On the numerical approximation of first order Hamilton Jacobi equations
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. On the numerical approximation of first order Hamilton Jacobi equations R. Abgrall. N° 6054 Décembre 2006. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6054--FR+ENG. Thème NUM.
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(14) ! "# "!$%"&'")(* +,-$+ ". -$/0-$. H(x, u, Du) = 0 x ∈ Ω ⊂ Rd u=ϕ x ∈ ∂Ω. cp[TVCEG54oeYg/C`Pk[]cv|EiGWF. ∂u + H(x, u(x), Du) = 0 x ∈ Rd , t > 0 ∂t u(x, 0) = u0 (x).. 6ÆX FoeXQP&kE[Vcp|iGWF Rcp)kECQPQR]\_goeiS\jX`T]GW[]G>RUT>ªpcpXG XEGG>qERT]cgcpF kEYSTVG T]CGJRcviYET]\jcpXce«RYg/C opX GW^`YoTV\cvX« LXEG ce T]CG"R]\jFkiG>RUTG#lEoeF kEijGWR\_RT]CEGg#cvF kEYSTVoeT]\jcpXceoqS\_RUT/oeXgGJYEXgÀT]\jcpX˪EF cp[VGLR]cpkECE\_RT]\_goTVGWq¢G#lEoeF kEijGWRg#cvXR]\jRT\jX GopiYoeT]\jXETVCEGLop[][V\opiET]\jF GJcpËo"[]cvX`T opG[VcpX`T>8ª 7opF Gf[VcpX`TWª`G#T/g 8\jX¢o&XEcpXCEcvF cppGWXEGcvYR8F GWqE\jo8NS\F \jijop[ kE[Vcp|EijGF oei_RcG#lS\_RUTf\jXg#cpX`TV[]cviTVCEGcv[]PvªQT]CEGW[]F cSqSPQXopF\_gRWªSG#T/ge GJ \jii)g#cvXg#GWX`T][/oT]GJcpXTVCEGLXQYEF GW[]\_gopi@opkEkE[VclQ\jFoTV\cvXcp«T]CGWR]GJGW^`YoeT]\jcpXRcpXg#cpXEcp[VF opiETV[]\_oeXEvYEi_oe[bTUPQk@G F GWR]CEGWRW&BCE\_R"\_RÙoF cp[VG.vGXEGW[VopiR]\T]YoeT]\jcpXT]CopXT]CG RTVopXqEop[Vq 4oe[]T]GWR]\_oeXF GWR]CEG>RL CG[VG&TVCE\jR"k[]cv|EiGWF ovR g#cvXR]\jqSGW[]G>q cp[V\v\XoeijiPvª|EYSTRT]\jiiijGWRVRfpGWXEG[/oeiËT]CopXT]CGgWopR]GcebXEcvXgcpXScv[]FoeiËFG>RCGWRWÙJcGWpG[>ªG.R 9pGTVg/C CEc T]cGl`TVGXqTVCEG&RVg/CEGF GWRfGqSG>R]g[]\j|@GCEG[VGÙT]cTVCEGF cvRTfpGXG[/oei«govRGv: 6ÆXqEGGWqTVCEG&R]\mTVYoTV\cvXGg#cvXR]\jqSGW[ \_RLopX6\jXvTVG[VF GWqS\_oTVG&cpXGpª<\mTÙ\jRÙvGXEGW[VopiËGXcpYEvCT]c|Gcp|Eij\jpGWqT]c¢\jFoev\XEGRcviYET]\jcpXRJT]CoeTÙop[]G.pGXG[V\jgGWXEcpYpC oeXq¢XEcpT T]cQcRk@GWg\m@gRcT]CoT T]CGRUTV[]Yg#T]YE[VG"ceT]CEGF GWR]CqScQGWRXEcpT kEijoPo&TVcQc[]\jp\_q¢[VcpijGp 6ÆXyT]CEG¢[/RUTkop[T>ªG!^`YE\_;g 9`ijP6[VGWgWoeiji8T]CG!XEceTV\cvX ceQ\jRVg#c`R\TUPR]cpijYST]\jcpXycv[&J6GW^`YoTV\cvXRWªGW\mTVCEG[cv[T]CG 4oeYg/C`PkE[Vcp|EijGF cp[cp[T]CEGÙRUTVGWovqSP cpXEGf \T]CzJ\j[V\jg/CEijG#T|cvYEXqEop[]P g#cvXqS\T]\jcpXRWBCEGX«ªQ\jXTUckoe[]T]\_g#Yijop[govRG>R GÙ[VGWgWoeiji@T]CEGGlSovgÀTfR]cpijYST]\jcpX«8BCEG"XEGlQTJRG>gÀT]\jcpX\_R qSGvceT]G>qTVc T]CEG"XQYEF G[V\jgWoei«opkEkE[VclQ\jFoTV\cvX!ceT]CE5G 4opYg/CQP kE[Vcp|EijGF < 6ÆXTVCEGJTVCE\j[Vq!R]GWg#T]\jcpX!GLqE\jRVg#YRVRT]CEG"oekEk[]clS\jF oeT]\jcpX ceËT]CEGÙzJ\j[]\_g/CEijG#TkE[Vcp|EijGF BCEGJcvYE[]T]C¢R]GWg#T]\jcpX g#cvXR]\jqSGW[VRopGWXEG[/oei@cv[]F.YEijoeT]\jcpX¢cp[ CE\jpCcp[/qSG[ qS\_RVg#[VG#T]\_RVoT]\jcpXËBCGÙ|[]\_qSpG"|@G#TUGGW=X 4op[TVGWR]\jopX¢F GWR]CEGWR opXq XEcvX g#cpXEcp[VF opiF GWR]CEG>R\_R.R 9pGTVg/CEG>q\jX TVCEG!EÚTVCR]GWg#T]\jcpX«BCEGijovRUTRG>gÀT]\jcpX¡\jR.qSGvceT]G>qyT]cyRcvFG¢X`YFGW[]\_gopi oekkEi\_goeT]\jcpXRW hJR&G¢CopG!opi[VGWovqSP6RVoe\_q)ªcpYE[k@cp\jX`T&cefQ\jG \jR^`YE\T]G!|E\_opR]GWq« BCG[VG¢oe[VG|opR]\_goeijijPTUcg#i_opRVRG>Rcefoek0> kE[VclS\FoTV\cvXT]G>g/CEXE\_^`YEGWRW8BCEGÙ[VRTcpXEGÙT][V\jGWRT]c qS\j[]G>gÀT]ijP!YR]GLT]CEG"XEcpT]\jcpX¢cpQ\_R]gcvR]\mTUPR]cpijYST]\jcpX«ªER]GG"R]GWgÀTV\cvX } ª T]C\jRcpY[k@cp\jX`TcpËQ\GW< 6ÆX!T]CG"R]GWgcpXq¢g#i_opRVRcpËF G#T]CcQqRªScvXEGJTV[]\jGWRT]cG#lSkEijcp\TT]CEGÙcp[VFoei@i\j0X 9|@G#TUGGWXR]cpF G R]PQRT]GWFRce8g#cpXRGW[]oTV\cvX¢i_ofRoeXqT]CG"LGW^`YoeT]\jcpXRW8BCEGij\0X 9¢\jRTVCoTf\cpXEGqS@\ ?@GW[]GWX`T]\_oT]G>RT]CEGG>^`YoTV\cvX. \T]C []G>Rk@GWg#TT]c x oeXq y ªqSGWXEceTV\X. ∂u + H(Du) = 0 ∂t pi =. ∂u ∂xi. oeXq. p = Du. ªEG"CopG. ∂H ∂pi + (p) = 0. ∂t ∂x. BB CE\_R\_R8TVCEGJk@cp\jX`Tcp«`\jG cp)T]CEGLkoek@G[/RT]CoTGlQT]GXq XE\T]GLpcpijYEF Gfcv[zL\jRVg#cvXvTV\XQYEcvYRA"opiGW[9Q\jXF G#T]CcQqRªQRGWG ~ C)cp[fopX G#lEoeF kEijGp F#G H , 0 3 #(+IF#((/KJ (# ¢#( D E ËG G"gcpXR]\_qSG[T]C5G 4oeYg/CQPkE[Vcp|EijGFMLXq u ∈ C (Ω) ªET]CEGR]kopgGÙcpgcpX`T]\jXQYEcpYRYEXgÀT]\jcpXcpXT]CGcpk@GX RYE|RGT ªR]Yg/C¢TVCoT Ω⊂R U~ H(x, u, Du) = 0 x ∈ Ω ⊂ R u=g x ∈ ∂Ω \jXTVCEGQ\jRVg#c`R\TUP!R]GXR]Gp< 6ÆX ~ Àª (x, s, p) ∈ Ω × R × R 7→ H(x, s, p) \_R YEXE\cp[VF iP¢gcpX`T]\jXQYEcpYR 0. d. d. d. Ê ÀÖÀ×/Ð. NON P0Q.
