(0)794 550 y Fw(\()p Ff(d)p Fw(\)\))31 b Fg(and)g Ff(d)25 b Fs(2)g Fw(\(0)p FD(;)14 b Ff(d)1422 520 y Fq(?)1460 550 y Fw(\))p Fg(,)31 b(then)g Fw(\000)f Fg(c)l(annot)g(touch)h Fs(f)p FD(u)2402 562 y Fq(")2461 550 y Fw(=)24 b(0)p Fs(g)29 b Fg(by)i(b)l(elow)g(in)g FD(B)3159 576 y Fe(d)p Fq(=)3228 527 y Fo(p)p 3282 527 133 3 v 3282 576 a Ft(tr)11 b Fq(M)3419 550 y Fg(:)456 657 y(mor)l(e)30 b(explicitly,)456 843 y Fw(\(7.2\))220 b Fs(f)p FD(u)937 855 y Fq(")995 843 y Fw(=)22 b(0)p Fs(g)32 b(\\)1285 726 y Fp(\032)1347 843 y FD(x)1394 855 y Fq(N)1481 843 y FD(<)1579 787 y Fw(1)p 1579 824 42 4 v 1579 900 a(2)1630 843 y FD(x)1677 808 y Fo(0)1719 843 y Fs(\001)19 b FD(M)9 b(x)1898 808 y Fo(0)1939 843 y Fw(+)19 b FD(V)37 b Fs(\001)18 b FD(x)2196 808 y Fo(0)2220 726 y Fp(\033)2315 843 y Fs(\\)2402 726 y Fp(\032)2465 843 y Fs(j)p FD(x)p Fs(j)23 b FD(<)2777 787 y Ff(d)p 2679 824 238 4 v 2679 840 a Fs(p)p 2748 840 169 4 v 71 x Fw(tr)14 b FD(M)2926 726 y Fp(\033)3035 843 y Fs(6)p Fw(=)45 b Fs(;)14 b FD(:)456 1047 y Fg(Pr)l(o)l(of.)43 b Fw(W)-7 b(e)27 b(will)h(apply)e(Lemma)h(6.1)g(b)n(y)f(making)h(use)g (of)g(the)h(follo)n(wing)e(c)n(hoice)g(of)h(parame-)456 1146 y(ters:)841 1275 y FD(l)d Fw(:=)1135 1219 y Ff(d)p 1011 1256 290 4 v 1011 1343 a FD(")1064 1272 y Fs(p)p 1133 1272 169 4 v 71 x Fw(tr)13 b FD(M)1325 1275 y(;)97 b(\016)26 b Fw(:=)c FD(\022)k Fw(:=)c Ff(d)1834 1241 y Ft(2)1886 1275 y FD(;)97 b(M)2087 1287 y Ft(1)2147 1275 y Fw(:=)2331 1219 y(1)p 2267 1256 V 2267 1332 a(tr)14 b FD(M)2459 1275 y(M)23 b(;)97 b(\030)27 b Fw(:=)2867 1219 y FD("l)p 2867 1256 66 4 v 2879 1332 a(\022)2955 1275 y(V)33 b(:)456 1442 y Fw(If,)i(b)n(y)e(con)n(tradiction,)h(the)g (claim)g(of)f(Prop)r(osition)f(7.1)h(w)n(ere)f(false,)j(b)n(y)e (scaling)g(bac)n(k)g(and)456 1542 y(using)39 b(the)g(ab)r(o)n(v)n(e)f (parameters,)j(w)n(e)e(w)n(ould)g(ha)n(v)n(e)f(that)i(\000)2389 1554 y Ft(1)2466 1542 y Fw(touc)n(hes)e(b)n(y)h(b)r(elo)n(w)g FD(u)g Fw(inside)456 1641 y([)p Fs(\000)p FD(l)r(;)14 b(l)r Fw(])658 1611 y Fq(N)719 1641 y Fw(,)28 b(where)723 1832 y(\000)775 1844 y Ft(1)835 1832 y Fw(=)923 1715 y Fp(\032)985 1832 y FD(x)c Fw(=)e(\()p FD(x)1222 1798 y Fo(0)1247 1832 y FD(;)14 b(x)1331 1844 y Fq(n)1376 1832 y Fw(\))23 b Fs(2)h Fr(R)1564 1798 y Fq(N)6 b Fo(\000)p Ft(1)1736 1832 y Fs(\002)18 b Fr(R)52 b Fw(s.t.)47 b FD(x)2130 1844 y Fq(n)2198 1832 y Fw(=)2328 1776 y FD(\022)p 2296 1813 106 4 v 2296 1889 a Fw(2)p FD(l)2365 1865 y Ft(2)2411 1832 y FD(x)2458 1798 y Fo(0)2500 1832 y Fs(\001)19 b FD(M)2623 1844 y Ft(1)2659 1832 y FD(x)2706 1798 y Fo(0)2749 1832 y Fw(+)2842 1776 y FD(\022)p 2842 1813 42 4 v 2849 1889 a(l)2893 1832 y(\030)j Fs(\001)d FD(x)3040 1798 y Fo(0)3064 1715 y Fp(\033)3154 1832 y FD(:)456 2018 y Fw(By)h(Lemma)g(6.1,)h(w)n(e)f(gather)f(that)i FD(\016)26 b Fs(\025)c Fw(tr)14 b FD(M)1868 2030 y Ft(1)1928 2018 y Fw(=)23 b(1,)e(whic)n(h)f(is)h(the)f(desired)g(con)n(tradiction) 82 b Fc(\003)555 2177 y Fw(Notice)27 b(that)g(Theorem)e(2.2)h(follo)n (ws)g(at)g(once)g(from)g(Prop)r(osition)f(7.1)h(b)n(y)g(taking)g FD(V)42 b Fw(:=)23 b(0.)1426 2365 y(8.)41 b Fv(Pr)n(oof)31 b(of)g(Theorem)h(2.1)555 2515 y Fw(W)-7 b(e)25 b(are)e(no)n(w)h(in)g(p) r(osition)g(to)h(pro)n(v)n(e)d(Theorem)i(2.1.)35 b(Let)24 b FD(x)2430 2484 y Fq(?)2493 2515 y Fw(b)r(e)h(a)e(p)r(oin)n(t)i(where) f Fs(S)30 b Fw(admits)456 2614 y(a)24 b(tangen)n(t)h(plane.)36 b(With)26 b(no)f(loss)f(of)h(generalit)n(y)-7 b(,)24 b(w)n(e)h(ma)n(y)f(assume)h(that)g(the)h(normal)e(v)n(ector)456 2714 y(of)j Fs(S)33 b Fw(at)27 b FD(x)781 2684 y Fq(?)847 2714 y Fw(is)g FD(e)969 2726 y Fq(N)1032 2714 y Fw(.)36 b(Let)28 b(b)r(e)f FD(P)39 b Fw(a)27 b(parab)r(oloid)e(touc)n(hing)i(b) n(y)f(b)r(elo)n(w.)37 b(W)-7 b(e)27 b(will)h(sho)n(w)e(that)h(the)456 2813 y(mean)k(curv)-5 b(ature)32 b(of)g FD(P)44 b Fw(is)32 b(non-p)r(ositiv)n(e:)45 b(an)31 b(analogous)f(pro)r(of)i(w)n(ould)f (giv)n(e)g(that,)j(if)f FD(P)44 b Fw(is)456 2913 y(a)27 b(parab)r(oloid)g(touc)n(hing)g(b)n(y)h(ab)r(o)n(v)n(e,)e(then)j(its)f (mean)g(curv)-5 b(ature)27 b(is)h(non-negativ)n(e,)e(and)i(this)456 3013 y(w)n(ould)f(end)h(the)g(pro)r(of)f(of)g(Theorem)g(2.1.)555 3195 y(By)h(construction,)e(if)i FD(P)40 b Fw(touc)n(hes)27 b Fs(S)34 b Fw(b)n(y)27 b(b)r(elo)n(w)h(at)f FD(x)2234 3165 y Fq(?)2273 3195 y Fw(,)g(w)n(e)h(ha)n(v)n(e)e(that)607 3381 y