Construction of simple, stable and convergent high order schemes for steady first order Hamilton Jacobi equations.

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(1)Construction of simple, stable and convergent high order schemes for steady first order Hamilton Jacobi equations. Remi Abgrall. To cite this version: Remi Abgrall. Construction of simple, stable and convergent high order schemes for steady first order Hamilton Jacobi equations.. [Research Report] RR-6055, INRIA. 2006, pp.34. �inria-00114888v4�. HAL Id: inria-00114888 https://hal.inria.fr/inria-00114888v4 Submitted on 7 Dec 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Construction of simple, stable and convergent high order schemes for steady first order Hamilton Jacobi equations R. Abgrall. N° 6055 Décembre 2006. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6055--FR+ENG. Thème NUM.

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(38) #%H$' : , $' # ' #> 7 .O ' . φ. @' !$. 4 $'" # !'"# !2$& '"# 687 .O"!$# . . H(σ, φ(σ), φ) = H(σ, φσ, Dφ(σ)).. X.sdmVBdmR1d.l¡ H lob(¬%VMu³ay`wshbmlobedfVhpdg H lob+bedfkmshTyz`5wshbmlzbdmVghdM¨ !$# &e[®shTsdmshTV¯¥ZuUWlzy}dfshTlouhb (*# + ! ,0# !# σi. . 7 ∈Σ. ui ≤ vj. 7.!$'2 !$'$,. 7 s∈R. H. $'$# $' 7 .O"!$# . H(σi , s, {uj }j∈Vi ) ≥ H(σi , s, {vj }j∈Vi ).. Q(RVkfV!V|clzbdfb.UWha`5haiTUWVkflzwguy bfwvRTVgU>VMb.cVgsudfVgdmsdmRTVkfVgbmsyzicdflshs¡ 3<4 ¨ 4-6 ¤V|TuUW{TyzVgbZukfV lVhp`kmV¡¢VkfVhwVgb B I c c 39p $; D c¬+RVkfV\dfRTV!{TkfsTyzVU 354 ¨ 476 lob+clzkmVMw;dfy`¯dvwv³ayzVg3¨ \hVnpiTldmVbdfhTkf©¬(`su¡Zwshbdmkfiw;dflhT¤bmwvRTVgUWV¡¢sk 354 ¨ 476 lobdfs¤wshbmlzTVk'dmRTViThbdmVgc` {kmsTyVgU ∂v + H(x, v(x), Du) = 0 x ∈ Ω ⊂ Rd , t > 0 ∂t v(x, t) = g(x, t) x ∈ ∂Ω, t > 0 v(x, 0) = v0 (x) x ∈ Ω, t = 0.. 354. ¨ . 6. ¡¢sk%bmsUWV\bmiTldfTyV\lhl}dflzyOwshcldmlzsh u biwvRWdfR1d?dmRV\bmsyzicdmlzshs¡ 354 ¨ 476 lobscdflhVg¯bdmRVZyzlUWldg ¬+RVh t → +∞ su¡*dfRTVbmsyzicdmlzsh®s¡ 3<4 ¨ 6 ¨.Q(RTVkfV!lzb\¯¬+RTsyV!lzhcibdmkf`su¡GhaiTUWVkflzwguy bfwvRTVUWVMb(¡¢sk 3<4 ¨ 6 ¨ \UWlddflhTWdfRTV!OsiThukf`5wshcldmlzshb+h5lzhdmRTVglk.bmlzU>{yVMbed+¡¢skmU admRV`kmVBs¡ dmRVBde`p{OV 0 . . un+1 = uni − ∆tH(σi , uni , {unj , j ∈ Vi }) i. +¬ ldmR u = u (σ ) ¨¼xZyshTdmRVgbmV±ylzhTVMb (shV®UW`§npiTsudfVdmRTV±¬?skf³@s¡MB T ?c 4 D huUWsh U¯hp`5sudmRVkvb B 4E4 4 D ¡¢sk /?kdfVgbmlzhUWVMbRTVMb.uh B 4 c 4 IT 4 a 4 D ¡¢sk.iThbdmkfiw;dfiTkfVgU>VMbRVgbg¨Q s dfRTVWVMbed!s¡siTkB³ahTs1¬+yzVgcV OdmRV>shTyz`±wshaVkfVhwVkfVgbmiTydfbg 3¬+l}dfR¤Vgkmkfsk°VgbdmlzU¯1dmVMb kmV¡¢sk°£kvbed skfTVkbfwvRTVUWVgbg *bVgV B 4 9p 4 ; D ¡¢sk'bdmkfiw;dfiTkmVMU>VMbRVgbh B 4 D ¡¢skihbedfkmiw;dmikmVMU>VMbRVgbg¨x VhTVgkfy UWVdfRTsc®¡¢skZ{Tkfs1alhTwshaVkfVhwV ¬+l}dfRTsiTd°Vgkmkfsk.VMbedflU¯udmVgbg Klzb\lzVgh®lh B D ¨Bx.yzy