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(16) Ecv[oeXQPYEXgÀTV\cv X ª@G.gcpXR]\_qSG[LT]CEGYEkEk@G[R]GF \ Qg#cvX`T]\jX`YcpYR Y« R g LoeXqijcGW[LR]GF \ Sg#cvXvTV\XQYEcvYR Ái° RW gfGWX`vGijcpk@GWRJce z \mTVC z[VGWR]kG>gÀTT]c!opii<TVCEGoe[V\jop|EiG>RBCEGWPoe[VG"qSG#XGWq|QP oeXq z (x) = lim inf z(y). z (x) = lim sup z(y) Ecviijc \jXE B } CΪEG"\X`T][VcSqSYgGLTVCEG"YEXg#T]\jcpX G ∗. ∗. x→y. BCEGgcpF kEYST/oT]\jcpXce G oeXq. G(x, s, p) =. . x→y. H(x, s, p) x ∈ Ω s − g(x) x ∈ ∂Ω.. \jRG>opR]P¢oeXqGCopGL. \ x ∈ Ω \m x ∈ ∂Ω °} \m x ∈ ∂Ω h icSgopiijP |cvYEXqSG>qY« RW g&YEXg#T]\jcpX qSGXEG>q6cvX \_RÙoQ\jRVg#c`R\TUP RYE0| >ÎR]cpijYST]\jcpX6ce ~ J\moeXqcvXEijP\mUª<cv[ oeXQP φ ∈ C (Ω) ªE\m x ∈ Ω \jRJo.ijcSgopi)F uoelS\F.YEFce u −Ω φ ªST]CEGWX ∗. 1. G∗. G∗ (x, s, p) = G∗ (x, s, p) = H(x, s, p) G∗ (x, s, p) = min(H(x, s, p), s − g(x)) ∗ G (x, s, p) = max(H(x, s, p), s − g(x)). 1 . 0. NQ\jF \i_oe[ViPvª ª`o"ijcSgopiijP.|@cpYXqSGWq«ªviA RW geYEXgÀTV\cvX¢qSGXEGWq cvX \jRo"`\_RVg#cvR]\TUP.R]YEkGW[>ÆR]cpijYST]\jcpX ce ~ 8\«oeXq cvXEiP \UªScp[fopX`P uφ ∈ C (Ω) ªE\ x ∈ Ω \_R oicSgopi)F \XE\jF&YF*ce u −Ω φ ªST]CEGWX x G (x , u(x ), Dφ(x )) ≥ 0. h Q\jRVg#c`R\TUPRcviYSTV\cvX6\_R"R\jF&YEiTVopXEGcvYR]iPoRYE| opXqoR]YEkGW[ >ÆR]cpijYST]\jcpXcp ~ BCE\jR"gWoeX6|@G pGWXEG[/oeij\ WGWq TVc ceTVCEG[TUPQk@GWRcp|@cpYEXqEoe[VP¢g#cvXqS\T]\jcpXR R]Yg/C opR HJGYEFoeXX«ªEG#T/ge 6ÆXT]CEGgWopR]GÙcpËTVCE5 G 4oeYg/CQP!k[]cv|EiGWF ª ∂u °w + H(x, u(x), Du) = 0 x ∈ R , t > 0 ∂t u(x, 0) = u (x) CEGW[]G u |GWicvXEvRTVcT]CGRGTfce8|cvYEXqSG>q oeXqYEXE\cp[VF iPg#cvXvTV\XQYEcvYRYEXgÀT]\jcpXRª BU C(R ) LXEGgopXopqEopkST GWovR\jijPT]CEGop[]vYEF GX`TVR[/oe\_RG>qcv[T]CEGRT]G>opqSP!k[]cv|EiGWF bfG[VGpª G \_R R\jF kEiP G∗ (x0 , u(x0 ), Dφ(x0 )) ≤ 0.. 1. 0. ∗. 0. 0. 0. d. 0. 2. 0. R]cT]CoeT G = G = G yNQY|RcviYET]\jcpXR [VGWR]k«R]YEk@G[/RcviYET]\jcpXR oe[VG!GWiGWFGWX`TVR.ce ªER]cT]CoeTf\XGW^`Yoeij\TUP 1 [VGWR]k« x CEcvijqR T >0 hfiji)TVCE\jR gWoeX |@GG#lQT]GWXqSGWq¢T]c TVCEG54oeYg/C`P SzJ\j[]\_g/CEijG#TkE[Vcp|EijGF ∗. ∗. G(x, s, p) = pt + H(x, s, px ), x ∈ Rd , s ∈ Rd , p = (pt , px ) ∈ R × Rd. ∂u + H(x, u(x), Du) = 0 x ∈ Ω ⊂ Rd , t > 0 ∂t u(x, 0) = u0 (x) x∈Ω u(x, t) = g(x, t) x ∈ ∂Ω, t > 0. BU C(Rd × [0, T ]). CG[VG Át . IfXqSG[&RTVopXqEop[VqopRVRYEF kSTV\cvXRcvX6TVCEG!cpk@GX R]YE|R]G#T ª oeXq oeXq ªcpXEGgWoeX kE[VcpGG#lS\jRT]GWXg#G!opXq EY XE\_^`YEGXGWRVRfceTVCEG&Q\_R]gcvR]\mTUP¢R]cpijYSTV\cvXRfcp ~ #ª w opXq t Ω#ª@gRGWG B } CÎ H6ÆX6koe[]T]u\_g#YEi_oe[>ªSTVCE\jRL\_R T][VYEG\T]CEGJopF\ji@> T]cvXE\_oeX H \_R g#cvXQpG#l!\jX p ∈ R oeXq\ ∂Ω ij\kR]g/CE\TgcpX`T]\jXQYEcpYR 6ÆX T]C\jR.koek@G[>ª8G¢opRVRYFG!T]CoeT ~ CovR&oYXE\j^`YEGWXEGWRVR.kE[]\jXg\kEijGpªTVCoT\jRoeXQP RYE| QR]cpijYST]\jcpX u oeXq opXQP R]YEkGW[ SRcviYET]\jcpX v ce ~ RVoTV\jRP Î ∀x ∈ Ω, u(x) ≤ v(x) oeXq ∀x ∈ R , t > 0, u(x, t) ≤ v(x, t) \jXTVCEGgovRGÙceT]C5G 4oeYg/CQPkE[Vcp|EijGF 0. d. d. Í N ÍÚ¹.
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(19) ! "# "!$%"&'")(* +,-$+ ". -$/0-$. lim. G"qEG#XEG"TVCEG « GWpGWXqS[VGJTV[VopXRUcv[]F*ce f |QP. ||x||→+∞. . f (x) = +∞ ||x||. . f ? (p) = sup p · y − f (y) .. ÎT]CEGR]YEkE[VGF.YEF\_R []G>opg/CEG>q¢oeT y ªEGÙCopGLT]CEG[VGi_oTV\cvX y∈Rd. 6. ?. f ? (p) + f (y ? ) = p · y ? .. BCE\_RJR]CEcfR TVCoT gWoeX|@G&RGWGX6ovRT]CG.oe|R]g\jRVR]o cpT]CEG&TVoeXpGX`TJcpT]CEG.p[/oekEC ce oT JBC\jRJp[/oekECE\_g j\ XvTVG[VkE[VG#TVoeT]\jcpX CEGWfikRf(p)T]c¢R]GGT]CoeTWª\ f \jRJ[]GWpYEi_oe[ GWXEcpYpC«ªT]CEGv[VopkEC ce f \jRfT]CEGGWXQpGijfcpk@Gcey8\TVRfTVoeXpGX`T>ªR]c T]CoT (f ) = f. J8g#cpY[VR]GJTVCE\_R []GWijoeT]\jcpX\jR vGXEGW[Vopi\ Wop|EijGfT]c!o gcpXQpGl f CEGWX\TJRVoT]\_RGWR À hfiji)TVCE\jRvGXEGW[Vopi\ G>RT]c!o g#cvXgovGfYEXg#T]\jcpXR ÁR]\XgG −f \_R g#cpXQvG#l Àª ?. ∗. ? ?. . f ? (p) = −(−f )? (p) = inf. y∈Rd. .
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(21) !"$#. − y · p − f (y) .. GopRVRYEF GÙT]CoT T]CGJoeF \jimTVcpXE\_oeX\_Rp\jpGX|QP H(x, u, p) = sup {−b(x, v) . p + λu − f (x, v)}. EC GW[]GLT]CEGR]kovg#GÙcegcpX`T][Vcpi_R V \_R g#cvFkopgÀT>ªEoeXq¢G"CovGLRTVopXqEoe[/q¢ovR]R]YEF kSTV\cvXRcpX b ª f oeXq λ > 0 ªER]GG B } CÎ Ecv[TVCEGLzL\j[]\_g/CEijG#TgcpXqE\mTV\cvX«ª`T]CEGÙRcviYSTV\cvX!ce ~ \jRp\jpGWX|QP.TVCEG"qSPQXoeF \_goeikE[Vcpp[/oeF F \jXEkE[V\Xg\kiGvªvcv[opXQP ª T >0 v∈V. u(x) = inf. v(.). BCEG"TV[VoedUGWgÀTVcp[VP. yx (.). RVoTV\jRGWR. Z. min(T,τ ). f (yx (t), v(t))e−λt dt. 0. + 1{T <τ } u(yx (T ))e−λT + 1{T ≥τ } ϕ(yx (τ ))e−λτ yx (0) = x ∈ Ω. opXq. . cv[ t > 0. BCEGWP¢oe[VGqSG#XGWq\m f _\ R [VGvYEijop[GWXEcpYpC«ªERVoP «\jkRVg/CE\Tg#cvXvTV\XQYEcvYRWBCGG#lS\mTTV\F G τ j\ R d yx (t) = b(yx (t), v(t)) dt. zLG#TVop\i_R gopX|@G"cpYXq\X B 1 ª } C. Ê ÀÖÀ×/Ð. NON P0Q. τ = inf{t ≥ 0, yx (t) 6∈ Ω}.. U~Wu .