FD(P)48 b Fw(=)810 3264 y Fp(\032)872 3381 y Fw(\()p FD(x)951 3347 y Fo(0)975 3381 y FD(;)14 b(x)p Fw(\))24 b Fs(2)f Fr(R)1247 3347 y Fq(N)6 b Fo(\000)p Ft(1)1420 3381 y Fs(\002)18 b Fr(R)52 b Fw(:)46 b FD(x)1725 3393 y Fq(N)1811 3381 y Fw(=)1909 3325 y(1)p 1909 3362 V 1909 3438 a(2)1974 3381 y(\()p FD(x)2053 3347 y Fo(0)2096 3381 y Fs(\000)18 b Fw(\()p FD(x)2258 3347 y Fq(?)2297 3381 y Fw(\))2329 3347 y Fo(0)2352 3381 y Fw(\))h Fs(\001)g(M)14 b Fw(\()p FD(x)2638 3347 y Fo(0)2680 3381 y Fs(\000)k Fw(\()p FD(x)2842 3347 y Fq(?)2881 3381 y Fw(\))2913 3347 y Fo(0)2936 3381 y Fw(\))h(+)f FD(x)3117 3347 y Fq(?)3117 3402 y(N)3180 3264 y Fp(\033)3270 3381 y FD(;)456 3572 y Fw(for)24 b(some)h Fs(M)e(2)g Fw(Mat\(\()p FD(N)g Fs(\000)14 b Fw(1\))g Fs(\002)g Fw(\()p FD(N)20 b Fs(\000)14 b Fw(1\)\).)35 b(Let)25 b Fr(I)2084 3584 y Fq(N)5 b Fo(\000)p Ft(1)2251 3572 y Fw(b)r(e)25 b(the)h(\()p FD(N)c Fs(\000)14 b Fw(1\))g Fs(\002)g Fw(\()p FD(N)20 b Fs(\000)14 b Fw(1\))24 b(iden)n(tit)n(y)456 3672 y(matrix)j(and)g(set)456 3834 y(\(8.1\))1465 3813 y Fp(c)1445 3834 y Fs(M)c Fw(:=)f Fs(M)c(\000)1963 3778 y Fw(tr)13 b Fs(M)p 1889 3815 325 4 v 1889 3891 a Fw(2\()p FD(N)27 b Fs(\000)18 b Fw(1\))2238 3834 y Fr(I)2276 3846 y Fq(N)6 b Fo(\000)p Ft(1)2432 3834 y FD(:)456 4019 y Fw(Notice)27 b(that)456 4179 y(\(8.2\))1040 b(tr)1766 4158 y Fp(c)1746 4179 y Fs(M)23 b Fw(=)1966 4123 y(1)p 1966 4160 42 4 v 1966 4236 a(2)2018 4179 y(tr)13 b Fs(M)h FD(:)456 4343 y Fw(Let)27 b(us)h(also)e(de\014ne)593 4513 y Ff(P)37 b Fw(:=)823 4396 y Fp(\032)886 4513 y Fw(\()p FD(x)965 4479 y Fo(0)989 4513 y FD(;)14 b(x)p Fw(\))23 b Fs(2)h Fr(R)1261 4479 y Fq(N)6 b Fo(\000)p Ft(1)1433 4513 y Fs(\002)18 b Fr(R)52 b Fw(:)46 b FD(x)1738 4525 y Fq(N)1825 4513 y Fw(=)1923 4457 y(1)p 1923 4494 V 1923 4570 a(2)1988 4513 y(\()p FD(x)2067 4479 y Fo(0)2109 4513 y Fs(\000)18 b Fw(\()p FD(x)2271 4479 y Fq(?)2310 4513 y Fw(\))2342 4479 y Fo(0)2366 4513 y Fw(\))h Fs(\001)2478 4492 y Fp(c)2458 4513 y Fs(M)14 b Fw(\()p FD(x)2651 4479 y Fo(0)2693 4513 y Fs(\000)k Fw(\()p FD(x)2855 4479 y Fq(?)2894 4513 y Fw(\))2926 4479 y Fo(0)2950 4513 y Fw(\))g(+)g FD(x)3130 4479 y Fq(?)3130 4534 y(N)3194 4396 y Fp(\033)3284 4513 y FD(;)456 4725 y Fw(Let)29 b(us)g(assume,)g(b)n(y)g(con)n (tradiction,)f(that)i(tr)14 b Fs(M)25 b FD(>)g Fw(0.)42 b(Then,)30 b(tr)2596 4704 y Fp(c)2576 4725 y Fs(M)c FD(>)f Fw(0)k(in)g(ligh)n(t)g(of)h(\(8.2\),)456 4825 y(and)h Ff(P)h Fw(is)f(also)g(touc)n(hing)g Fs(S)38 b Fw(from)32 b(b)r(elo)n(w)f(at)h FD(x)2004 4794 y Fq(?)2042 4825 y Fw(.)50 b(Let)32 b FD(r)g Fs(2)f Fw(\(0)p FD(;)14 b Fw(1\),)32 b(to)g(b)r(e)g(c)n(hosen)f(suitably)456 4924 y(small)25 b(in)g(the)h(sequel,)f(and)g(let)h(us)f(consider)f(the)i (cub)r(e)g FD(Q)d Fw(:=)f Fs(fj)p FD(x)14 b Fs(\000)g FD(x)2662 4894 y Fq(?)2700 4924 y Fs(j)2723 4936 y Fo(1)2817 4924 y Fs(\024)22 b FD(r)r Fs(g)k Fw(Notice)f(that,)456 5024 y(b)n(y)i(the)h(tangency)f(b)r(et)n(w)n(een)g Fs(f)p FD(x)1476 5036 y Fq(N)1562 5024 y Fw(=)c FD(x)1697 4994 y Fq(?)1697 5047 y(N)1760 5024 y Fs(g)28 b Fw(and)f Fs(S)6 b Fw(,)28 b(if)g FD(r)j Fw(is)c(small)g(enough,)g(w)n(e)h(ha)n(v)n(e)e (that)456 5192 y(\(8.3\))809 b Fs(S)25 b(\\)19 b FD(Q)k Fs(\022)1762 5100 y Fp(n)1817 5192 y Fs(j)p FD(x)1887 5204 y Fq(N)1969 5192 y Fs(\000)18 b FD(x)2099 5158 y Fq(?)2099 5212 y(N)2162 5192 y Fs(j)23 b(\024)2307 5136 y FD(r)p 2306 5173 V 2306 5249 a Fw(4)2357 5100 y Fp(o)2440 5192 y FD(;)p eop %%Page: 32 32 32 31 bop 456 251 a Ft(32)648 b(BERARDINO)23 b(SCIUNZI)g(AND)f(ENRICO)h (V)-7 b(ALDINOCI)456 450 y Fw(Also,)27 b(b)n(y)g(\(2.2\),)h(\014xed)f (an)n(y)g FD(\021)f(>)d Fw(0,)k(w)n(e)g(ha)n(v)n(e)g(that)456 601 y(\(8.4\))1050 b(dist)14 b(\()p FD(x;)g Fs(S)6 b Fw(\))25 b Fs(\024)e FD(\021)17 b(;)456 752 y Fw(for)27 b(an)n(y)f FD(x)e Fs(2)f(f)p FD(u)978 764 y Fq(")1036 752 y Fw(=)g(0)p Fs(g)17 b(\\)i FD(B)1362 764 y Ft(1)1399 752 y Fw(\()p FD(x)1478 722 y Fq(?)1517 752 y Fw(\),)28 b(pro)n(vided)f FD(")g Fw(is)g(small)h(enough.)36 b(W)-7 b(e)28 b(also)e(claim)i(that)456 904 y(\(8.5\))869 b FD(B)1559 916 y Fq(\021)1599 904 y Fw(\()p FD(x)1678 869 y Fq(?)1717 904 y Fw(\))33 b Fs(\\)f(f)p FD(u)1959 916 y Fq(")2017 904 y Fw(=)22 b(0)p Fs(g)36 b(6)p Fw(=)h Fs(;)14 b FD(;)456 1055 y Fw(if)29 b FD(")f Fw(is)h(small)f(enough.)40 b(T)-7 b(o)28 b(pro)n(v)n(e)f(this,)j(let)f(us)f(argue)g(b)n(y)g(con)n (tradiction)g(and)g(assume,)g(sa)n(y)-7 b(,)456 1154 y(that)31 b FD(u)687 1166 y Fq(")752 1154 y FD(<)d Fw(0)j(in)h FD(B)1082 1166 y Fq(\021)1122 1154 y Fw(\()p FD(x)1201 1124 y Fq(?)1240 1154 y Fw(\).)49 b(Then,)32 b(since)g FD(u)1844 1166 y Fq(")1910 1154 y Fw(con)n(v)n(erges)d(in)i FD(L)2443 1124 y Ft(1)2443 1178 y(lo)r(c)2562 1154 y Fw(to)g FD(\037)e Fw(:=)g FD(\037)2917 1166 y Fo(E)2983 1154 y Fs(\000)20 b FD(\037)3120 1171 y Fk(R)3167 1154 y Fn(N)3216 1171 y Fo(nE)3294 1154 y Fw(,)33 b(w)n(e)456 1254 y(ha)n(v)n(e)924 1427 y(0)83 b Fs(\025)134 b Fw(lim)1196 1485 y Fq(")p Fo(\000)-11 b(!)p Ft(0)1367 1468 y Fd(+)1428 1314 y Fp(Z)1474 1502 y Fq(B)1524 1510 y Fn(\021)1561 1502 y Ft(\()p Fq(x)1625 1486 y Fn(?)1659 1502 y Ft(\))1703 1427 y Fs(j)p FD(u)1774 1439 y Fq(")1828 1427 y Fs(\000)18 b FD(\037)p Fs(j)1049 1665 y Fw(=)134 b(lim)1196 1723 y Fq(")p Fo(\000)-11 b(!)p Ft(0)1367 1707 y Fd(+)1428 1552 y Fp(Z)1474 1741 y Fq(B)1524 1749 y Fn(\021)1561 1741 y Ft(\()p Fq(x)1625 1724 y Fn(?)1659 1741 y Ft(\))p Fo(\\E)1789 1665 y Fs(j)p FD(u)1860 1677 y Fq(")1913 1665 y Fs(\000)18 b Fw(1)p Fs(j)g Fw(+)2162 1552 y Fp(Z)2208 1741 y Fq(B)2258 1749 y Fn(\021)2295 1741 y Ft(\()p Fq(x)2359 1724 y Fn(?)2393 1741 y Ft(\))p Fo(\\)p Ft(\()p Fk(R)2537 1724 y Fn(N)2585 1741 y Fo(nE)5 b Ft(\))2704 1665 y Fs(j)p FD(u)2775 1677 y Fq(")2828 1665 y Fw(+)18 b(1)p Fs(j)1049 1853 y(\025)82 b Ff(L)p Fw(\()p FD(B)1346 1865 y Fq(\021)1387 1853 y Fw(\()p FD(x)1466 1818 y Fq(?)1505 1853 y Fw(\))19 b Fs(\\)g(E)7 b Fw(\))14 b FD(;)456 2004 y Fw(where)27 b(w)n(e)g(denoted)h(b)n(y)f Ff(L)h Fw(the)g FD(N)9 b Fw(-dimensional)26 b(Leb)r(esgue)h(meausure,)g(from)g(whic)n(h)1597 2155 y Ff(L)p Fw(\()p FD(B)1747 2167 y Fq(\021)1788 2155 y Fw(\()p FD(x)1867 2121 y Fq(?)1906 2155 y Fw(\))19 b Fs(\\)f(E)7 b Fw(\))24 b(=)f(0)14 b(;)456 2306 y(this)27 b(con)n(tradicts)g(\(2.3\))g(and)h(pro)n(v)n(es)d(\(8.5\).)555 2406 y(W)-7 b(e)26 b(will)f(no)n(w)g(consider)f(the)h(touc)n(hing)g(of) g(a)f(suitable)h(sliding)g(of)g Ff(P)g Fw(with)h Fs(f)p FD(u)3008 2418 y Fq(")3065 2406 y Fw(=)d(0)p Fs(g)h Fw(in)i FD(Q)p Fw(.)456 2505 y(In)h(order)g(to)g(formalize)g(the)h(argumen)n (t,)e(let)i(us)g(de\014ne)513 2702 y Ff(P)582 2714 y Fq(t)648 2702 y Fw(:=)772 2585 y Fp(\032)835 2702 y Fw(\()p FD(x)914 2667 y Fo(0)938 2702 y FD(;)14 b(x)p Fw(\))23 b Fs(2)h Fr(R)1210 2667 y Fq(N)6 b Fo(\000)p Ft(1)1382 2702 y Fs(\002)18 b Fr(R)52 b Fw(:)46 b FD(x)1687 2714 y Fq(N)1774 2702 y Fw(=)1871 2645 y(1)p 1871 2682 42 4 v 1871 2759 a(2)1937 2702 y(\()p FD(x)2016 2667 y Fo(0)2058 2702 y Fs(\000)18 b Fw(\()p FD(x)2220 2667 y Fq(?)2259 2702 y Fw(\))2291 2667 y Fo(0)2315 2702 y Fw(\))g Fs(\001)2427 2681 y Fp(c)2407 2702 y Fs(M)13 b Fw(\()p FD(x)2599 2667 y Fo(0)2642 2702 y Fs(\000)18 b Fw(\()p FD(x)2804 2667 y Fq(?)2843 2702 y Fw(\))2875 2667 y Fo(0)2898 2702 y Fw(\))h(+)f FD(x)3079 2667 y Fq(?)3079 2722 y(N)3161 2702 y Fs(\000)g FD(t)3274 2585 y Fp(\033)3364 2702 y FD(;)456 2903 y Fw(and)27 b(let)i FD(t)23 b Fs(2)h Fr(R)34 b Fw(so)28 b(that)g Ff(P)1310 2915 y Fq(t)1358 2903 y Fs(\\)19 b(f)p FD(u)1522 2915 y Fq(")1580 2903 y Fw(=)k(0)p Fs(g)18 b(\\)h FD(Q)k Fs(6)p Fw(=)h Fs(;)p Fw(,)j(while)h Ff(P)2400 2915 y Fq(s)2454 2903 y Fs(\\)19 b(f)p FD(u)2618 2915 y Fq(")2677 2903 y Fw(=)k(0)p Fs(g)18 b(\\)h FD(Q)k Fw(=)g Fs(;)28 b Fw(for)f(an)n(y)456 3002 y FD(s)c(>)f(t)p Fw(.)37 b(Notice)28 b(that,)g(from)f(\(8.4\))g(and)h(\(8.5\),)456 3153 y(\(8.6\))1168 b Fs(j)p FD(t)p Fs(j)23 b(\024)g Fw(2)p FD(\021)17 b(:)456 3304 y Fw(Let)32 b FD(x)656 3274 y Fq(])719 3304 y Fs(2)f Ff(P)874 3316 y Fq(t)925 3304 y Fs(\\)21 b(f)p FD(u)1091 3316 y Fq(")1157 3304 y Fw(=)31 b(0)p Fs(g)20 b(\\)i FD(Q)p Fw(.)51 b(W)-7 b(e)33 b(no)n(w)f(sho)n(w)f(that)i FD(x)2342 3274 y Fq(])2406 3304 y Fw(is)f(in)h(the)f(in)n(terior)f(of)i FD(Q)p Fw(;)h(more)456 3404 y(precisely)-7 b(,)26 b(w)n(e)i(will)f(sho)n(w)g(that)456 3576 y(\(8.7\))891 b FD(x)1565 3541 y Fq(])1619 3576 y Fs(2)1698 3483 y Fp(n)1753 3576 y Fs(j)p FD(x)19 b Fs(\000)f FD(x)1972 3541 y Fq(?)2011 3576 y Fs(j)2034 3588 y Fo(1)2127 3576 y Fs(\024)2226 3519 y FD(r)p 2225 3556 V 2225 3632 a Fw(2)2276 3483 y Fp(o)2359 3576 y FD(:)456 3755 y Fw(First)27 b(of)h(all,)f(b)n(y)g(\(8.3\))h(and)f (\(8.4\),)456 3931 y(\(8.8\))679 b Fs(f)p FD(u)1396 3943 y Fq(")1453 3931 y Fw(=)23 