(39) dmRTVMbV wshbdmkfiwdmlzshbWukfV®bedfkmshTyz`«kfVyo1dfVg@dmsdmRV±cl}²KVkfVhpd¯dfVgwvRThTlonpiTVgb¯dmRudRVOVVh cVgalzbmVg@¡¢sk wshbdmkfiwdmlzhT'RlRskfTVk+wgwiTkv1dfV °scciThTs1¯deà{OV bfwvRTVgU>VMb?¡¢sk.wshbmVkf11dmlzshyz¬.bg¨ ¥ZVkfV KbedvukmdmlzhT¡¢kfsU 5Tl}²KVkfVhpd!wshbedfkmiwdmlzsh O¬?VV|c{Tyoulzh±RTs1¬»ld°lob\{spbmbmlyV!dfswshbdmkfiwd .wshaVkfVghpdg RlR§skfTVkwgwiTkv1dfV±bmwvRVUWVgb>¡¢sk¯dmRTVE{TkmsTyzVU 3<4 ¨ 4-6 ¨ VEuyobs©bmRTs1¬ V|TuUW{TyzVgb3¡¢sk ¬+RTlowvRBdmRV%wsUW{TicdfudmlzshybdmVghwlzylobdfRTVUWspbed {OsbfbmlTyzVwsUW{wdg¨ \iTkwshbdmkfiw;dflsh kfVyzlzVgb+shdfRTVTyzVhclhsu¡rWyzs1¬ skfcVgk.wgwiTkv1dfVbmwvRTVgUWVuh¯RTlzRskfcVgkZbdfuyVbfwvRTVUWV¨?Q(RTV bdmkfiwdmiTkfVsu¡*dfRTVTyzVhclh5{kfU>VdmVgk+lzb°uhyJ` KgVgbsdmRud\RTlzR®skfcVgkZwgwiTkvw`5lob.scdflhTVM±b ¬?Vyzyb\5wshaVkfVhwVB{Tkfsps¡e¨ZQ(RTVMbV>bmwvRTVgUWVgbZkmVsu¡rwsiTkfbmVhTsd\UWshsudmshTV TicdBUWshTsdmshlzwl}de` {kmVMbVgkmalzhTŸZ{Bdms+sik³ahTs1¬+yVMcV ld lobKdmRTVr£kfbd3dmlzUWVr¬+RTVgkmVGOsudfR!{Tkms{VgkdflVMbwh!blzUiTydfhTVsibmy` OV¯wvRTlzVVg3¨ V¯uyobmsbdmic`±dmRTVW{Tkvwdmlowuy*lzUW{TyVgUWVhpdfudmlzshEs¡dfRTV¯bmwvRTVgUWV uh¤cVgUWshbdmkv1dfV ldfb.V²KVgw;dflVhTVMbmb(shshTV!clzUWVhbmlshuyhde¬%s FÝTlUWVhblzshyKV|TuUW{TyzVgbg¨ 8ÖhdmRTlob+{{Vgkg c¬?VB¡¢scwib sh@iThbedfkmiwdmiTkfVg@dmkflouhTiTyzkdeà{OVU>VMbRVgbg¨ 8Ýd¯lzbWwyVMukWRTs1¬?VVgkdfR1dWdfRTVUWlhªkmVMbiy}d¯su¡\dmRTV 0 i. 0. i. . . NON. Ú$P"Q;å;æ;æ. .

(40) . . H!$. {u{OVk 3 lÄ¨ V¨

(41) dmRV(¡¢skfUsu¡KdfRTVZbfwvRTVUWV+uh>dfRTV.wshaVkfVghwV?{Tkfsasu¡ 6 wguhWOV.ibmVg>lhBUWskmV(VghTVkvuy wshpdmV|adg¨ Q(RVZbedfkmiw;dmikmV+s¡OdfRTlob{{VgkrlobGdmRV+¡¢syzys1¬+lzhT1*¬?V(£kfbd%bdfkdra`W!VghTVkvuyTcVgkmlz11dflsh¯su¡KdmRTV bfwvRTVgU>V¨ VWTlzbfwibfb°lhcVdfulzy