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(24) $#. BCEGopXoeijP`T]\_goei<GlQk[]G>R]R]\cvX!cv[T]CEGR]cpijYST]\jcpXce w Àª CEGX H cpXEijP¢qSGWkGWXqER cvX ~v CEGX H \jRYEX\mcv[]F ijP «\jkRVg/CE\T g#cvXvTV\XQYEcvYRoeXq u g#cpXQvG#l)ª. p ∈ Rd. ªE\jRv\vGX\X B x CΪ. 0. . u(x, t) = sup x · p −. CEGX u \_R g#cvXgovGpª. p∈Rd. 0. u(x, t) = inf. . u?0 (p). −x·p+. − tH(p) ,. u?0 (p). U~>}. . − tH(p) .. }E CEGX u \_RYEXE\cp[VF iPg#cvXvTV\XQYEcvYRWª`G"CopGÙcv[fogcpXQpGl¢JopF\jiT]cpX\jopX p∈Rd. U~p~ . . 0. oeXq¢cp[fo g#cvXgovGÙLoeF \jimTVcpXE\_oeX. ?. . y−x t. #. ,. U~ 1 . u(x, t) = sup u0 (y) + tH ?. . x−y t. #. .. U~x . u(x, t) = inf. p∈Rd. p∈Rd. ". u0 (y) + tH. ". BCEGcp[VF&Yijo ~v~ "oeXq ~>} Ù[VG-7G>gÀT"T]CG!fYEPQpGWXR R"kE[V\jXg#\jkEijGpª« CE\jijG ~ 1 "oeXq W~ x "oe[VGg#cpXRG>^vYGXgG.cpT]CG qSPQXopF\_gopi)kE[Vcpp[/oeF \jXE&k[]\jXg#\jkEijG ~Wu À HfcpT]GÙT]CoTf\ u \jR ij\jXEGWop[ \X x ª u (x) = A + p · x ªEG"CopG 0. 0. u(x, t) = u0 (x) − tH(p).. BCEG>RG[VGWR]YEimT/Roe[VGcvXEijPopi\_q.cp[bR]kG>g#\_oeiS\jXE\T]\_oeiEg#cvXqS\T]\jcpXRcv[koe[]T]\_g#YEi_oe[8LoeF \iT]cvXE\_oeX« G CopGT]CEG F cp[VG pGWXEG[/oei<[VGWR]YEiTVR ¤ ¡ oe[/qS\@> ÙR]CEG[ B w C u = u + u /0 u -$# u $ ; " -$# ;" /0 & /0 $ w $ "# $ +$ u. .
(25) . . ψ1 (x, t) = inf sup. q∈Rd p∈Rd. ". ψ2 (x, t) = sup inf. d p∈Rd q∈R. LBCEGkE[VcQceYRG>RT]CEG"ÁovgÀTT]CoT. . conc 0 . ∀(x, t) ∈ Rd × [0, T ],.
(26) / . conv 0. 0. . . conv 0 . . conc 0 . . . . . ψ2 (x, t) ≤ u(x, t) ≤ ψ1 (x, t). ? ? x · p − uconv (p) − uconc (q) − tH(p − q) 0 0. . ? ? − x · q − uconv (p) − uconc (q) − tH(q − p) 0 0. uconc (x) 0. . = supp∈Rd. . x·p−. ? uconc (p) 0. ? uconc (x) ≥ vp,0 (x) := x · p − uconc (p). 0 0. . . RcT]CoTcp[fopX`P p ª. BCEGWX6GR]cpijpG&T]CEG 4oeYg/CQP kE[Vcp|EijGF cv[ C\jg/C\jR"g#cvXQpG#l)ª<YR]G&TVCEG g#cpF kop[]\_R]cpXkE[V\jXg#\jkEijG #ª«opXq TVo89pGÙT]CGF oelS\F.YEF BC\jR v\vGWRT]CG"[VRTf\jvXEGW+^`Yuopi\TUPp8BCEGR]GWgcpXq¢cpXEG\_Rcp|ST/oe\jXEGWq\jX o R]\F \jijop[ oPv ¤ ¡ B t C H = H + H /!- H +$# H $ ;" " +$# ;" K -"+ ;"K
(27) K!" $
(28) /! $ .
(29) /! (* " ;/ @ w $ "# $ -$ 0. . . 0. .
(30) . conv. . conc #!. conv $. conc. . . . . "!. . . Φ2 (x, t) ≤ u(x, t) ≤ Φ1 (x, t). Í N ÍÚ¹.
(31)
(32) /
(33) ! "# "!$%"&'")(* +,-$+ ". -$/0-$. . #. ". . y−x ? ? Φ1 (x, t) = inf sup u0 (y) + tHconv +q (q) + tHconc d t q∈R y∈Rd " # x − y ? ? (q) + tHconv Φ2 (x, t) = sup inf u0 (y) + tHconc +q d t p∈Rd y∈R. BCEG"kE[VcQce\_RfR]\F \jijop[ oeXq\_Rp\jpGX\jX B t CÎ À() + fW
(34) "#1% & J 2 #f
(35) 6ÆX¢cv[VqEG[T]cR\jF kEij\mP TVCEGÙT]G#lQT>ªSG"opRVRYFGJ[]cvF XEc cvX ªQ|EYSTfoeiji@T]CEG"[VGWR]YEiTVRgoeX|G"G>opR]\ijP pGXG[/oeij\WGWq T]cceTVCEG[ qS\jF GXR]\jcpXRWªQ\Xkoe[]T]\_g#YEi_oe[ d = 3 GgcpXR]\jqEG[do=TV[]\_2oeXEvYEi_oT]\jcpX¢ce R ªQT]CEG"vG[]T]\_g#GWRoe[VG {M } ª T]CG.TV[]\_oeXpijGWR"oe[VG {T } GqSGWXEceTVG.|QP T o¢vGXEGW[]\_gTV[]\_oeXpijGp&BCEG vG[]T]\_g#GWRLce T oe[VG M ª M opXq ªcp[R\jF kEi\_g#\TUPyGcpÚT]GX qEGXEcpT]GTVCEGF |QP ª ª cp[&|`P 1 ª 2 ª 3 CEGX T]CEGW[]G\_RXEcoeF&|\vYE\mTUPvBCG M ÁoeF \jiPceT][V\jopXEpYEi_oTV\cvXRGgcpXR]\jqEG[ \_R RCopkG[ViGvYEii_oe[> i IfkT]c¢cvYE[ 9`Xc iG>qSpGvªET]CG&[/RTLkopkGW[JTVcqS\_R]gYRVRf\jXqSGTVoe\jiT]CEG oekEk[]clS\jF oeT]\jcpX ce w J\_R B CÎÙhLRJ\jXTVCE\jR [VG#G[VGXg#Gp ª w \_RfoekEk[]clS\jF oeT]G>q!|QP U~>w u = u − ∆tH , i = 1, · · · n , n ∈ N u = u (M ) CEGW[]G ∆t > 0 \jR8T]CEG TV\F GfRT]GWk!oeXq u \_RbopX opkEkE[VclS\FoTV\cvX&cp u(M , n∆t) ª`oeXq.T]CGfXQYEF G[V\jgWoeiEJopF \iT]cvXE\jopX qEGk@GXqER&cpX ªTVCEG!oeijYEG>Rce CEG[VG j ∈ V V \jRT]CEGRGT&ce XEGW\vC`|@cp[/Rcp M \jXg#ijYqS\jXE M |`P H g#cvXQpGX`TV\cvX ÀªSoeXqu\XEGWgGWRVR]op[]PcvX Muª U~Wt ). H := H(M , u , {u } 6ÆXyT]CE\_R[VG#GW[]GWXg#G \_R\jXvTV[]cSqSYg#GWqTVCEGXEceTV\cvX ce g#cpXR\_RUTVGXgPp¢BCG!XQYEF G[V\jgWoeibJopF \iT]cvXE\jopX H \jR&g#cpXR\_RUT/oeX`T CEGWX«ªE\ v = A + p · OM~ ªQT]CEGWX CoT GWpG[ M oeXq s ∈ R ª ~ H(M , s, {v } ) = H(M, s, p). h ijGWRVR[]G>RUTV[]\_gÀTV\vGÙqSGXE\T]\jcpX«ª CE\jg/C \_RCEGijkSYEiËcp[TVCEGkE[Vc`cpUªS\jRv\vGX\X B CΪ $ -¤G ° ¨ ; 8
(36) $; /0 ":0$
(37) /!% 5 "K " H $ ;0$+ $+, ! <-" x ∈ Ω " φ ∈ C (Ω) /0 C ~ lim sup H(y, φ(y) + ξ, φ + ξ) ≤ G (x, φ(x), Dφ(x)) 2. i i=1,ns. j j=1,nt. k. i3. 1. n+1 i 0 i. 0. i2. 3. i. ∗. s. i. n i n j. n i. i. n i. 2. i1. i. i. i. i. i. i. i. i. n i. n i j∈Vi. i. i. i. j. j. !. . ∞. i j∈Vi. . !. . . . ∞ b. . . . . ∗. h→0,y→x,ξ→0. ". ~ . lim inf. $ ; /0
(38) /! " $ "# $$-$. H(y, φ(y) + ξ, φ + ξ) ≥ G∗ (x, φ(x), Dφ(x)).. $$+" "! ; 0$+ $+, ! . ρ→0,y→x,ξ→0. $+ "!;0$+ $+, ! $ /!- $ B C 6. . ~ . ; "! ; "0$+ $+ -K . B CEGRT][VYgÀTVYE[]G cpTVCEG!RcviYET]\jcpXyce w ÙopRÙ[VB RTYRG>qy\X w ª ~ Î ÆX koe[]T]\_g#YEi_oe[>ª)TVCEGPYRG>q6TVCEG[VGWR]YEimT/R"ce ~ T]cqSGXEG o AÙcQqEYEXEc `ij\@9vGRVg/CEGF Gv 6ÆX t CAªp\mTb\_RRCEc X.T]CoTWªp\XpGWXEG[/oeiAªcp[8XEcvX.RT][VYgÀTVYE[VGWq.FG>RCGWRWª T]CGvGXEGW[Vopi\ WoeT]\jcpXcpT]CGAÙcSqSYEXc Qi\ 9pGR]g/CGF G.cp B w CbijGWopqRJTVcoXEcvXg#cvXR]\jRTVopXvTJopF \iT]cvXE\jopX«&JcGWpG[>ª «GWF F o } kE[VcQ\jqSG>R o RcviYSTV\cvX« hJRVR]YEF GfT]CoT } \_:R 9QXEc X¢opXq!qSGWXEceTVGJ|QP u TVCEGJk\G>g#G \_R]GJij\XEG>oe[\X`T]GW[]k@cpi_oTV\cvX!ce)TVCE\_RqEoeTVo Ecv[oeXQPF G>RCk@cp\j{u X`T M ªvGfgcpXR]\_qSG[R]G#T {Ω } cp<opXEpYEi_oe[bR]GWg#T]cv[VR8oeT ª`RGWGpY[]G ~ ovg/CoeXEvYEijop[ R]GWgÀTVcp[ Ω gcp[V[]G>Rk@cpXqR T]ccpXEG.cebTVCEGTV[]\_oeXpijGWRJT]CoeTÙR]Coe[VG M oeXq6GqSGXceT]MG.|QP U TVCEGp[/opqS\jGX`TJcp u \jX T]CoT T][V\_oeXEviGvBCGLYXgÀTV\cvXR Φ oeXq Φ GWopiYoT]G>qoeT x = M oe[VG «GWF F o. . H. n j j=1,··· ,ns. . n h. i. i i=1,··· ,ωi. i. i. i. 1. 2. i. i. Ui · z − H1∗ (z − q) − H2∗ (q) q∈R 1≤k≤ωi z∈Ωk Φ2 (Mi , ∆t) = uni − ∆t max2 min inf Ui · z − H1∗ (z − q) − H2∗ (q). Φ1 (Mi , ∆t) = uni − ∆t min2 max sup. q∈R 1≤k≤ωi z∈Ωk. Ê ÀÖÀ×/Ð. NON P0Q. . n h.