b(0)p Fs(g)17 b(\\)i FD(Q)k Fs(\022)1892 3838 y Fp(n)1948 3931 y Fs(j)p FD(x)2018 3943 y Fq(N)2100 3931 y Fs(\000)18 b FD(x)2230 3896 y Fq(?)2230 3951 y(N)2293 3931 y Fs(j)23 b(\024)2438 3874 y FD(r)p 2437 3912 V 2437 3988 a Fw(2)2488 3838 y Fp(o)2571 3931 y FD(;)456 4128 y Fw(pro)n(vided)18 b FD(")i Fw(is)f(small)h (enough.)33 b(Hence,)22 b(to)d(pro)n(v)n(e)f(\(8.7\),)j(it)g(remains)d (to)i(sho)n(w)f(that)h Fs(j)p FD(x)3147 4088 y Fq(])3147 4151 y(i)3181 4128 y Fs(\000)s FD(x)3296 4098 y Fq(?)3296 4149 y(i)3334 4128 y Fs(j)j(\024)456 4227 y FD(r)r(=)p Fw(2)k(for)g(1)c Fs(\024)f FD(i)h Fs(\024)g FD(N)k Fs(\000)18 b Fw(1.)555 4327 y(F)-7 b(or)27 b(this,)h(recalling)e(\(8.4\),)i(let)33 b(~)-47 b FD(x)23 b Fs(2)h(S)34 b Fw(b)r(e)28 b(so)f(that)456 4482 y(\(8.9\))1090 b Fs(j)5 b Fw(~)-47 b FD(x)19 b Fs(\000)f FD(x)1936 4448 y Fq(])1968 4482 y Fs(j)23 b(\024)g FD(\021)17 b(:)456 4633 y Fw(Then,)27 b(since)h Fs(S)34 b Fw(is)28 b(ab)r(o)n(v)n(e)e FD(P)12 b Fw(,)456 4819 y(\(8.10\))531 b(~)-48 b FD(x)1241 4831 y Fq(N)1328 4819 y Fs(\025)1425 4763 y Fw(1)p 1425 4800 V 1425 4876 a(2)1477 4819 y(\()5 b(~)-47 b FD(x)1556 4785 y Fo(0)1598 4819 y Fs(\000)18 b Fw(\()p FD(x)1760 4785 y Fq(?)1799 4819 y Fw(\))1831 4785 y Fo(0)1855 4819 y Fw(\))g Fs(\001)h(M)p Fw(\()5 b(~)-47 b FD(x)2126 4785 y Fo(0)2168 4819 y Fs(\000)18 b Fw(\()p FD(x)2330 4785 y Fq(?)2369 4819 y Fw(\))2401 4785 y Fo(0)2425 4819 y Fw(\))g(+)g FD(x)2605 4785 y Fq(?)2605 4839 y(N)2683 4819 y FD(:)456 5006 y Fw(Since,)27 b(on)h(the)g(other)f(hand,)g FD(x)1448 4976 y Fq(])1503 5006 y Fs(2)c Ff(P)1650 5018 y Fq(t)1679 5006 y Fw(,)1033 5192 y FD(x)1080 5152 y Fq(])1080 5216 y(N)1166 5192 y Fw(=)1264 5136 y(1)p 1264 5173 V 1264 5249 a(2)1315 5192 y(\(\()p FD(x)1426 5158 y Fq(])1459 5192 y Fw(\))1491 5158 y Fo(0)1533 5192 y Fs(\000)18 b Fw(\()p FD(x)1695 5158 y Fq(?)1733 5192 y Fw(\))1765 5158 y Fo(0)1789 5192 y Fw(\))h Fs(\001)1901 5171 y Fp(c)1881 5192 y Fs(M)p Fw(\(\()p FD(x)2092 5158 y Fq(])2124 5192 y Fw(\))2156 5158 y Fo(0)2198 5192 y Fs(\000)f Fw(\()p FD(x)2360 5158 y Fq(?)2399 5192 y Fw(\))2431 5158 y Fo(0)2455 5192 y Fw(\))g(+)g FD(x)2635 5158 y Fq(?)2635 5212 y(N)2717 5192 y Fs(\000)g FD(t)c(;)p eop %%Page: 33 33 33 32 bop 743 251 a Ft(MEAN)29 b(CUR)-7 b(V)g(A)i(TURE)29 b(PR)n(OPER)-5 b(TIES)28 b(F)n(OR)h Fq(p)p Ft(-LAPLA)n(CE)h(PHASE)e (TRANSITIONS)221 b(33)456 450 y Fw(whic)n(h,)27 b(from)g(\(8.1\))h(and) f(\(8.6\),)h(implies)456 629 y(\(8.11\))40 b FD(x)756 589 y Fq(])756 653 y(N)843 629 y Fs(\024)22 b(\000)1078 573 y Fw(tr)14 b Fs(M)p 1005 610 325 4 v 1005 686 a Fw(4\()p FD(N)27 b Fs(\000)18 b Fw(1\))1340 629 y Fs(j)p Fw(\()p FD(x)1442 594 y Fq(])1474 629 y Fw(\))1506 594 y Fo(0)1531 629 y Fs(\000)r Fw(\()p FD(x)1677 594 y Fq(?)1715 629 y Fw(\))1747 594 y Fo(0)1771 629 y Fs(j)1794 594 y Ft(2)1833 629 y Fw(+)1910 573 y(1)p 1910 610 42 4 v 1910 686 a(2)1961 629 y(\(\()p FD(x)2072 594 y Fq(])2104 629 y Fw(\))2136 594 y Fo(0)2162 629 y Fs(\000)r Fw(\()p FD(x)2308 594 y Fq(?)2346 629 y Fw(\))2378 594 y Fo(0)2402 629 y Fw(\))r Fs(\001)r(M)p Fw(\(\()p FD(x)2672 594 y Fq(])2703 629 y Fw(\))2735 594 y Fo(0)2761 629 y Fs(\000)r Fw(\()p FD(x)2907 594 y Fq(?)2945 629 y Fw(\))2977 594 y Fo(0)3001 629 y Fw(\))r(+)r FD(x)3149 594 y Fq(?)3149 649 y(N)3214 629 y Fw(+)r(2)p FD(\021)e(:)456 814 y Fw(Subtracting)27 b(\(8.10\))g(and)g(\(8.11\),)g(and)g(making)g(use)h(of)f(\(8.9\),)h (one)f(gets)g(that)874 960 y(const)14 b FD(\021)86 b Fs(\025)i Fw(~)-48 b FD(x)1399 972 y Fq(N)1481 960 y Fs(\000)18 b FD(x)1611 920 y Fq(])1611 984 y(N)1205 1130 y Fs(\025)1436 1074 y Fw(tr)13 b Fs(M)p 1362 1111 325 4 v 1362 1187 a Fw(4\()p FD(N)27 b Fs(\000)18 b Fw(1\))1697 1130 y Fs(j)p Fw(\()p FD(x)1799 1096 y Fq(])1831 1130 y Fw(\))1863 1096 y Fo(0)1905 1130 y Fs(\000)g Fw(\()p FD(x)2067 1096 y Fq(?)2106 1130 y Fw(\))2138 1096 y Fo(0)2162 1130 y Fs(j)2185 1096 y Ft(2)1352 1351 y Fw(+)1427 1294 y(1)p 1427 1332 42 4 v 1427 1408 a(2)1478 1351 y(\()5 b(~)-47 b FD(x)1557 1316 y Fo(0)1600 1351 y Fs(\000)18 b Fw(\()p FD(x)1762 1316 y Fq(?)1801 1351 y Fw(\))1833 1316 y Fo(0)1856 1351 y Fw(\))h Fs(\001)g(M)p Fw(\()5 b(~)-47 b FD(x)2128 1316 y Fo(0)2170 1351 y Fs(\000)18 b Fw(\()p FD(x)2332 1316 y Fq(?)2371 1351 y Fw(\))2403 1316 y Fo(0)2426 1351 y Fw(\))1352 1550 y Fs(\000)1427 1494 y Fw(1)p 1427 1531 V 1427 1607 