(42) dmRTV¯bdmkfiw;dfiTkfV's¡dmRV'TyzVhclh{ukvuUWVdfVkM¨ V>dmRVh{Tkms1alocV' wshaVgkmVhwV\{kmsasu¡e¨(Q(RTVhV|ad\bmVgw;dflsh®lzb\cVsudfVgdms5bmRTs1¬+lzhT¯bsUWVV|TuUW{TyzVgb.su¡rbmwvRVUWVgbg uh ¬?Vyzbms5clobmwibfb.dfRTV{Tkvwdmlowy lzU>{yVgU>Vghpdf1dflsh±su¡GdfRTV'bfwvRTVgU>V¨\Q(RTV'yzbed°bVMw;dmlzshElzb°cVsudfVgdms haiTUWVgkmlowyOV|TuUW{TyzVgbg¨ ¹ µ M µ µ ) VW£kvbedTlzbfwibfb°dmRVbmwvRTVgUWV'¡¢skBdmRTV¯UWVMbR{Oslzhdvb!lzh ) µ $ ,TT- , ¹ + dfRTV!s{OVhbmVd Ω ¨rQ(RTVBOsiThukf`5wshcldmlzshb+kmV!clobmwibfbVMud?dfRTV!Vhs¡ dfRTlzbZbVMw;dmlzsh ¨ VrwshbmlzcVgk H := H (σ , u , {u } ) (U>shTsudfshTVrwshbmlobedfVhpd¥ZuUWlzy}dfshTlouh!h H := RTlzRskfTVkwshblobedfVhpd?¥\uUWlzy}dfshTlouh¨ :?`WRTlR5skfcVgk¬?VZUWVgh¯dmR1d%l¡ u H (σ , u , {u } lob+WbUWsasudfR®bsyicdflsh) s¡ 3<4 ¨ 476 TdmRTVgh 3 c¨ 476 H (σ , u , {u } ) = O(h ) ¡¢sk k > 1 XZV|ad.¬?VBwshbmlocVk?¡¢sk.bmsUWV ` ∈ R dmRTVB¡¢syyzs1¬+lh>¥\uUWlydmshTlouh 3 c¨Æ 6 H(σ , u , {u } ) = ` H (σ , u , {u } ) + (1 − ` )H (σ , u , {u } ) + ε(h) V!RVZdfRTVblzUW{TyzV!yVgU>U¯>¬+RTlowvR{Tkmsas¡lob(lUWUWVgclo1dfV .

(43) #%!$# H !$'2 H !$H # $' :, ' # '"# 7 H 2E ' 52 , 3 T¨ 6 . !*0$1, ' # '"# V'bmbmiTUWVdfR1d ε(h) = O(h ) ¨ 8ÖhEskvcVkZdmsTV£hTV ` K¬%V'lhpdmkfscciwVdmRVkv1dmlzs r := hkmVg¬+kmldmVMb 3 c¨Æ 6 b 3 c¨ 6 ) + ε(h) H(σ , u , {u } ) = ` + (1 − ` )r H (σ , u {u } hwvRTsasbmV ` biwvR5dfR1d 3 c¨ I 6 ` + (1 − ` )r ≥ ε (h). ¬+RVkfV ε (h) ε(h) = o(1) ¨±Q(RTV5yzscwibsu¡(dmRV5{Oslzhdvb (r, `) dfR1d'bf1dflzb¡¢`wshcl}dflsh 3 c¨ I 6 ylzVgb OVde¬?VVghdmRTV!de¬?s¯TkvuhwvRVgb(su¡*dfRTVRaà{VgkmOsyo ` + (1 − `)r = 0 clob{Tyo`VMlh =

(44) lziTkfV'c¨ 4 ¨?Q(RVh ¬?V!whkfV¬+kfl}dfV

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55) `

(56)

(57)

(58)

(59)

(60) 3 c¨Æ 6

(61) H(x, t, {u }

(62) ≤

(63) + 1 − `

(64)

(65) H (x, t, {u }

(66) +

(67) ε(h)

(68) . ) )

(69)

(70)

(71) r

(72)

(73)

(74)

(75)

(76) _clhwV ε(h) = O(h ) hl}¡ ` ∈ [0, 1] cl¡dfRTVkfV!V|clobedvb C > 0 bmiwvRdmRud

(77)

(78)

(79) `

(80) 3 c¨ 6

(81)

(82) ≤C

(83)

(84) H. M i. i. i. M. i. i. j j∈Vi. H. i. H i. j j∈Vi. i. j j∈Vi. i. . M. i. M. i. i. i. k. j j∈Vi. j j∈Vi. H. i. i. i. j j∈Vi. H. k. i. i. i. j j∈Vi. i. i. i. 0. i. M. i. i. i. HH i HM i. j j∈Vi. 0. i. −1. M. j j∈Vi. j j∈Vi. k. r. Ü< N ÜèÊ.

(85) 9.

(86)

(87) "!$#%&$'". `. r.