(39) . Duh|Ωi+1/2 = Ui+1/2 Mi Ωi+1 θi ~ni+1/2 Ωi. \vYE[VG~L 6ÆijijYRUTV[VoeT]\jcpX ceT]CGoeXEvYEijop[. RG>gÀT]cv[VR Ω ª θ oeXq¢TVCEGpG>gÀTVcp[/R ~n T]CoeTJoe[VG"XEGG>qSGWq\jXX`YFGW[]\_gopi JopF \iT]cvXE\jopXR qSGXE\T]\jcpXRW BCE*G 9vGPL[]GWF op[9J\jR<TVCoToeXQPJcpvTVCEG8T]GW[]FRËF&YEiT]\jkEij\G>q"|`P ∆t ªRVoP min max sup U ·z−H (z−q)−H (q) ª [VGWqSYgGWRTVc CGX U ≡ p ∀i fGWXg#GvªopX`PceT]CGWR]G"T]G[VFRfqSGXEGWRLog#cvXR\_RTVoeX`T XQYEF G[V\jgWoeiËLoeF \jimTVcpXE\_oeX«ª cp[ GlEoeF kEijGpH(p ª °}eu H := max min inf U · z − H (z − q) − H (q) , T]CGqSGWkGWXqSGWXg#P\jX u ª j ∈ V oekEk@GWop[VRL\jXT]CG.v[VovqS\GWX`TVR U BCE\_RLcp[VF&YEi_o!gWoeX6G>opR]\ijP |GG#lQTVGXqSG>qTVc!T]CG F cp[VG"pGXG[/oei)govRG H = H(x, u, Du) oeXqR]\jFki\GWR CGX H \_RfgcpXQpGl<ª`cv[ G#lEoeF kEijG °}S~ H = max max U · z − H (z) . h RG>g#cvXq¢[VGFop[ 9\_RT]CoTWª|`Pg#cpXRUTV[]Yg#T]\jcpX«ª H qEG#XEG>q|`P }pu cp[ }S~ \_RF cpXEcpT]cvXEGpªQTVCoTf\_R ¤ ° ¨ ÁKcvXEceTVcpXEGLJoeF \jimTVcpXE\_oeXR
(40) $
(41) /! " H $ ", /! " "- M ∈ Σ u ≤ v " i. i. i+1/2. ∗ 1. i. q∈R2 1≤k≤ωk z∈Ωk. ∗ 2. i. i. n j. ∗ 1. i. q∈R2 1≤k≤ωi z∈Ωk. i. ∗ 2. i. i. 1≤k≤ωi z∈Ωi. ∗. i. i. -" "!. . . !. . i. . s ∈ R. j. . j. BCEGJoeF \jimTVcpXE\_oeX }eu f\_RJF cvXEceTVcpXEG|QPg#cvXRT][VYgÀTV\cvX\m ∆t/h max LT]CE\_RÙ\jR ||D H(p)|| ≤ 1/2 o g#cvXRG>^`YEGXgG"ce opXqT]CEG\jXEG>^vYoeij\mTV\G>R hfXceT]CG[ 9pGP[VGFop[9\jRTVCoTT]CGLoeijYEGÙce qSGXEG>q¢|QP }eu cp[ }E~ qScQGWRXEceT qSGWkGWXq¢cvX!TVCEGÙRT][VYg#T]YE[VG ce)TVCEGfF GWR]C«ªQ|EYSTcpX TVCEGJ\jX`T]G[Vk@cpi_oeX`T u 6ÆXcpT]HCEGW[cp[/qERWªp\«cpXEGJR]kEij\mT/RopX!oeXEvYEi_oe[R]GWg#T]cp[ Ω \X!TUcª
(42) /0" ;/0 "
(43)
(44) /! ! ªeTVCEGfXQYEF G[V\jgWoeiELoeF \iT]cvXE\_oeX \jRbXEceTbF cSqS\GWq) GfR]oPT]CoeTbT]CEGfRVg/CEGWFG \_R
(45) K+
(46) !$;. oeXq¢GCopGLT]CG"cpijic U\jXEG[V[]cv[G>RUTV\FoeT]Gpª B t C - H : R → R ;"K
(47) K!" $ " u ∈ BU C(R ) " K$ /
(48) ;!#
(49) !!" $ ¤Q¨ ¤ H(Mi , s, {uj }j∈Vi ) ≥ H(Mi , s, {vj }j∈Vi ).. p. ||Du|T ||∞ ≤L. i. n. . ∞. k. . . . k.
(50) . . . . . . ; + 8@ "# " /0 h $ /0 @ "0 -$+ "
(51) $ /0
(52) @+$ ; ! - M L T ;"K,
(53)
(54) /0#+ " +$ /0
(55) $ "-+ "$$+
(56) /! +$/ $/0 ;0i i= 1, · · · , n M s i 8@ "
(57) /!
(58) !
(59) @ α
(60) /!#+ " +$ T $ ! -+ "! ;" ; ; 8*' ! ~>w /0
(61) /!- $+$ ;"0$+ "K - ;G /0 $ ; $+
(62) ! $ w " {un} / / u8 0 " " ! " " $+i;/j=1,··· /0 "-,n"5 ! .
(63) / c α L T >0 H M n 0 ≤ n∆t ≤ T.
(64)
(65) / ; "!$; "K. -. 2. 2. 2.
(66). . s. . . i. . .
(67)
(68) √
(69) n
(70)
(71) ui − u(Mi , n∆t)
(72) ≤ c ∆t
(73)
(74). Í N ÍÚ¹.