a(2)1478 1550 y(\(\()p FD(x)1589 1516 y Fq(])1621 1550 y Fw(\))1653 1516 y Fo(0)1696 1550 y Fs(\000)g Fw(\()p FD(x)1858 1516 y Fq(?)1896 1550 y Fw(\))1928 1516 y Fo(0)1952 1550 y Fw(\))h Fs(\001)f(M)p Fw(\(\()p FD(x)2255 1516 y Fq(])2287 1550 y Fw(\))2319 1516 y Fo(0)2361 1550 y Fs(\000)g Fw(\()p FD(x)2523 1516 y Fq(?)2562 1550 y Fw(\))2594 1516 y Fo(0)2618 1550 y Fw(\))g Fs(\000)g Fw(2)p FD(\021)1205 1754 y Fs(\025)1436 1697 y Fw(tr)13 b Fs(M)p 1362 1734 325 4 v 1362 1810 a Fw(4\()p FD(N)27 b Fs(\000)18 b Fw(1\))1697 1754 y Fs(j)p Fw(\()p FD(x)1799 1719 y Fq(])1831 1754 y Fw(\))1863 1719 y Fo(0)1905 1754 y Fs(\000)g Fw(\()p FD(x)2067 1719 y Fq(?)2106 1754 y Fw(\))2138 1719 y Fo(0)2162 1754 y Fs(j)2185 1719 y Ft(2)2240 1754 y Fs(\000)32 b Fw(const)14 b(\(1)k(+)g Fs(kMk)p Fw(\))c FD(\021)i(:)456 1956 y Fw(Since)31 b(tr)14 b Fs(M)29 b FD(>)g Fw(0,)j(this)g(sho)n(ws)e(in)i(particular)e (that)i Fs(j)p FD(x)2225 1917 y Fq(])2225 1980 y(i)2277 1956 y Fs(\000)21 b FD(x)2410 1926 y Fq(?)2410 1978 y(i)2449 1956 y Fs(j)29 b(\024)g FD(r)r(=)p Fw(2)i(for)g(1)e Fs(\024)g FD(i)g Fs(\024)g FD(N)h Fs(\000)21 b Fw(1,)456 2056 y(pro)n(vided)26 b FD(\021)31 b Fw(\(and)d(so)f FD(")p Fw(\))g(is)h(suitably)f(small,)h (and)f(th)n(us)h(completing)f(the)h(pro)r(of)f(of)h(\(8.7\).)555 2156 y(With)j(no)f(loss)g(of)g(generalit)n(y)-7 b(,)30 b(w)n(e)f(ma)n(y)h(no)n(w)g(assume)f(that)i FD(x)2553 2126 y Fq(])2612 2156 y Fw(=)c(0.)44 b(Notice)31 b(that,)g(with)456 2255 y(this)25 b(assumption,)h(b)n(y)f(\(8.7\),)g Ff(P)1477 2267 y Fq(t)1532 2255 y Fw(touc)n(hes)f Fs(f)p FD(u)1917 2267 y Fq(")1975 2255 y Fw(=)f(0)p Fs(g)h Fw(from)h(b)r(elo)n(w)g(in)h Fs(fj)p FD(x)p Fs(j)2828 2267 y Fo(1)2921 2255 y Fs(\024)d FD(r)r(=)p Fw(2)p Fs(g)p Fw(.)35 b(Then,)456 2355 y(either)28 b FD(u)740 2367 y Fq(")800 2355 y FD(<)c Fw(0)k(or)g FD(u)1110 2367 y Fq(")1170 2355 y FD(>)c Fw(0)k(b)r(elo)n(w)h Ff(P)1635 2367 y Fq(t)1664 2355 y Fw(:)38 b(w)n(e)29 b(will)g(consider)e(the)i(\014rst)f(p)r(ossibilit)n(y)-7 b(,)29 b(the)g(second)456 2455 y(one)e(b)r(eing)h(analogous.)34 b(Also,)27 b(since)h(0)22 b Fs(2)i Ff(P)1873 2467 y Fq(t)1902 2455 y Fw(,)j(the)h(equation)f(of)h Ff(P)2598 2467 y Fq(t)2655 2455 y Fw(tak)n(es)e(the)i(form)1483 2625 y FD(x)1530 2637 y Fq(N)1617 2625 y Fw(=)1714 2569 y(1)p 1714 2606 42 4 v 1714 2682 a(2)1780 2625 y FD(x)1827 2591 y Fo(0)1869 2625 y Fs(\001)1930 2604 y Fp(c)1910 2625 y Fs(M)p FD(x)2057 2591 y Fo(0)2099 2625 y Fw(+)18 b FD(V)38 b Fs(\001)18 b FD(x)2356 2591 y Fo(0)2394 2625 y FD(;)456 2785 y Fw(with)1467 2886 y Fs(j)p FD(V)h Fs(j)k(\024)g Fw(const)13 b Fs(k)1955 2865 y Fp(c)1936 2886 y Fs(M)o(k)h FD(r)25 b Fs(\024)e Fw(tr)2339 2865 y Fp(c)2320 2886 y Fs(M)456 3005 y Fw(pro)n(vided)456 3206 y(\(8.12\))970 b FD(r)40 b Fs(\024)d Fw(const)2033 3149 y(tr)2132 3128 y Fp(c)2112 3149 y Fs(M)p 2031 3186 183 4 v 2031 3280 a(k)2092 3259 y Fp(c)2073 3280 y Fs(M)o(k)2237 3206 y FD(:)456 3403 y Fw(W)-7 b(e)28 b(no)n(w)f(tak)n(e)f Ff(d)e FD(>)f Fw(0)k(appropriately)e(small)j(and)f(de\014ne)1704 3574 y FD(r)f Fw(:=)1977 3518 y(2)p Ff(d)p 1888 3555 262 4 v 1888 3571 a Fp(p)p 1971 3571 179 4 v 92 x Fw(tr)2069 3642 y Fp(c)2049 3663 y Fs(M)2173 3574 y FD(:)456 3796 y Fw(Notice)d(that)h(if)g(\000)f(:=)g Ff(P)1215 3808 y Fq(t)1268 3796 y Fw(and)g FD(M)32 b Fw(:=)1668 3775 y Fp(c)1648 3796 y Fs(M)p Fw(,)25 b(the)f(h)n(yp)r(otheses)f(of)g(Prop) r(osition)f(7.1)h(are)g(ful\014lled.)456 3895 y(Ho)n(w)n(ev)n(er,)37 b(the)g(result)g(of)f(Prop)r(osition)g(7.1)g(is)g(in)h(con)n (tradiction)f(with)h(the)g(fact)g(that)g Ff(P)3415 3907 y Fq(t)456 3995 y Fw(touc)n(hes)27 b Fs(f)p FD(u)844 4007 y Fq(")901 3995 y Fw(=)c(0)p Fs(g)k Fw(from)g(b)r(elo)n(w)g(in)h Fs(fj)p FD(x)p Fs(j)1763 4007 y Fo(1)1856 3995 y Fs(\024)23 b FD(r)r(=)p Fw(2)p Fs(g)p Fw(.)555 4095 y(This)36 b(con)n(tradiction,) g(caused)f(b)n(y)h(the)g(assumption)f(that)h(tr)13 b Fs(M)36 b FD(>)g Fw(0,)i(sho)n(ws)c(that)i(the)456 4194 y(mean)27 b(curv)-5 b(ature)27 b(of)g FD(P)40 b Fw(m)n(ust)28 b(b)r(e)g(non-p)r(ositiv)n(e.)555 4294 y(This)g(completes)f(the)h(pro)r (of)f(of)h(Theorem)e(2.1.)613 4468 y(9.)41 b Fv(Comments)32 b(on)f(the)h(unif)n(orm)g(limit)f(pr)n(oper)-6 b(ty)32 b(\(2.1\))g(and)f(pr)n(oof)g(of)1741 4568 y(Theorem)h(2.3)555 4717 y Fw(W)-7 b(e)37 b(w)n(ould)g(lik)n(e)f(to)g(sho)n(w)g(that)h (uniform)g(limit)g(conditions)f(of)h(the)g(t)n(yp)r(e)g(in)g(\(2.1\))f (are)456 4817 y(someho)n(w)c(natural.)56 b(First)34 b(of)g(all,)h(a)f (condition)f(of)h(this)h(kind)f(is)g(necessary)e(to)i(a)n(v)n(oid,)g (for)456 4917 y(instance,)g(p)r(erio)r(dic)f(one-dimensional)f (solutions.)52 b(T)-7 b(o)33 b(motiv)-5 b(ate)33 b(the)h(deca)n(y)e (prop)r(ert)n(y)g(w)n(e)456 5016 y(assume,)27 b(ev)n(en)h(if)h(w)n(e)f (do)g(not)g(need)h(the)g(follo)n(wing)e(result)h(in)g(the)h(presen)n(t) f(pap)r(er,)g(w)n(e)g(think)456 5116 y(it)e(is)g(con)n(v)n(enien)n(t)e (to)i(p)r(oin)n(t)g(out)g(that)g(global)f(solutions)g(of)h(\(1.5\))f (attaining)h(a)f(uniform)h(limit)456 5216 y(alw)n(a)n(ys)f(p)r(ossess)i (an)g(exp)r(onen)n(tial)g(deca)n(y:)p eop %%Page: 34 34 34 33 bop 456 251 a Ft(34)648 b(BERARDINO)23 b(SCIUNZI)g(AND)f(ENRICO)h (V)-7 b(ALDINOCI)456 450 y FE(Prop)s(osition)30 b(9.1.)40 b Fg(L)l(et)29 b FD(u)23 b Fs(2)g FD(W)1544 410 y Ft(1)p Fq(;p)1532 475 y(loc)1635 450 y Fw(\()p Fr(R)1721 420 y Fq(N)1790 450 y Fw(\))30 b Fg(b)l(e)g(a)g(we)l(ak)h(solution)f(of)48 b Fw(\(1.5\))29 b Fg(such)h(that)456 632 y Fw(\(9.1\))1307 519 y Fp(Z)1353 708 y Fk(R)1400 691 y Fn(N)1466 632 y Fs(jr)p FD(u)p Fs(j)1629 598 y Fq(p)1681 632 y FD(dx)20 b Fw(+)1874 519 y Fp(Z)1920 708 y Fk(R)1967 691 y Fn(N)2033 632 y FD(h)2081 644 y Ft(0)2118 632 y Fw(\()p FD(u)p Fw(\))14 b FD(dx)24 b(<)e Fw(+)p Fs(1)456 814 y Fg(Assume)29 b FD(h)814 784 y Fo(0)814 835 y Ft(0)880 814 y Fg(nonde)l(cr)l(e)l (asing)i(in)f Fw([)p Fs(\000)p Fw(1)p FD(;)14 b Fs(\000)p Fw(1)i(+)i FD(\022)1928 784 y Fo(\003)1967 814 y Fw(])29 b Fg(and)i(supp)l(ose)f Fs(j)p FD(u)p Fs(j)23 b FD(<)f Fw(1)30 b Fg(and)986 950 y FD(u)23 b(<)g Fs(\000)p Fw(1)17 b(+)h FD(\022)1393 915 y Fo(\003)1601 950 y Fg(for)31 b(any)85 b FD(x)170 b Fg(such)30 b(that)85 b FD(x)2626 962 y Fq(N)2713 950 y Fs(\024)22 b FD(K)e(:)456 1085 y Fg(Then,)31 b(if)h(we)e(set)g FD(\014)t Fw(\()p FD(x)p Fw(\))40 b(:=)e Fs(\000)p Fw(1)18 b(+)g FD(k)1612 1097 y Ft(1)1663 1085 y FD(e)1702 1055 y Fq(k)1737 1063 y Fd(2)1770 1055 y Fq(x)1808 1063 y Fn(N)1865 1085 y Fg(,)31 b(we)g(\014nd)f(p)l(ositive)i(c)l(onstants)d FD(k)2919 1097 y Ft(1)2987 1085 y Fg(and)i FD(k)3192 1097 y Ft(2)3229 1085 y Fg(,)g(such)456 1185 y(that)936 1285 y FD(u)p Fw(\()p FD(x)p Fw(\))24 b Fs(\024)e FD(\014)t Fw(\()p FD(x)p Fw(\))341 b Fg(for)31 b(any)85 b FD(x)99 b Fg(such)30 b(that)99 b FD(x)2677 1297 y Fq(N)2763 1285 y Fs(\024)23 b FD(K)c(:)456 1436 y Fg(Pr)l(o)l(of.)43 b Fw(Let)27 b(us)h(start)f(with)h(the)g(follo)n(wing)f(easy)f(calculations)1262 1583 y(\001)1331 1595 y Fq(p)1369 1583 y FD(\014)i Fw(=)22 b(\()p FD(p)d Fs(\000)f Fw(1\))p FD(k)1827 1540 y Ft(2\()p Fq(p)p Fo(\000)p Ft(1\))1824 1605 y(2)2035 1583 y Fw(\()p FD(k)2110 1595 y Ft(1)2162 1583 y FD(e)2201 1549 y Fq(k)2236 1557 y Fd(2)2268 1549 y Fq(x)2306 1557 y Fn(N)2363 1583 y Fw(\))2395 1549 y Fq(p)p Fo(\000)p Ft(1)1444 1769 y Fw(=)k(\()p FD(p)d Fs(\000)f Fw(1\))p FD(k)1827 1726 y Ft(2\()p Fq(p)p Fo(\000)p Ft(1\))1824 1791 y(2)2035 1769 y Fw(\(1)g(+)g FD(\014)t Fw(\))2293 1735 y Fq(p)p Fo(\000)p Ft(1)2427 1713 y FD(h)2475 1683 y Fo(0)2475 1733 y Ft(0)2512 1713 y Fw(\()p FD(\014)t Fw(\))p 2427 1750 201 4 v 2427 1826 a FD(h)2475 1797 y Fo(0)2475 1848 y Ft(0)2512 1826 y Fw(\()p FD(\014)t Fw(\))1444 1948 y Fs(\024)k FD(h)1579 1913 y Fo(0)1579 1968 y Ft(0)1617 1948 y Fw(\()p FD(\014)t Fw(\))456 1760 y(\(9.2\))456 2083 y(for)27 b FD(k)626 2095 y Ft(2)691 2083 y Fw(su\016cien)n(tly)g (small.)456 2183 y(Moreo)n(v)n(er,)e(w)n(e)i(tak)n(e)g FD(k)1189 2195 y Ft(1)1254 2183 y Fw(suc)n(h)g(that)h FD(\014)t Fw(\()p FD(x)p Fw(\))c(=)f Fs(\000)p Fw(1)17 b(+)i FD(\022)2144 2153 y Fo(\003)2210 2183 y Fw(on)27 b Fs(f)p FD(x)2414 2195 y Fq(N)2500 2183 y Fw(=)c FD(K)6 b Fs(g)p Fw(,)26 b(hence)456 2318 y(\(9.3\))901 b FD(u)23 b(<)f(\014)88 b Fw(on)82 b Fs(f)p FD(x)2080 2330 y Fq(N)2166 2318 y Fw(=)23 b FD(K)6 b Fs(g)456 2454 y Fw(By)28 b(\(1.5\))f(and)h(b) n(y)g(\(9.2\))g(w)n(e)f(get)456 2632 y(\(9.4\))839 2519 y Fp(Z)885 2707 y Fk(R)932 2691 y Fn(N)985 2632 y Fw(\()p Fs(jr)p FD(u)p Fs(j)1180 2597 y Fq(p)p Fo(\000)p Ft(2)1303 2632 y Fs(r)p FD(u)18 b Fs(\000)g(jr)p FD(\014)t Fs(j)1687 2597 y Fq(p)p Fo(\000)p Ft(2)1811 2632 y Fs(r)p FD(\014)t Fw(\))i Fs(\001)e(r)p FD(')c(dx)24 b Fs(\024)2363 2519 y Fp(Z)2409 2707 y Fk(R)2456 2691 y Fn(N)2508 2632 y