(88) . ` + (1 − `)r = 0 . dfRTVhdfRTVbmwvRVUWV!cV£hVga`¯dmRTV¥\uUWlzy}dfshTlouh 3 T ¨ 6 bf1dflzb£VMb. H(σi , ui , {uj }j∈Vi ) = O(hk ). ¡¢sk.uha`5bmUWspsdmRbsyiTdmlzshsu¡ 3<4 ¨ 476 ¨ _clhwVdfRTV5dvuhTVhpd>ud>skmlzlzh©su¡ZdmRTVRa`p{OVkfOsyo lob * bmsyzicdflshs¡ dmRV!{TkmsTyzVU 3 c¨ 6 lob ` = max(` , ε (h)) ¬+{(r,RTVkf`)V ∈ R , ` + (1 − `)r = 0} −1 l}¡ 3 c¨:9 6 0 VyobrV¨≥ 0 ` = min 1, α|r| ¡¢sk.uha` α ≥ 1 ¨ V(RV(uh>TTl}dflshuycwshbdmkvulzhd*sh ` O¨ 8ÝdrwsUWVgb¡¢kfsU»dfRTV(l}dfVkv1dmlzV(bfwvRTVUWV%dmRud*lob*hTVgVgcVM dfs¯wsUW{TiTdmVBdmRVbsyicdflshs¡ 2. ∗. 0. ∗. H(σi , ui , {uj }j∈Vi ) = 0. ¬+RVkfV H lob.cV£hTVga` 3 T ¨ 6 ¨ Ö8 hdmRTlob+{{Vgkg c¬?V!VUW{Tyzs1`WdmRVB¡¢syzys1¬+lzhTWV|c{Tyzlzwl}dZbfwvRTVgU>V ¡¢sk n ≥ 1 3 c ¨ ; 6 ) = u − ∆tH(x , t, u , {u } u u = u (M ) YZ{dfs\siTk ³ahTs1¬+yzVgcV 1uyzyp£kvbed

(89) skvcVkU>shTsudfshTV?¥ZU>lzydmshlzhbbf1dflzb¡¢ùh L bdfTlyzlde`BwshTl}dflsh ihcVk.Wwshbdmkvulzhpd(sh5dfRTVBdmlzUWVbedfV{su¡dmRTVBdeà{OV 3 c¨ 6 ∆t ≤ Ch n+1 i 0 i. n i. 0. i. n i. n j j∈Vi. i. ∞. . NON. Ú$P"Q;å;æ;æ.

(90) ;. . H!$. ¬+RVkfV lob.¯wshbedvuhpd(dmRudZcV{OVhTb+shy`sh h ThTsd.sh dmRV!UWu|clU'iTU-clouUWVdmVk+s¡ dfRTV!UWVgCbmRVgyVgUWVhpdfbg¨ =TkfsU 3 c¨ ; 6 3 c¨ 6 h 3 c¨ uI 6 ¬?V uyoHbs>RV h 0. M n ` + (1 − ` )r H (σ , u , {u } ) + ε(h) un+1 = u − ∆t i i i i i j j∈Vi i i. bms>dmRud+dmRTV!bfwvRTVUWV!lob ÖhskfcVgk?dmsWRV 8. ∆t. L∞. bedvuTyzV!l}¡. ∆t ì + (1 − ì )ri ≤ Ch.. hsud+dmsas¯bmUWyyP c¬?VwshblocVk. C0 > 0. uh. (ri , ì ). h. 0 ≤ ì ≤ 1. bmiwvRdmRud. 0 ≤ ì + (1 − ì )ri ≤ C 0 .. Q(RlzblUW{OsbmVgbh«TTl}dflshuywshTl}dflsh©shTyzÈ¡¢sk ¨®Q(RTVkfV5kmV¯U¯uha`¤¬?`cb!s¡(lzUW{spblzhT dfRTlob!wshbedfkflhpdBlh©wsh1teiThwdmlzsh¬+l}dfR 3 c¨ I 6 ¨ =Tsk!bmrlUW>{Tyzlo0wldeÈkfVgbmshb KlzVghwshbdfhpdfb α ≥ 1 uh β > 0 T¬%V!wvRTsaspbV\dmRTVB¡¢syyzs1¬+lzhT¡¢skmU8¡¢sk ` α >0  l}¡ r ≤ 0 min(1, α |r|)  3 c¨ 4 6 l}¡ 0 ≤ r ≤ β 0 ` = V o y b V ¨  min(1, α (r − β)) Q(RV>kf{TREs¡(bmiwvR5¡¢ihw;dflsh ¡¢sk C = 1 lzb!clobm{Tyz`VgEsh =lzikmVT¨ a¨ ldmR ` TV£hTVMbBlzh i. −. ∗. +. −. ∗. +. `. β. . 3. c¨ 4 6 cdmRVkfV°V|clzbdfb C bmiwvRdmR1d(¡¢sk. `(r) .

(91) . 0. 0 < ∆tC 0 h. dfRTVbfwvRTVUWV!cV£hTVga` 3 T ¨ ; 6 3 c¨ 6 uh 3 c ¨ I 6 lzb. r. L∞. OsihcVg¨. 3. c¨ 4 4 6. Ü< N ÜèÊ.