(75) BCEG kE[VcQce@\jRbopXopqEopkSTVoeT]\jcpXce@T]CEG Foe\jX []G>RYimTbcp \T]CR]cpF GT]G>g/CEXE\_gopi\T]\jGWR Á\Xkop[TV\jgYEi_oe[cp[R]CEc \jXE T]CoT T]CG"T]\jFGRT]GWk cpXEijP¢qSGWkGWXqcpX u Rk@GWg\mgÙTVc YEXRT][VYgÀTVYE[]G>q¢F G>RCEG>R G[VG#G[TVc B t CA BCEGkE[/opg#T]\_gopiGWoeijYoTV\cvXycpTVCEG «GWpGWXqS[VG.TV[VopXRcp[VF \jR"XceT&oeijoPQR"opXyG>opR]PTVovR9<ªRc ceTVCEG[&X`YFGW[]\_gopi JopF \iT]cvXE\jopXRG#lS\_RUT/R8BCEG"R\jF kEijGWRTcvXEGJ\_RT]CG ol E[V\jGWqS[V\jg/CRcvXEGpªQ CE\_g/C¢\_R\XRkE\j[VGWq!|QP TVCEG ol []\jGWqS[V\_g/CR RVg/CEGF GÙcp[fgcpXR]G[VoT]\jcpXijofRW< 6ÎTfCopR R]GvG[/oei<pGW[VR]\jcpXRWBCGL[VRTfcpXEG"\_R I A}p}eo ¯) − [u (M ) − u (M )]dl, H (Du , · · · , Du ) = H(U h CEGW[]G C [VGWR]k« D \_Rfo g#\j[/g#ijG °qS\_R 9 ceg#GWXvTVG[ M opXq[VovqS\YR h ª
(76) ! "# "!$%"&'")(* +,-$+ ". B -C $/0-$. . 0. LF i. h. h|Ω1. h. h|Ωki. h. i. Ch. h. i. b= U. R. Dh. Duh dxdy. ,. oeXq \_R8i_oe[VpGW[ËTVCoeX oeXQP Ë\kRVg/CE\T g#cvXRTVoeX`T8ce H qE\Q\_qSGWq|QP 2π BCE\_R8qSG#XGWRboLF cpXEcpT]cpXGRVg/CEGWFGkE[VcQ\jqEGWq T]CoT ∆t/h ≤ h qE\ ?<G[VGX`TLvG[/R\jcpX cp8T]CEG Ëol E[V\jGWqS[V\jg/CRJJopF \iT]cvXE\jopX«ª@T]CoeTL\_RfF cpXceT]cvXEG&YXqSG[JT]CEG RVoeF G&g#cvXRT][/oe\jXvT>ª \_RT]CEG"cviijc \X L ε 2π. HhLF (Duh|Ω1 , · · ·. BCE\_R pGW[VR]\jcpX¢gWoeX |G[VG [V\TT]GWXovR. , Duh|Ωki ) =. HhLF (Duh|Ω1 , · · · , Duh|Ωki ) =. X. Z. πh2. H(Duh ) Dh. πh2. − h. I. Ch. A}p}| . [uh (M ) − uh (Mi )]dl.. X ~nil−1/2 + ~nil+1/2 θli H(Duh|Ωil ) + ε tan θli · Duh|Ωil . 2π 2. B CEGfvGWg#T]cp[ ~n \_R8TVCEGfYEX\mTvGWg#T]cp[8cp<T]CEGfG>qSpGTVCoTRGWkoe[/oTVGWRT]CGJoeXEvYEi_oe[bR]GWgÀTVcp[/R Ω oeXq Ω ªvT]CEGJopXEpijG \_RTVCEGoeXEviGLceT]CEG"oeXEvYEi_oe[RG>gÀT]cv[oeT M ªER]GG \jpY[]G ~ bBCEGÙkoe[/oeF GT]G[ ε \_RTVCEG"R]opFG"opR\jX!TVCEG"kE[]GWQ\cvYR θ pGW[VR]\jcpX« h T]CE\j[/qvG[/R\jcpX\jR 0≤l≤ki. 0≤l≤ki. l. l+1/2. i l. l+1. i. H. LF. (Duh|Ω1 , · · · , Duh|Ωki ) =. X. T 3Mi. |T |H(Duh|T ) + α X. |T |. X. Mj ∈T. (ui − uj ). A}p}eg. eo Xq α ≥ h max ||D H|| CEG[VG h \_RT]CGijop[]vGWRTG>qSpG"ce T BCEG Foe\jX!qS\@?@GW[]GWXg#GJ|GTUGWGX T]CGWR]GJqS\@?@GW[]GWXvTcp[VF&YEi_opR\_R8TVCoT }p}po 8oeXq
(77) }p}e| bop[]G\jXvTV[]\jXR]\jgf\X TVCEGJR]GXR]G p\jpGWX\jX B t CË CE\jiG }p}pg \_R XEceT>JGXgGpªScviijc \X.TVCEGRVoeF G[]GG[VGXgGpª }p}eo oeXq }v}| op[]GgcpXQpGW[]vGX`TopXq T]CGÙGW[][Vcp[GWRT]\jFoTVGÙB \_R O(h ) Ecv[ }p}pg #ªERYg/C oeXG[V[Vcp[GWRT]\jF oeT]GÙ\jRXceTfooeB \jijop|EiG ÁoeT iG>opRT CGXcpijic \jXE T]CG"T]GWg/CXE\j^`YEG"cp t CΪS|EYSTf\Tf\jRJg#cpXQvG[VpGX`T LT]CE\_R \_R oR\jF kEiGopkEkEij\jgWoT]\jcpXce CA BCEG"opqSopXvT/oevGJce }v}eg cvG[bTVCEG"ceTVCEG[TUc.vG[/R\jcpXR\jR\TVRR]\F kEij\_g#\TUP!\X g#cSqS\jXE g#cvFkoe[VGWq TVc }p}eo opXq }v}| À##-2f
(78) BCEGyoekEkE[VclS\FoeT]\jcpXceT]CGzL\[V\jg/CiGT¢kE[Vcp|EijGF \jRXEcpTopRR\jF kEiGyopR¢\mTicQ8c 9SRW hfX \jijiYRT][/oTV\cvX \jR¢T]c Xq R]Yg/C¢TVCoT u : [0, 1] → R \jX opXq u(1) = 2 |u | − 1 = 0 x ∈ [0, 1], u(0) = 1 CE\_g/C CopR Xc!g#i_opRVR\_gopi«RcviYSTV\cvX«ªE|EYETJ CE\_g/CQ\_R]gcvR]\mTUPRcviYSTV\cvX«ªqSGXEG>qcvXEijP¢\jX [0, 1[ \_R u(x) = x GCovG 6ÆXceTVCEG[bgWopR]GWRWªpRVoP u(0) = u(1) = 0 ªpG CovG u(x) = |x − 1/2| CE\_g/CFoeTVg/CEG>R8RUTV[]cvXEpijP lim u(x) = 1 6= 2 T]CG|cvYEXqEop[]P¢gcpXqE\mTV\cvXR T 3Mi. T. p. p. T. 1/2. 0. x→1−. Ê ÀÖÀ×/Ð. NON P0Q.
(79) W~ u 6ÆXcp[/qSG[TVc qEG#XEGoRVg/CEGWFGvªSGRTVop[T [VcpF ~Wu ªopXqgcpXR]\_qSG[fo.T][V\_oeXEvYEijoeT]\jcpXce Ω \j[VRTfGovR]R]YEF G T]CoT M ∈ ∂Ω 6ÆX ~>u Àª<TVCEG RGTÙcpg#cvXvTV[]cvijRLgopX6|@GR]kEij\mT]T]GWq\jXvTVc!TUc¢kop[T/R LfT]CEG RGT V cp[Ù CE\_g/C T < τ ª oeXq V cp[ C\jg/C T ≥ τ fGWXg#Gvª i. 1. 2. u(x) = min( inf [· · · ] , inf [· · · ]).. «GT. @| G T]CEG\jX`T]G[V\jcp[ XEcp[VFoeiT]c oT NS\XgG \_R!oe[V|E\T][/oe[VPpªb\T¢gopX|@Gg/CEc`RGWXopRR]FoeijifopR k@cvRVR\j|EijGp~n6ÆXT]CEG ij\F \T T → 0 ª<TVCEGRGT ΩV xcpYE∈i_qΩ|@GT]CEGR]G#T"Tceg#cpX`TV[]cvijRJcp[" CE\_g/C b(x, v) . ~n > 0 ª)\A GpTVCEG g#cvX`T][Vcpi`cp[8 CE\_g/C.T]CEGTV[VoedUGWgÀTVcp[VPpcQGWR\jX`T]c Ω BCGfqSPQXoeF \_goeiQkE[Vcpv[VopF \XELkE[V\Xg\kiG inf [· · · ]−u(x) = 0 g#cv[][VGWR]k@cpXqERTVc.TVCEGJopF \iT]cvXE\jopX v∈V2. v∈V1. 1. v∈V1. Hb (x, t, p) = sup {b(x, v) . p + λt − f (x, v)}.. G oei_RcCopGÙTVCEG[]GWijoeT]\jcpX H ≤ H BCEG cpX gopX |GopkEkE[VclQ\jFoTVGWq)ªS\ T \_RfR]Foeiji°ªE|QP ϕ(y (τ )) fNQ\jXgG T ≤ τ ª@oeXq\GgopXg/CEcQcvR]G g#cvX`T][Vcpi_Rinfcv[ CE\_g/C V T ' τ ªG"pGT |@GWgopYR]G ϕ \jRJg#cpX`TV\XQYEcvYR GRGWGLTVCoT ~Wu gWϕ(y oeX |@G(τoe))kEkE'[Vcϕ(x) lS\FoeT]GWq«ªQoeTfo.|@cpYXqEoe[VP!k@cp\jX`TWªE|QP v∈V1. b. 2. x. x. 0 = max(Hib , u(x) − ϕ(x)). CEGW[]G H \_R og#cvXR]\jRTVopXvT oekkE[]clS\jFoT]\jcpXce H CEGWX ªvT/o9Q\jXE RFoeiji<GXEcvYEpC˪`G"goeXRGWGJcv[]FopiijPT]CoeTTVCEG"|cvYEXqEop[]P kEi_oPSRXc.[VcpijGÙR]cTVCoT G"gWoeXTVo89pMGoe6∈XQP¢∂ΩgcpXR]\jRTVopX`T TLoeF \jimTVcpXE\_oeX«ªQcv[ G#lEoeF kEijGÙT]CEc`RGqSGXEG>q\jXT]CEGk[]GW`\jcpYRR]GWg#T]\jcpX« BCEGRVg/CEGF G"\jR T]CEGWX A} 1 S(M , u , {u } )=0 ∀i \T]C \ x 6∈ ∂Ω ) H(x, s, {u } °}x S(x, s, {u } )= W G j i ] R p G BCEGRVg/CEGF G } 1 }ex gop||@GG#lQTVGXqSG>qmax(H T]c ceTVCEG(x,[TUs,PQk@{uGWR }cpÙ|@cp),YEsXqE−oe[Vϕ(x)) P¡g#cvXqS\T]\jcpXRW BCEGW[]G \_R!oeX\F kEij\_g#\T qSGWkGWXqSGXg#P cp S \mTVC [VGWR]k@GWgÀTT]c h G G#lQT]GWXq TVCEG qSG#XE\mTV\cvX ce S T]c oeXQP y ∈ Ω |QP R]oPQ\jXE TVCoT \m |GWicvXEvRTVcT]CEGqSYoeiËgcpX`T][Vcpi<pcviYFGovR]R]cSg#\_oT]G>qTVc M S(x, s, {u } GÙCopGÙT]CEG")=cviijS(M c \jXE., s,[VGW{uR]YEiT } ) x ¤Q¨ ¤ B C $;$+ /0 " b. b. i. i. i. j j∈Vi. j j∈Vi. j j∈Vi. b. i. j j∈Vj . . . j j∈Vi. j j∈Vj. i.