Fw(\()p FD(h)2588 2597 y Fo(0)2588 2652 y Ft(0)2626 2632 y Fw(\()p FD(\014)t Fw(\))19 b Fs(\000)f FD(h)2891 2597 y Fo(0)2891 2652 y Ft(0)2928 2632 y Fw(\()p FD(u)p Fw(\)\))p FD(')c(dx)456 2821 y Fw(for)27 b(an)n(y)f(nonnegativ)n(e)h FD(')c Fs(2)h FD(C)1422 2791 y Fo(1)1416 2842 y Ft(0)1492 2821 y Fw(\()p Fr(R)1578 2791 y Fq(N)1647 2821 y Fw(\).)456 2921 y(W)-7 b(e)28 b(consider)e(no)n(w)h FD(w)j Fw(de\014ned)e(as)f (follo)n(ws)456 3110 y(\(9.5\))674 b FD(w)40 b Fw(:=)1524 2968 y Fp(\()1591 3054 y Fw(\()p FD(u)18 b Fs(\000)g FD(\014)t Fw(\))1855 3023 y Ft(+)2160 3054 y Fw(if)84 b FD(x)2339 3066 y Fq(N)2425 3054 y Fs(\024)23 b FD(K)1591 3173 y Fw(0)527 b(if)84 b FD(x)2339 3185 y Fq(N)2425 3173 y FD(>)23 b(K)456 3320 y Fw(Since)38 b FD(u)i(<)h(\014)i Fw(on)38 b Fs(f)p FD(x)1182 3332 y Fq(N)1285 3320 y Fw(=)j FD(K)6 b Fs(g)37 b Fw(w)n(e)h(ha)n(v)n(e)f(that)i(supp)14 b FD(w)43 b Fs(\032)e(f)p FD(x)2555 3332 y Fq(N)2659 3320 y FD(<)f(K)6 b Fs(g)p Fw(.)68 b(Moreo)n(v)n(er,)38 b(in)456 3420 y(ligh)n(t)e(of)43 b(\(1.2\))37 b(and)g(\(9.1\))o(,)j(w)n (e)c(get)h(that)g(1)24 b(+)g FD(u)38 b Fs(2)h FD(L)2238 3390 y Fq(p)2276 3420 y Fw(\()p Fr(R)2362 3390 y Fq(N)2431 3420 y Fw(\))e(and)g Fs(jr)p FD(u)p Fs(j)h(2)h FD(L)3023 3390 y Fq(p)3061 3420 y Fw(\()p Fr(R)3147 3390 y Fq(N)3216 3420 y Fw(\),)h(and)456 3520 y(th)n(us)32 b FD(w)j Fs(2)c FD(W)913 3489 y Ft(1)p Fq(;p)1005 3520 y Fw(\()p Fr(R)1091 3489 y Fq(N)1160 3520 y Fw(\).)52 b(Therefore,)33 b(there)f(exists)h(a) f(sequence)g(of)h(nonnegativ)n(e)e(functions)456 3619 y FD(')510 3631 y Fq(n)587 3619 y Fs(2)h FD(C)739 3589 y Fo(1)733 3640 y Ft(0)810 3619 y Fw(\()p Fr(R)896 3589 y Fq(N)965 3619 y Fw(\))i(con)n(v)n(erging)c(to)j FD(w)j Fw(in)d FD(W)1843 3589 y Ft(1)p Fq(;p)1934 3619 y Fw(\()p Fr(R)2020 3589 y Fq(N)2089 3619 y Fw(\).)53 b(W)-7 b(e)34 b(ma)n(y)e(therefore)g(tak)n(e)g FD(')3124 3631 y Fq(n)3203 3619 y Fw(as)g(test)456 3730 y(function)k(in)g(\(9.4\))o(.)61 b(Moreo)n(v)n(er,)35 b(since)g(\()p FD(h)1837 3700 y Fo(0)1837 3751 y Ft(0)1875 3730 y Fw(\()p FD(\014)t Fw(\))25 b Fs(\000)e FD(h)2151 3700 y Fo(0)2151 3751 y Ft(0)2188 3730 y Fw(\()p FD(u)p Fw(\)\))37 b Fs(2)f FD(L)2564 3666 y Fn(p)p 2527 3680 104 3 v 2527 3713 a(p)p Fm(\000)p Fd(1)2645 3730 y Fw(\()p Fr(R)2731 3700 y Fq(N)2800 3730 y Fw(\))g(and)g(\()p Fs(jr)p FD(u)p Fs(j)3233 3700 y Fq(p)p Fo(\000)p Ft(1)3380 3730 y Fs(\000)456 3841 y(jr)p FD(\014)t Fs(j)622 3811 y Fq(p)p Fo(\000)p Ft(1)746 3841 y Fw(\))d Fs(2)h FD(L)1003 3777 y Fn(p)p 966 3791 V 966 3824 a(p)p Fm(\000)p Fd(1)1084 3841 y Fw(\()p Fr(R)1170 3811 y Fq(N)1239 3841 y Fw(\),)i(b)n(y)d(H\177)-42 b(older's)33 b(inequalit)n(y)h(and)f(letting)h FD(n)f Fs(\000)-14 b(!)33 b Fw(+)p Fs(1)p Fw(,)i(w)n(e)e(deduce)456 3940 y(that)456 4098 y(\(9.6\))829 3984 y Fp(Z)875 4173 y Fk(R)922 4157 y Fn(N)974 4098 y Fw(\()p Fs(jr)p FD(u)p Fs(j)1169 4063 y Fq(p)p Fo(\000)p Ft(2)1293 4098 y Fs(r)p FD(u)18 b Fs(\000)g(jr)p FD(\014)t Fs(j)1677 4063 y Fq(p)p Fo(\000)p Ft(2)1801 4098 y Fs(r)p FD(\014)t Fw(\))p Fs(r)p FD(w)27 b Fs(\024)2195 3984 y Fp(Z)2241 4173 y Fk(R)2288 4157 y Fn(N)2341 4098 y Fw(\()p FD(h)2421 4063 y Fo(0)2421 4118 y Ft(0)2458 4098 y Fw(\()p FD(\014)t Fw(\))20 b Fs(\000)e FD(h)2724 4063 y Fo(0)2724 4118 y Ft(0)2761 4098 y Fw(\()p FD(u)p Fw(\)\))p FD(w)f(dx)456 4275 y Fw(Therefore,)26 b(b)n(y)h(construction,)1009 4335 y Fp(Z)1055 4524 y Fo(f)p Fq(u)p Fo(\025)p Fq(\014)s Fo(g\\f)p Fq(x)1372 4532 y Fn(N)1425 4524 y Fq()456 1403 y(\014)t Fs(g)16 b(\\)h(f)p FD(x)726 1415 y Fq(N)812 1403 y FD(<)22 b(K)6 b Fs(g)26 b Fw(w)n(ere)g(non-empt) n(y)-7 b(,)26 b FD(c)h Fw(is)f(actually)g(strictly)h(p)r(ositiv)n(e,)f (and)h(a)f(con)n(tradiction)456 1502 y(arises)g(from)h(\(9.3\).)2307 b Fc(\003)555 1656 y Fw(W)-7 b(e)32 b(no)n(w)e(sho)n(w)h(that)g(Class)f (A)i(minimizers)f(of)g Fs(F)39 b Fw(ful\014ll)32 b(\(2.1\),)g(pro)n (vided)f FD(h)3045 1668 y Ft(0)3113 1656 y Fw(is)g(con)n(v)n(ex)456 1756 y(near)26 b Fs(\006)p Fw(1.)456 1855 y(Here,)h(it)g(is)g(crucial)g (the)g(use)g(of)h(some)e(densit)n(y)h(estimates)g(for)g(minimizers)g (of)g Fs(F)8 b Fw(,)28 b(pro)n(v)n(ed)d(in)456 1955 y([15)o(].)37 b(In)27 b(our)g(setting,)h(w)n(e)f(ma)n(y)g(state)g(this)h(result)g(as) f(follo)n(ws:)456 2074 y FE(Theorem)41 b(9.2.)k Fg(L)l(et)38 b FD(u)g Fg(b)l(e)h(a)f(Class)i(A)e(minimizer)h(of)g Fs(F)8 b Fg(.)65 