(92)

(93)

(94) "!$#%&$'" . ZdfRTVk.wvRslowVgb(kmVB{OsbfblzTyzVBbmiwvRb |r| 1 + |r|. l∗ = ϕ(r) :=. sk(U>skmVBVhTVgkfyyz`. l∗ = ϕ(ψ(r)). ¬+ldmR ≥ r uh ψ (0) = 1 ¨¯x.hV|TU>{yVWlob ψ(r) = r + r ¨Q(RVgbmV>{OsbfbmlTlzyzl}dflVMbBRV>hTsud OVVghψ(r) V|c{TyzskfVg3¨ _clhwVVMni1dmlzsh 354 ¨ 4-6 csaVgb\hTsudBcV{OVh±sh±dmlzU>V KsiThTkm`®wshcldmlzshb\Uibed°VWb{OVgwl}£Vg3 sdmRTVgkm¬+lobmV'dfRTV¯{TkfsTyzVU lzbU>VMuhTlzhTyVMbmbg¨¥.VgkmV¯¬?VW¡¢syzys1¬dmRTVdmVgwvRhTlznpiTVTVgbfwkflOVglzh B 4 D ¨ 8Öh skfTVk3dfs\bmlUW{Tyzl}¡¢`BdfRTVrdfV|adg 1¬?VwshbmlzTVk shTyz`\£kfbdskfcVgk wgwiTkv1dfVclobmwkmVdmlobmudmlzsh!s¡pdfRTV?siThTkm` wshTl}dflshb¨ VwshbmlzcVgk+WsiThTkm`¯haiTUWVkflowuy¥\uUWlzy}dfshTlouh dmR1d.lob.wshblobdfuhpd(¬+l}dfR±>siThTkm` ¥\uUWlzy}dfshTlouh H ¨ 8Ýd.lob.cV£hTVg¡¢sk x ∈ ∂Ω s ∈ R h Hp ∈ R uhyzbms¯bmudmlobe£VMb 0. 2. . b. b. ∀x ∈ ∂Ω, s ∈ R, p ∈ Rd ,. Q(RV°kfVgbmiTydmlzhT¯bmwvRVUWVBlzb *1 £h {u } l¡ σ ∈ Ω, ` H σ , u , {u }. Hb (x, s, p) ≤ H(x, s, p).. bmiwvR5dfR1d. j j=1,··· ,nΣ. i. M. i. i. i. j j∈Vi. . d. + (1 − ì )HH σi ui , {uj }j∈Vi + ε(h) = 0. c¨ 6. 3 4. 3 T¨ 4 1 6 l ¡ σ ∈ ∂Ω, max H σ , u , {u } , u − g(σ ) = 0. Q(RVbsyicdflshsu¡ 3 T¨ 4 6 lob+hTsdZuh®Vgb`dfbm³K¨.=syzys1¬+lzhT¯bedvuhukvdfVgwvRThTlonpiTVgbg ¬?VwsUW{TicdmV ld5b'dfRTV®ylzUWl}d¬+RVh n → +∞ # J @ # E # %s¡ {u } n ∈ N cV£hTVgªa` uh u = u (x ) l}¡ σ ∈ Ω, u = u − ∆t` H σ , u , {u } + (1 − ` )H σ u , {u } b. i. i. i. j j∈Vi. . 0 i. 0. i. i. n j j=1,··· ,nΣ. i. n+1 i. i. n i. i. M. i. i. i. j j∈Vi. + ε(h). H. . i i. j j∈Vi. T¨ 6. 3 4. l ¡ σ ∈ ∂Ω, max u − u + H σ , u , {u } , u − g(x ) = 0. 3 T¨ 4 u 6 ∆t 8Öh 3 c¨ 4 6 uh 3 c¨ 4 6 lob+cV£hTVga` 3 c¨ 4 6 ¨ 8ÖhdfRTV¯¡¢syyzs1¬+lh `¬?V¯V|adfVhdfRTV5cV£hTldmlzsh©su¡ dms±hp` b!¬%V5RV¯cshTVW¡¢skdmRTV ¥\uUWlzy}dfshTlouhalz'dmRV!V|c{Tyzlzwl}dZcVg{VghThw`s¡dfRTV!kv1dm`lzs r lzh xx¨ ∈ Ω i. NON. Ú$P"Q;å;æ;æ. n+1 i. n i. b. i. i. j j∈Vi. n+1 i. i.

(95) 4. . ). ¹ + µ µ + µ ¹¹ ) VcVghTsudfV°a` S dfRTV!s{OVkv1dfsk.  `(x)HM x, uh (x), uh + (1 − `(x))HH x, uh (x), uh + ε(h)    S(h, x, uh ) =    max Hb x, uh (x), uh , uh (x) − g(x) = 0. x ∈ ∂Ω.. l}¡. V!RV. $

(96) #%!# + $'" %2E #" 3 T¨ 4 6 !'32 ! G M 7 H !$'2 !$H #%H$' :, $'" # '"# HHM !H'32 Hb !$H

(97) b$'$# $' ! A# '"&!$'" 57 7 H "b ≤ H '32$' !H!$ # ` ' # [0, 1] !'32 !# > r=. HH (x, uh (x), uh ) , HM (x, uh (x), uh ). H!$. l}¡ x ∈ Ω. #"!$#. `(x) + (1 − `(x))r ≥ ε0 (h). .O"