(80) . H Hb "% , "
(81) /! $ 0$% } 1 % ! -"+ "! ; 8 - "% φ ∈ C ∞ (Ω) !/ " b "% ! x ∈ Ω, lim H(y, ϕ(y) + ξ, ϕ + ξ) = H(x, ϕ(x), Dϕ(x)) Hb ≤ H . A}pweo . h→0,y→x,ξ→0. !
(82) / ;"/! % " %. ∂Ω,. lim. h→0,y→x,ξ→0. Hb (y, ϕ(y)+ξ, ϕ+ξ) = Hb (x, ϕ(x), Dϕ(x)). . /!5; ! "# " /0 $5 ! !-'+$;$+
(83) - @ /0 & /0- " "! u : ! ;"- -$@ ; "! ! -+ ! ,
(84) /! $ .
(85) . °}vw| . . LBCEG v9 GP¢op[]vYEF GX`TcpTVCEG"kE[Vc`cp\_Rfo g#cvX`vG[VpGWXg#GL[]G>RYEiT |QP oe[ViG>RoeXq NQcpYvoeX\jqS\_R B CA. . h. . . . . Ω. Í N ÍÚ¹.
(86) fI XEcp[]T]YEXoeT]GWiPvªvTVCE\_R[VGWR]YEiTVR\_RXceTGXEcvYEpCTVc.vYoe[/oeX`TUP o pcQcSq g#cvXQpG[VpGWXg#GvBo9vGfT]CGÙGlEoeF kEijGLoTT]C~pG~ |@Gp\jXEXE\jXE ceT]CGRG>gÀT]\jcpX˪Eo.[VGvYEi_oe[F GWR]C 1/∆x = N + 1#ªST]CEG%AÙcSqSYXEc¢RVg/CEGF GÙT]CoeTf[]G>qSYgGWRCEGW[]GÙT]c
(87) ! "# "!$%"&'")(* +,-$+ ". -$/0-$. Hi = H(ui−1 , ui , ui+1 ) = max. . |ui+1 − ui | |ui−1 − ui | , ∆x ∆x. . −1. oeXq bBCE\jR opF cpYEX`TVRTVc!RGTTV\XE ª bBCEGcv[]GWF } oekEki\jGWR|EYETfX`YFGW[]\_gopi)G#lSk@G[V\F GX`T/R \jXqS\_gHoTVGÙ=T]C−∞ oeTTVCEGp[/opqE\GWXvTceT]CEGR]cpijYST]\jcpXu\_R=XEce0T |@ucpYEX=qEGW2qRcT]CoTT]CEGW[]G"\_RXEc CEcpk@GÙT]c CovGÙog#cvX`vG[VpGWXg#G ij@\ 9vG ∆xB \mTVC α > 0 [VGWopR]cpXoe|EijGp 6ÆX CAªvT]C\jRkE[Vcp|EijGF \jRRUTVYqS\jGWq!opXq \mT\jRRCEc X TVCoT\m \_RgcpXQpGl<ªp\m)TVCEG AÙcSqSYEXEc RVg/CEGF Gfg#cvXRT][VYgÀTVGWq cpX TVCEG!|@cpYEXqEoe[VPyLoeF \jimTVcpXE\_oeX H \_R&gcpXRT][VYg#T]GWq«ªopXq \mHJo6g#cQG[/g#\jQ\mTUPyopRVRYEF kSTV\cvX CEcpi_B qERcv[ H ª H opXq T]CG ovR]R]cSg#\_oTVGWqXQYEF G[V\_goeiJopF \iT]cvXE\jopX«ª)TVCEGXcpXEGgopXgcpX`T][Vcpi Du ª«oeXq6\mT\_RÙR]CEc X\X B ~>u CT]CoeT"T]CEG G[V[Vcp[ |@GCovGWRi\ 9pG h NQ\jF \i_oe[ GW[][Vcp[GWRT]\jFoTVGWR cp[ 4oe[]T]G>R\_oeXF GWR]CEGWR GW[]GÙcp|ETVoe\jXEG>q¢\jX ~v~ CA , 1+- (# Ifk¢T]c.XEcªSoeijiTVCEGLGlEoeF kEijGWRGLCopGfv\vGX¢op[]GLcpXEijP[/RUTcp[/qSG[opggYE[/oT]GÙR]g/CGF GWRWBCG[VGLoe[VGLR]GvG[/oeiEoPSR g#cvXRT][VYgÀTV\XE CE\jpCcp[/qSGW[RVg/CEGWFG>R qSGk@GXqSGXgP \jX T]CG LXEGkc`R]R]\|iG g#cvXRUTV[]YgÀT]\jcpX \jRog#cpXRG>^vYGXgG6ce.T]CEGcpijijc \XE¡Áopg#TW*BCEG JopF \iT]cvXE\jopX¢g#cvF GWRcp[VF T]CEGLT]GW[]F {u , j ∈ V } \jX ~Wt 8Kcp[VGJk[]G>g#\_RGWiPvªv\jXopii@T]Du CG 9QXEc X!GlEoeF kEijGWRWªpTVCE\jR qSGWkGWXqSGXg#P!cSggYE[/RbTVCE[]cvYEpCqS\ ?<G[VGXg#GWRWª cp[ 8BCEG>RGLT]GW[]FRgopX¢|@G"[]GW []\TTVGX¢\jX!TVG[VFRcp«T]CG p[/opqE\GWXvT/RJcp u \XT]CEG.T][V\_oeXEviG>RLR]YE[V[]cvYEXqSu\jXE −B Mu T]CE\_jR"∈[]GWVFoe[ 9CopRÙoeij[]G>opqSP |@GGXYR]GWq\jX }p}po Àª }p}e| LopXq }v}eg À LXEGgopXGlQkicv\mT TVCE\jRJ[]GWFoe[ 9@ªSovR \X ~ C«cv[ G#lEoeF kEijGpª|`PF cQqE\mPQ\jXE T]CEGGWoeijYoTV\cvXcpTVCEGp[/opqE\GWXvT/R \jXTVCEG T][V\_oeXEviG>R 6ÆXRT]G>opqce"B i\jXEGWop[\X`TVG[VkcvijopXvT>ªcvXEGgopXYR]G CE\jpCEGW[!qSGv[]GWGk@cpijPQXEcpF \_oei_R.T]Cop0X 9SR T]cyT]CG H H F G#TVCEcSqScpijcpvPpª t ª ~} ª ~ 1 CA h ceTVCEG[ R]cpijYST]\jcpX\jRTVCEGzJ\_RVg#cpX`TV\XQYEcvYR A"oeijG[ 9Q\XRUTV[VoeT]GvP B ~Wx ª ~>w ª ~Wt CÎ GqSc XEcpTfqSG#T/oe\ji<T]CE\_RT]G>g/CEXE\_^`YEG CEGW[]Gv h i_opRTfF G#T]CcQq \_RJo |iGWXqS\jXE¢RT][/oT]GWpPvª B ~ CÎBCEG&\jqSG>o\_RT]c!|iGWXqo ic cp[/qSGW[WªEF cvXEceTVcpXEGLoeF \iT]cvXE\_oeX H \T]C o&CE\jpCcp[/qSG[JopF\jiT]cpX\jopX gcpXR]\_RUTVGX`TLoeF \iT]cvXE\_oeX H À PCE\jpC¢cv[VqEG[GÙFG>oeX¢T]CoeT\ u \_R o RF cQceTVCR]cpijYSTV\cvXcp ~ ÀªET]CEGWX °}et ) = O(h ) H (M , u , {u } cp[ k > 1 8BCEGRVg/CEGF G" []\T]G>R A}v ) + ε(h) ) + (1 − ` )H (M , u , {u } ) = ` H (M , u , {u } H(M , u , {u } CEGW[]G ε(h) = Ch cp[fR]cpF G"kc`R\T]\jpGgcpXRTVopXvT C oeXq ` \_R g/CEcvR]GX RYg/C¢TVCoT>ªE\m r := ªG"CovG A} ` + (1 − ` )r ≥ ε (h). CEGW[]G ε (h) ε(h) = o(1) GCopGLT]CEGR]\F kEijGijGF Fo. C\jg/C kE[VcQceË\_R \jFF G>qS\joeT]Gvª ¤ ¡ H H "$+#" ;0$+ $+, ! H 8 *' }` $ ; ;0$+ $+, ! BCEGdUYRT]\goeT]\jcpX ce } gcpF GWR[VcpF TVCEGR]\F kEijG[]GWijoeT]\jcpX A} ) + ε(h) H(M , u , {u } ) = ` + (1 − ` )r H (M , u {u } [VcpF CE\_g/C«ªQYR]\XEcpXg#GÙFcv[]GJT]CEGÙTVGWg/CEXE\_^`YEGÙce«TVCEGg#cpXQvG[VpGXg#Gf[VGWR]YEiTce B CAªScvXEGÙgWoeXRCEc cv[T]CEG"R]g/CGF G } 1 > }x CEGW[]G H \jRfp\jpGX|QP }v TVCEG"cpijic \jXE[]G>RYimT ¤Q¨ ¤
(88) ;"!$+ 8
(89) /! $ ;/0 } 1 /0
(90) }x H $ 8 *' }`
(91) % $;$+ /0 b. 0. N. α. . b. b. −1/2. . . j. i. j. i. j. i. . M. H. H. i. i. i. j j∈Vi. M. i. i. i. i. . M. .