b(Supp)l(ose)38 b(that)h(the)f(hyp)l(othe-)456 2173 y(sis)45 b Fw(\(1.2\))37 b Fg(and)47 b Fw(\(1.3\))37 b Fg(ar)l(e)i(ful\014l)t(le)l(d.)64 b(Assume)37 b(also)i(that)e FD(h)2408 2143 y Fo(0)2408 2194 y Ft(0)2483 2173 y Fg(is)i(monotone)f (incr)l(e)l(asing)g(in)456 2273 y Fw(\()p Fs(\000)p Fw(1)p FD(;)14 b Fs(\000)p Fw(1)i(+)i FD(\022)879 2243 y Fo(\003)918 2273 y Fw(\))h Fs([)f Fw(\(1)h Fs(\000)f FD(\022)1259 2243 y Fo(\003)1297 2273 y FD(;)c Fw(1\))p Fg(.)555 2373 y(Fix)31 b FD(\017)25 b Fs(2)h Fw(\(0)p FD(;)14 b Fw(1\))30 b Fg(and)i(supp)l(ose)f FD(x)26 b Fs(2)g(f\000)p Fw(1)17 b(+)i FD(\017)25 b(<)g(u)g(<)g Fw(1)19 b Fs(\000)f FD(\017)p Fs(g)p Fg(.)42 b(Then,)32 b(ther)l(e)f(exist)g(p)l(ositive)456 2472 y FD(r)493 2484 y Ft(0)530 2472 y Fw(\()p FD(\017)p Fw(\))f Fg(and)h FD(c)p Fw(\()p FD(\017)p Fw(\))e Fg(such)h(that)456 2613 y Fw(\(9.7\))41 b Ff(L)p Fw(\()p FD(B)818 2625 y Fq(r)855 2613 y Fw(\()p FD(x)p Fw(\))18 b Fs(\\)f(f)p FD(u)22 b(>)g Fs(\000)p Fw(1)16 b(+)g FD(\017)p Fs(g)p Fw(\))23 b Fs(\025)f FD(c)p Fw(\()p FD(\017)p Fw(\))p FD(r)1850 2579 y Fq(N)1999 2613 y FD(and)85 b Ff(L)p Fw(\()p FD(B)2371 2625 y Fq(r)2408 2613 y Fw(\()p FD(x)p Fw(\))18 b Fs(\\)f(f)p FD(u)22 b(<)h Fw(1)16 b Fs(\000)g FD(\017)p Fs(g)p Fw(\))22 b Fs(\025)h FD(c)p Fw(\()p FD(\017)p Fw(\))p FD(r)3339 2579 y Fq(N)456 2751 y Fg(for)30 b(any)g FD(r)c Fs(\025)d FD(r)935 2763 y Ft(0)972 2751 y Fw(\()p FD(\017)p Fw(\))p Fg(,)31 b(b)l(eing)f Ff(L)g Fg(the)g FD(N)9 b Fg(-dimensional)31 b(L)l(eb)l(esgue)f(me)l(asur)l(e.) 456 2869 y FE(Corollary)37 b(9.3.)42 b Fg(If)33 b FD(h)1216 2839 y Fo(0)1216 2890 y Ft(0)1286 2869 y Fg(is)g(monotone)g(incr)l(e)l (asing)h(in)f Fw(\()p Fs(\000)p Fw(1)p FD(;)14 b Fs(\000)p Fw(1)19 b(+)h FD(\022)2693 2839 y Fo(\003)2731 2869 y Fw(\))h Fs([)h Fw(\(1)e Fs(\000)g FD(\022)3081 2839 y Fo(\003)3120 2869 y FD(;)14 b Fw(1\))p Fg(,)34 b(then)456 2969 y(any)c(Class)g(A)g(minimizer)g(of)h Fs(F)38 b Fg(satis\014es)30 b(\(2.1\).)456 3123 y(Pr)l(o)l(of.)43 b Fw(Supp)r(ose)36 b Fs(f)p FD(u)g Fw(=)h(0)p Fs(g)23 b(\\)h(fj)p FD(x)g Fs(\000)g Fw(\()p FD(!)j Fs(\001)d FD(x)p Fw(\))p FD(!)s Fs(j)38 b(\024)e FD(l)r Fs(g)g(\032)h(f)p FD(!)26 b Fs(\001)f FD(x)37 b Fs(\025)g(\000)p FD(l)r(=)p Fw(10)p Fs(g)c Fw(and)j(let)g FD(x)i Fw(=)456 3222 y(\()p FD(x)535 3192 y Fo(0)559 3222 y FD(;)14 b(x)643 3234 y Fq(N)706 3222 y Fw(\))23 b Fs(2)h Fr(R)894 3192 y Fq(N)6 b Fo(\000)p Ft(1)1065 3222 y Fs(\002)17 b Fr(R)33 b Fw(satisfying)26 b FD(!)20 b Fs(\001)d FD(x)23 b(<)g Fs(\000)p Fw(3)p FD(l)r(=)p Fw(5)i(and)h Fs(j)p FD(x)18 b Fs(\000)f Fw(\()p FD(!)j Fs(\001)d FD(x)p Fw(\))p FD(!)s Fs(j)24 b(\024)e FD(l)r(=)p Fw(2.)35 b(Fixed)27 b FD(\017)c(>)g Fw(0)456 3322 y(assume)j(b)n(y)i(con)n(tradiction)e FD(u)p Fw(\()p FD(x)p Fw(\))e FD(>)e Fs(\000)p Fw(1)c(+)g FD(\017)456 3422 y Fw(By)29 b(Theorem)f(9.2,)i(with)g FD(l)h Fw(su\016cien)n(tly)e (large,)g(so)f(that)i FD(l)r(=)p Fw(2)25 b Fs(\025)h FD(r)2542 3434 y Ft(0)2579 3422 y Fw(\()p FD(\017)p Fw(\),)31 b(w)n(e)e(get)g(that)h FD(B)3240 3437 y Fq(l=)p Ft(2)3332 3422 y Fw(\()p FD(x)p Fw(\))456 3521 y(con)n(tains)d(p)r(oin)n(ts)h (where)f FD(u)d Fw(=)f(0.)38 b(This)28 b(is)g(a)g(clear)f(con)n (tradiction)g(with)i(the)f(assumption)g(on)456 3621 y(the)34 b(set)g FD(u)g Fw(=)f(0)h(since)g FD(B)1270 3636 y Fq(l=)p Ft(2)1363 3621 y Fw(\()p FD(x)p Fw(\))h Fs(\032)e(fj)p FD(x)23 b Fs(\000)g Fw(\()p FD(!)i Fs(\001)e FD(x)p Fw(\))p FD(!)s Fs(j)35 b(\024)e FD(l)r Fs(g)g Fw(and)h Fs(f)p FD(!)25 b Fs(\001)e FD(x)35 b(<)e Fs(\000)p FD(l)r(=)p Fw(10)p Fs(g)f Fw(for)h(an)n(y)456 3721 y FD(x)23 b Fs(2)h FD(B)668 3736 y Fq(l=)p Ft(2)760 3721 y Fw(\()p FD(x)p Fw(\).)456 3821 y(This)i(con)n(tradiction)g(sho)n(ws)g(that)h FD(u)p Fw(\()p FD(x)p Fw(\))c Fs(\024)g(\000)p Fw(1)16 b(+)g FD(\017)27 b Fw(for)f FD(l)j Fw(large)c(suc)n(h)i(that)g FD(l)r(=)p Fw(2)21 b Fs(\025)i FD(r)3106 3833 y Ft(0)3143 3821 y Fw(\()p FD(\017)p Fw(\).)37 b(The)456 3921 y(second)27 b(part)g(of)34 b(\(2.1\))27 b(follo)n(ws)g(with)h(the)g(same)f(argumen) n(ts.)963 b Fc(\003)555 4074 y Fw(The)28 b(pro)r(of)f(of)g(Theorem)g (2.3)g(is)g(no)n(w)g(completed)h(via)f(Corollary)e(9.3.)1708 4253 y Fv(References)491 4385 y FC([1])35 b(G.)20 b(Alb)r(erti,)f(L.)h (Am)n(brosio,)e(X.)i(Cabr)n(\023)-33 b(e)20 b Fb(On)i(a)g 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