(98) # !H!$ # !$'32 0 !# > ε0(h)−1 ε(h) = o(1) 7 3 T¨ 4 6 @G#%&$' <ε (h) # !# !$#% > L∞ $G'2 # !$# G' $ " H E # ! G' &" ε(h) uh ' 7 "h "!$#%&$' 4 6 !$ !G' &" ' ' " ' #" >!$M:, uh 2E ' 52 , 4 6 '

(99) - !:, G' $M:, # # $@G#%&$' 5 4 6 ' Ω

(100) !$# ' '"#%&!$ ' H 52$& '"# 'M#" $H # !# #" 3 T¨ 4 6 "! >! G' &" @G# &' .OJ& G'32 52 ' ∞ G'" 1, ' 'H!E #%& !$. ! GL!$#%&$' 57 #" $AG# &' L . L-*0$h# A # 7 .O" ' # !$' 52 , !$' # H!$#% 0 " 7 <C#" # H!# 3 c¨ 4 6 $ J!$ 3 c¨ 4 6 687 l¡biwvR ylzUWl}dV|clzbdfb8 C " n →! +∞ @M# E #& 57 #" ∞ G'32 $ H4 # $'32# &' 3 c¨ 4 46 7 G## E # ' < #" @M# M'$# ! # L&!$ # !$# '"# ! H-< ¨B qrkmsas¡su¡GQ(RTVgskfVU T¨ 4D . . . . V!{TkfscwVVMb(lh B 4D r¨ Q(RTVbVMnpiTVhwV u lob(siThcVMbsW¬?VBwguhcV£hTV h u(x) = lim inf u (y) u(x) = lim sup u (y) Q(RV`'kmV+cV£hTVM'sh Ω OVgwibmV u Rb*OsiThbGlhcV{OVhTVhpdsu¡ h ¨ V.bRs1¬«dmR1dGdmRV+¡¢iThwdmlzshb uh ukfV!kmVMb{OVgwdmlzVgy`biTG ?uh®bmiT{OVHk FÝbmsyzicdflshb+su¡ 3<4 ¨ 476 ¨ V{kmscwVgVglzh®de¬%s¯{ukmdf

(101) b 1%£kvbed u ¬?V!wshublocVgk%dfRTVwbVBs¡*hlzhpdmVkflzsk({slhpdM cdmRTVghdmRTVwgbmV°s¡

(102) >OsiThukf`{slhpdM¨ !$ 5!$' '"# &$ '"# 8Öh±¡Ôwdg K¬?V>bmRTs1¬¼£kvbed\dmRud°l¡ x ∈ Ω lzb°yzscwuy*U¯1|clzUiTU s¡ ¡¢sk.bsUWV φ ∈ C (Ω) TdmRTVgh u−φ 3 T¨ 476 H(x , ϕ(x )Dϕ(x )) ≤ 0, ¬+RlyzVBl}¡ x ∈ Ω lob.>yzsawguy3UWlzhTlU'iTU8s¡ u − φ 3 T¨Æ 6 H(x , ϕ(x )Dϕ(x )) ≥ 0 h. h. y→x,h→0. y→x,h→0. h. h. 0. b ∞. 0. 0. 0. 0. 0. 0. 0. Ü< N ÜèÊ.

(103) 4E4.

(104)