(92) . . HM HH HM. Ê ÀÖÀ×/Ð. NON P0Q. i. j j∈Vi. i. HH i HM i. i. 0. i. −1. . . . . Hb. . . !. H. i. . i. i. i. . H. i. j j∈Vi. k. 0. k. j j∈Vi. . . . !. "!. i. . . j j∈Vi. i. i. i. . " $+#" "! ;0$+ $+, ! "'" "' 5 " ,! 0$. M. . i. i. j j∈Vi. . Hb. . . . . . . !.
(93) ~>}. . . Hb ≤ H . /! @
(94) 0 " , . `. r=. @ $,. [0, 1]. . HH (x, uh (x), uh ) , HM (x, uh (x), uh ). /0 /0 ! " , ;$ " 0 $ "# $ ε(h) ε (h) /!-5 $;$ K -! $- # " u 1 h /!5; ! "# " /0 $5 ! !-'+$;$+
(95) - 0/ & /0- " "! : ! /0$ ;/0 ;" . }. . $- $$ +$ `(x) + (1 − `(x))r ≥ ε0 (h). + $ 0 −1 ε (h) ε(h) = o(1)
(96) /! "$ "# $ +$ ∞ ; /0 $ K -"+
(97) L h @ +$ ; ! ! -+ "! , /0$ # "
(98) . . lEoeF kEijGWRcpËTVCEG|EijGXqS\jXEkop[VopF G#T]GW[ oe[VGpªSp\jpGWXgcpXRTVopXvT/R α ≥ 1 ª α > 0 oeXq β > 0 ª ` \m r ≤ 0 min(1, α |r|) \ 0 ≤ r ≤ β 1 u 0 `= G _ i ] R p G min(1, α (r − β)) BCE\_RfTVCEGcpXEG.GCopG&g/CEc`RGWX\jXkE[/opgÀTV\jgWoeioekEkEij\_goTV\cvXRf \T]C β = 0 opXq α = 1 6ÆF kEijGF GWXvT/oTV\cvX6qSG#T/oe\jijR gopX|@G"cpYXq\X B ~ CÎ (2yK H Ë (#1+1&$%
(99)
(100) ( ( # 6ÎT \jRXceTfqS\ gYEiTTVcg#cpXRUTV[]Yg#TXQYEF GW[]\_gopi<LoeF \jimTVcpXE\_oeXRTVCoT cp[ 9 cpXpGXG[/oei@XEcpX g#cvXScp[VFoei@FG>RCGWRWbBCEG cpXiP 9vGP k@cp\jXvT\_R.T]c g#cvXRUTV[]YgÀT " "' JopF \iT]cvXE\jopXRW¡BCEG g#cpXQvG[VpGXg#G![VGWR]YEiTVRce B C opXqBCEGcv[]GWF } gWoeX G>opR]\ijP |GopqEopkST]G>q L!og#ijcvR]G¢\jXR]kG>gÀTV\cvX ce TVCEG¢kE[VcQceJR]CEcfRTVCoT.T]CEG RT][VYgÀTVYE[]G¢ce T]CGF GWR]C kEi_oPSR []cviGv CoeT&FoT]T]G[/R\_RTVc6qSG#XEGpªcp[oeXQP6F GWR]C kcv\X`T ª8o icSgWoei\jX`T]GW[]k@cpi_oTV\cvX«ª π T]CoT&cvkGW[VoeT]GWRcpX M v c ` X ] T c V T E C G R k p o # g G e c L E k \ > G # g W G j \ ] R G j i j \ E X W G p o & [ E Y X À g ] T j \ p c X R ªoeXqRYg/C¡T]CoeT\m u ≤ v ª j ∈ V ª8T]CEGWX U := {u } π(U ) ≤ π(V ) 4cpXR\_qSG[ \jpYE[VG } BCEGXEGW\vCQ|cv[VRce oe[VG } ª[VcpF CE\jg/CGgcpXRT][VYg#ToÙijcQgWoei`T][V\_oeXEvYEijoeT]\jcpX ÁqEceTTVGWq6i\jXEGWR TVCoTÙ\jRLYR]GWq TVcqSGXEGokEM\jGWg#G ` {P \_RG&i\jXEGWop[J\jX`T]G[Vk@cpi_oeX`TW 6ÎTÙqScQGWRLXEcpTLXEGWGWq TVc!|G g#cvX`T]\jX`YcpYRW BCEGWXGgWoeX YR]GcpYE[fLoeF \jimTVcpXE\_oeXR T]c¢qSGXEG&RVg/CEGWFG>RT]CoeTLoe[VGg#ijGWop[]ijP¢gcpXR]\jRTVopX`TWªEF cpXceT]cvXEGpBCEGTV[]\_;g 9QP kop[T\_R"T]CEG!g/Ccp\_g#G ceT]CEG!XG\jpCQ|cv[VRW \vYE[]G } R]CEcfRoeXyG#lQT][VGF GgovRGv!h kE[Vcp|op|EiP|@G#T]T]GW[&g/CEcp\_g#G cpYEi_q CovGL|@GGWXTVc!g/CEcQcvR]GLcvXEijP {P , P , PB , P , P } |@GWgopYR]GLTVCEGopR]k@GWgÀTf[/oTV\cceT]CG"T][V\jopXEpijGWR\_Rijop[]vG[> HfcpT]GÙT]CoT T]CGJoeF \jimTVcpXE\_oeXRce ~ CΪSTVCoe0X 9SRTVc.TVCE\_RfRGTfce[]GWF op[ 9SRWªSoe[VGÙkoe[]T]\_g#YEi_oe[ govRG>RcecpYE[cv[]F.YEijo
(101) # (Ë /( 6ÆX pGWXEG[/oeiAªS\mTL\jRfqE \ !g#YimTfTVc!g#cvFkYST]GopXoeijP`T]\_goeijijPT]CEGR]cpijYSTV\cvXcpo.[/RTfcp[/qSG[fLoeF \iT]cvX Q`opgcp|E\<G>^vYoT]\jcpX˪ oeXqT]CEG&R]\T]YoeT]\jcpX \jRJGvGXcp[/RG" CEGWX T]CEG&LoeF \jimTVcpXE\_oeX\_R XEcpTLg#cvX`vG#l ÁXEcp[Ùg#cvXgovG |G>gopYRG"TVCEG.opXoeijcpvP \T]C!CQPQkGW[]|@cpij\_gfRPSRT]GWF R|@GWg#cvF GWRbijcQcvR]G[b\jX!vGXEGW[Vopi°fGWXg#Gvªv\T|@GWgcpF GWRFcv[]GLqS \ !gYEimTT]cfdUYqSpGJT]CEGÙ^`Yoeij\mTUP ceXQYEF G[V\jgWoei[]G>RYEiTVRW&Bc cpG[/g#cvF GT]CE\_RqS\ !g#YEiTUP\X oRk@GWg\jopibgopR]Gpª<Gg#cpXR\_qSG[ H(p) = (||p|| − 1) opXq T]CGkE[]cv|EijGF cpX Ω, H(Du) = 0 cvX Γ , 1 ~ u=0 cpX Γ u = 10 CEGW[]G Ω \_RJqSGWkE\jg#T]G>q \X vYE[VG 1 fNQ\Xg#G t 7→ t \_R F cpXEcpT]cpXG\Xg[]G>opR]\Xª u \_RJR]cpijYSTV\cvXcp 1 ~ \mbopXq cpXEijP¢\ \Tf\jR oR]cpijYST]\jcpXce cvX Ω, ||Dv|| − 1 = 0 cvX Γ , 1 } v=0 p c X Γ . BCEGR]cpijYST]\jcpXce 1 ~ oeXq 1 } \_RT]CEGqE\jRTVopvX=g#G"10TVc Γ uh. . . −. . Ω. ∗. +. −. +. +. i. i. j j∈Vj. i. j. j. j. i. i. 2. 3. 4. 5. i i=1,··· ,8. 7. 3. 1. 2. 3. 1. 2. 1. Í N ÍÚ¹.
(102) ~1.