(105) "!$#%&$'". Qs\bmRTs1¬dfRTV%lzhTVMniuyzl}de` 3 ¨ 6 1¬?V%kfV{OVgud9:(ukfyzVgb h'_csiTpuhTloclob ukfiTUWVghdvb gdmRV%lzhTVgnpiylde` T¨Æ 6 ol bBscdvulzhTVg¤lh¤dmRVbmU>V>¬(`¨ VWU¯ÈbfbmiTUWVdfR1d x lzb!bdmkflzwdBUWlzhTlzUiTU u(x ) = siTdfbmlzcV!s¡ B(x , r) ¬+RTVkfV r lob+bmiwvR5dfR1d φ(x ) φ ≤ 2 inf ||u || lh u(x) − φ(x) ≥ u(x ) − φ(x ) = 0 B(x , r). Q(RVkfV'V|alobdfbBbVMnpiTVhwVgb h uh y ∈ Ω bmiwvR±dfR1d n → +∞ h → 0 y → x u (y ) → uh lob*ZysuyaUWlzhTlU'iTUs¡ −φ ¨ V+cVghTsudfV?a` ξ dmRV?npihdfl}de` u (y )−φ(y ) ¨ u(x ) V!RV ξ y→ 0 u (y) ≥ φ(y) + ξ ulh B(x ¨ , r) \V£hTlh r = H y , u (y ), u ¬%VBVd 3. 0. 0. h. h ∞. 0. n. 0. n. n. h. n. n. n. 0. hn. hn. n. n. n. 0. h. yn , uh (yn ), uh. 0 = `(yn )HhMn yn , uh (yn ), uh + (1 − `(yn ))HH yn , uh (yn ), uh + ε(hn ) = `(yn ) + (1 − `(yn ))rn HM yn , uh (yn ), uh + ε(hn ). ¬?VBRVBbmlhwV OVgwguibmV. hn H hn M H hn. 0. n. hn. n. _clhwV. 0. n. n. n. 0. 0. 0≤. 3. T¨ 6. ε(hn ) ε(hn ) ≤ 0 = o(1) `(yn ) + (1 − `(yn ))rn ε (hn ). `(yn ) + (1 − `(yn ))r ≥ ε0 (h) > 0. ¨G¬%VBVd.uyzylzhyy. ε(hn ) = o(1). `(yn ) + (1 − `(yn ))rn. lob°UWshTsdmshV2h. c¬?VBVd. HM ε0 (h) > 0 0 `(yn ) + (1 − `(yn ))rn > ε (hn ) > 0. ¨!¥ZVhwV 3l¡¬%V>clzplocV'dmRTV>yobd°VMniuyzl}de`su¡ 3 ¨ 6 a` T¨. 3 I 6. 0 ≤ HhMn (yn , uh (yn ), uh ) + o(1).. bdg cibmlzhT>dmRTV!UWshsudmshTlowlde`su¡. HhMn. ¬%VBVhi{dfs. 0 ≤ HhMn yn , φ(yn ) + ξn , φ + ξn + o(1).. XZsudfV\dfR1d.lzh{bfbmlhT'¡¢kfsU 3 T ¨ I 6 dms 3 T ¨Æ 6 ¬%VBRV\ibmVgdfRTV!iThTl¡¢skfU8wshpdmlzhpil}de`s¡ H ¨ Q(Raib 0 ≤ lim sup HM (yn , φ(yn ) + ξn , φ + ξn n. ≤ H(x0 , ϕ(x0 ), Dϕ(x0 )). . 3. T¨Æ 6. Q(Rlzb(bmRTs1¬.b%dmRud lob+'bmiT{OVk<?abmsyzicdmlzsh5su¡ 3<4 ¨ 4-6 ¨rQ(RVBbfuUWVBukfiTUWVghdvb%{T{TyzlVMdfs u bmRTs1¬ dfR1d ld.lob+>bmiTGFÖbmsyzicdmlzsuhsu¡ 354 ¨ 4-6 lzh Ω ¨ NON. Ú$P"Q;å;æ;æ.

(106) 4. . . . G'32E!$ ,M"$'"# BX.s1¬. !$ C5 !. dfR1d+¡¢sk. Thha` x ∈ ∂Ω lim sup. ϕ ∈ Cb∞. ¬%V¯wshbmlzTVkBdmRVWwgbmVWsu¡ (Ω). x0 ∈ ∂Ω. H!$. ¨WQ(RV'£kfbd!kmVgUWkm³®lob. S(h, x, ϕ + ξ) = max(H(x,ϕ(x), Dϕ),. h→0,y→x,ξ→0. max(Hb (x, ϕ(x), Dϕ(x)), ϕ(x) − g(x))). lim inf. h→0,y→x,ξ→0. dfRTVh. S(h, x, ϕ + ξ) = min(H(x,ϕ(x), Dϕ),. Q(RV.{Tkfsps¡¡¢skbmRTs1¬+lhBdmRudl}¡ x. ¬+RlyzVBl}¡ x. lzb!yscwguyTU¯1|clzUiTU su¡ u − φ ¡¢skbsUWV ∈ ∂Ω. . max(Hb (x, ϕ(x), Dϕ(x)), ϕ(x) − g(x))). 0. b φ ∈ C∞ (Ω). lob.>yzsawguy3UWlzhTlU'iTU8s¡ u − φ . T¨ 6. 3. min(H(x0 , ϕ(x0 )Dϕ(x0 )), max(Hb (x0 , ϕ(x0 ), Dϕ(x0 )), ϕ(x0 ) − g(x0 ))) ≤ 0,. T ¨:9 6 wguhVgblzy`5V!scdflhTVM5a`5wsU'TlhlhT>dfRTVbmUWVBkmiTUWVhpdfb(hdfRTsbmV!su¡>B 4 D Q(RTVskmVgU-c¨Æc¨ _clhwV S lob(UWshTsdmshTV c¬?VBVd 0. ∈Ω. max(H(x0 , ϕ(x0 )Dϕ(x0 )), max(Hb (x0 , ϕ(x0 ), Dϕ(x0 )), ϕ(x0 ) − g(x0 ))) ≥ 0. 3. . 0 ≤ lim sup S(hn , yn , φ(yn ) + ξn ) ≤ n. lim sup. S(h, y, ϕ + ξ). h→0,y→x,ξ→0. = max(H(x0 , ϕ(x0 ), Dϕ(x0 )), max(Hb (x0 , ϕ(x0 ), Dϕ(x0 )), F (x, ϕ(x0 ), Dϕ(x0 ))).. XZs1¬¬%V+RV+dms'wvRTVgwv³dmRuddmRVZwshcldmlzsh 3 ¨ 6 3 kfVgbm{¨ 3 ¨ 9 6<6 lzUW{TyzlVMbrdfRTVZbmiT{Vgk<FÖbmsyzicdmlzsh 3 kfVgbm{¨ bmiTG FÖbmsyzicdmlzsh 6 wshcldmlzsh ¨ ¶ ) O¨ 8Ý¡ F (x , u(x ), Dφ(x ) ≤ 0 MdfRTVkfVrlob hTsdmRTlzhT+dfs.{Tkfs1V¨ V?bfbiU>V *+ µ