(103) ! "# "!$%"&'")(* +,-$+ ". -$/0-$. P7 P6 P5. P8. M. P1 P2 P3 P4. \jpY[]G} L 4opR]G"ceoXEcvXg#cpXEcp[VF opi<FG>RCË. . \vYE[VG 1 L.4cpF kEYETVoTV\cvXoeiEqEcpFoe\jXcp[bkE[Vcp|iGWF 1. ~ Γ \_RT]CEGJ\XEXG[g\[/g#ijGce)g#GX`TVG[ (0, 0) opXq[VovqS\jYR r = 1 ª _ \ R ] T C G p c S Y V T G f [ # g j \ / [ # g j i G °. # g W G v X V T G [ S ª V [ v o S q j \ Y R À Γ (0, 0.5) r=3 6ÆX¡cv[VqEG[.T]c qS\jRVg#[VG#TV\WG 1 ~ #ªbG [V\mTVG \mTVC H (p) = max(|p|| − 1, 0 opXq H (p) = H = H +H BCEG>RG.YEXg#T]\jcpXR"oe[VG&[VGWR]k@GWgÀTV\vGijPg#cpXQvG#loeXqg#cpXgopGvLBCEGXQYEF G[V\jgWoeiJopF\jiT]cpX\jopX6\_R min(||p|| − 1, 0 T]CG Ëoel []\jGWqE[]\_g/ C RoeXq T]CEGL|cvYEXqEop[]P JopF \iT]cvXE\jopX!\_RAÙcSqSYEXEc RBCEGLX`YFGW[]\_gopiR]cpijYST]\jcpX¢\jRqS\jR]kEi_oPpG>q cpX \vYE[VG x > Áo #JBCEG&R]cpijYSTV\cvXce 1 } \T]CT]CE G AÙcQqEYEXEc JoeF \jimTVcpXE\_oeX\_RLkE[]cQ\_qSGWqcvX \jpY[]G x > Á| ÀLh g#ijcvR]G g#cvF koe[V\jR]cpXRCc T]CoeT T]CEGWPop[]G ÁopiF c`RUT \_qSGWXvTV\jgWoeiA 1. 2. 3. 1. 3. Ê ÀÖÀ×/Ð. NON P0Q. 2. 1. 2.
(104) ~x. . °o. Á| . \vYE[VGx!L °oL NQcpijYST]\jcpX cpk[]cv|EiGWF 1 ~ #ªF \jX 0 ªFol 1.48 Á| L NQcpijYST]\jcpX cpkE[Vcp|iGWF 1 } ÀªF\jX 0 ª Fol 1.504 hfX ceT]CG[opkEkEij\jgWoTV\cvX ceTVCEG6|@cpYEXqEoe[VPgcpXqE\mTV\cvXR!\_R¢v\vGX |QPT]CEG oekkE[]clS\jFoT]\jcpX cpÙTVCEGcpijic \jXE kE[Vcp|EijGF ªScvX¢TVCEGR]opF GÙvGcvFGT][VPpª cpX Ω, H(Du) = 0 u(x, y) = 0 u(x, y) = 3 cos(2πx). 181 . (x, y) ∈ Γ1 , (x, y) ∈ Γ2 .. NQ\jXg#G \_R"XEcpX g#cvXQpG#l)ª)\T\jRqE\ !g#YimTTVc=9`Xc '+ "+ CoTcpYijq|G TVCEGoeijYEG ceT]CEG!R]cpijYST]\jcpXycpXT]CG |@cpYEXqHoe[VPpBCEG¢gcpF kEYSTVGWq R]cpijYST]\jcpX \_Rv\vGX cpX \jpYE[VG w > Áo À6ÎTgWoeX |@G¢RGWGX TVCoTTVCEGR]cpijYSTV\cvX R]oeT]\_RUG>R RT][VcpXEviP"TVCEGf|@cpYEXqoe[VP&g#cvXqS\T]\jcpXcpX opXqcpXEijPGWo9QijPcpX Ág#cvX`T][/oe[V\ijP"T]c"TVCEG kE[VGQ\cvYRG#lEopFkiG ÀHfceTVG CEcGvG[T]CoeTbT]CGP&CopG|GWGXXQYEF G[V\_ΓgoeijijP ; \F k@cvR]GWq cpX Γ Γ oeXq Γ 8BCEGfRcviYET]\jcpX \jRoei_Rc"\jXvG[VPpcQcSq oev[]GWGF GX`T \mTVCT]CEGcvXEG"cp|ST/oe\jXEGWq[VcpF TVCEGqS\_R]g[]GT]\ WoTV\cvX¢cp cpX Ω, ||Dv|| − 1 = 0 1 x v(x, y) = 0 (x, y) ∈ Γ , CE\jg/C\_RfqS\_RkEi_oPpG>q!cvX \jpYE[VG w > | #v(x, y) = 3 cos(2πx) (x, y) ∈ Γ . G¢oei_RcR]CEc CEc T]CG!CE\jpC cv[VqSGW[G#lQTVGXR]\cvX ceJR]GWgÀTV\cvX t cp[ 9QR" CGX TVCEG AÙcSqSYEXEc R]cpijpG[& \mTVC \jXvTVG[VkcvijoeT]\jcpXcp[¢T]CEG6CE\vC cp[/qSG[R]g/CEGWF Gp BCEG Wc`cvF \_R¢qS\_R]kEijoPvGWqcpX \jpYE[VG t 4ijGWop[]ijPpª o vG[VP¡i_oe[VPpG cpGW[VR]CEcQceTÙG#lS\_RUT/R" CEG[VG u \jR"XceT C ªËT]CEGW[]G TVCEG!R]cpijYST]\jcpXyceT]CEG|iGWXqSGWq RVg/CEGF G \_R"FcvXEceTVcpXEGopXq6\_RvG[VP R]\F \jijop[&TVcT]CG¢[/RTcv[VqSGW[&cvXEGp. 6ÆX TVCEGR]F cQceT]C¡koe[]T.cpfT]CEG R]cpijYST]\jcpX«ªTVCEG RG>g#cpXq cp[/qSG[oeXq T]CEG|EijGXqSG>q RVg/CEGF G"oe[VGLvG[VP R]\jF\ji_oe[ ÚTVCEG"[]G>RYEiTVR|QP T]CGÙ|iGWXqSGWqRVg/CEGF GÙ\jR R]i\jpC`TViPF cp[VGLqE\jRVR\jkoeT]\jpGLT]CopX¢TVCEcvR]GLcp«T]CG R]GWg#cvXqcp[/qSG[YEXEij\jF\T]G>qR]g/CGF G # BCEGi_opRT \vYE[VGRCEc T]CoeTJcpY[f\F kEijGF GWXvT/oTV\cvXceT]CEG&|cvYEXqEop[]Pg#cpXqS\mTV\cvXR \_R -G ?<GWg#T]\jpGv 6ÎbG\F k@cvR]G RT][VcpXEviP TVCEG|@cpYEXqEoe[VPgcpXqS\T]\jcpXRªopR\jX \vYE[VG °o BCE\_RCovR T]c |@G6gcpF koe[VGWq \T]C \jpYE[VG w BCEG \vYE[VG RCEc T]CoTfTVCEG[VG\jRLoRUTV[]cvXE|@cpYEXqoe[VP!ijoPvG[ cvXkoe[]TVR cpTVCEGcvYST]GW[f|cvYEXqEop[]P CEGW[]G"TVCEG\_Rcvi\jXEG>R oevpijcpF G[/oTVG #JBCE\_RL\_RLXEceTÙT]B [VYEGcv[JTVCEG \jpY[]G °o # 6ÆX6ÁopgÀT>ª<cpXR]cpF G.koe[]TVRJcebTVCEG.cvYST]GW[L|@cpYXqEoe[VPpªT]CG g#cvF koTV\|E\jij\mTUP g#cvXqS\T]\jcpXce 1 C\jRfT][VYEGpª<R]cT]CoeTLcvXEG&gopX\F k@cvR]GT]CG&|@cpYEXqoe[VPgcpXqS\T]\jcpXRJRT][VcpXEviPvªoeXq cpX ceTVCEG[koe[]TVRLT]CE\_R"\_R"XEceT"T][VYEGoeXq6G CopG.T]coekkEiPT]CEGWF GWo 9QijPpBCE\jR"koe[]T]\T]\jcpXyceTVCEG |cvYEXqEop[]P6\jR"XEcpT 9QXEc X o.k[]\jcp[V<\ L8cpYE[ \jF kEiGWF GX`TVoeT]\jcpXTV8o 9pGÙT]CE\_R\jXvTVc!opgWg#cpYXvT oeYET]cpFoeT]\_goeijijPp &# (Ë#( G.CovG.qSG>R]g[]\j|@G.R]GpGW[Vopi«TVGWg/CEXE\_^`YEG.cp[LT]CEG RcviYET]\jcpX6cpb[/RUTÙcp[/qSG["JopF \iT]cvX6vovg#cv|E\G>^`YoTV\cvXR G.CopG T][V\jGWqTVcLGlSkEijop\XT]CEGCE\_qEqSGWX.qSGTVoe\ji_RopXqTVCEGcp[V\v\XRceT]CEGR]g/CGF GWRWNQGWpGW[VopieT]CGcp[VG#TV\jgWoei`[]G>RYimT/Rop[]GkE[Vc`\_qSG>q)ª 2. "!. . 1. 1. 2. 1. 2. 2. 1. Í N ÍÚ¹.
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(115) ' 0 "# " ª@}euvupt B ~p-~ C zLGW;g 9vGijXE\_;g 9oeX& q 4L K iji\jceT>fIfXE\_^`YEGXGWRVRfoeXq GW[][Vcp[fopXoeijPSR\_Rcv[LJopF \iT]cvX Q`opgcp|E\)G>^vYoT]\jcpXR \mTVC qS\_RVg#cpX`TV\XQYE\T]\jGWRW ! - ;+$% ; 8 ª!t L 1 } 1 x Eªv}pupuex. . . . . . . . . . . .
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(122) Unité de recherche INRIA Futurs Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY Cedex (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France).
(123). . ISSN 0249-6399.
(124)
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