(107) &,Ô • ¨ V!RVBVldmRVk F (x , u(x ), Dφ(x ) > 0 3 T¨ ; 6 H(x , ϕ(x ), Dϕ(x )) ≤ 0 sk 0. 0. 0. 0. 0. 0. 0. 0. 0. Öh>dfRTVZbmVgwshWwmax(H bV u¬%V+R(xV(, u(x hTVgwVg),bfbmDφ(x kmlzyz` 3 T),¨ ; F6 (xh,>u(xlh¯O),sudfDφ(x RwbmVg))b

(108) df≤RTV.0.lzhTVgnpiylde`RTsyzb¨ ¶ ) ¨ 8Ý¡ F (x , u(x ), Dφ(x ) ≥ 0 OdmRTVgkmV>lzb\hTsdmRTlzhTdms5{kms1V¨BxZbfbiTUWV *+ µ

(109) &,Ô • adfRTVh¬?V°U'ibd.RV°VldmRTVgk H(x , u(x ), Dφ(x )) ≥ 0 sk F (x , u(x ), Dφ(x ) < 0 b. 8. 0. 0. 0. 0. _clhwV. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. max(H (x , u(x ), Dφ(x ), F (x , u(x ), Dφ(x ))) ≥ 0. a m d R lzb+lzhTVMniuyzl}de`lzUW{TyzlVMb H ≥ 0 Tbs>dfR1d F <0 b. 0. 0. 0. 0. 0. 0. b. Dφ(x )) ≥ H (x , u(x ), Dφ(x )) ≥ 0. Q(Raibg clzhOsudfR®wH(x bmVgbg a¬%, u(x V!Vd ),H(x c¬+RlzwvRlob(¬+R1d.¬?VB¬?hdfVg3¨ , u(x ), Dφ(x )) ≥ 0 0. 0. 0. 0. b. 0. 0. 0. 0. 0. Ü< N ÜèÊ.

(110) 4.

(111)

(112) "!$#%&$'". . ' @J &$' BxZyyrdmRTlob!bRs1¬.b.dfR1d lob!bmiT{OVHk FÝbmsyzicdmlzshuh u lzb!bmiTGFÖbsyicdflshEs¡ 354 ¨ 4 6 ¨ Q(RVBbdmkfsh>ihTlznpiTVghTVgbfb({Tkflhwlz{TyVBVghuTuyzVgb?dfswshwyzicV¨

(113) # ¹ ,Ä µ¼¹. +º. . 7. ' . MJ # "! . *2 7. 7. 32. !'32. 2. 0 k ε(h) ε(h) = O(hk ) ` = !$O(h ) ! ε7 (h) "$& $ $! ' ' 0 −1 0 k+1 ε (h) $ε(h) = o(1) ε(h) ε (h) ε(h) = Ch ε(h) = !' $'" # !$' &# 0 C' 0 h#k" '"G & !!E@& C C !$#%&$'"

(114) " ! " ' 0 #"!$#

(115) C# ε (h) = 0 '. $,. . . 0 ¹ + 7,Ôg µ + ' :, M H. 57. .. 57 .. ¶§¹ H ) " H75 2 7

(116) '$#> !*0 !$' , J < # $' # ' , < H !# # -. 57 ' !$#% > 3 c¨ 4-6 7#MJ # ' # !'"# H O) µ ¹ g µ µ ' + º u-,Ä , µ µ + gc7, ¹ +) O)&() ,¢1 ¹ º µ + µ $,Ä %g µ µ ) V!wshblocVk?de¬?s'³alzhTb(s¡ £kvbed(skvcVgk(bfwvRTVgU>VMb dfRTV °scciThs1¤bfwvRTVUWVuhdmRTV 1| =TkflVMckmlowvRbbfwvRTVgU>V¨ VkfVgwguyzyrTkflV;`±dfRTVlzk'wshbdmkfiw;dflsh RVkfV¨ V(cVgbfwkflOVrdfRTVlzk*wshbedfkmiwdmlzsh!¡¢sk VgyVgU>Vghpdfbg uuhdmRTV?cVkmVgVgbs¡T¡¢kfVVgTsU»ukfVGdmRV%VkmdmlowVgb s¡(dmRTVMbVVgyVgUWVhpdfbg¨®Q s±U¯³VWdmRVdmV|ad'blzUW{TyzVkM ¬?VlzU>{ylowldmyz`©bfbiTUWVWdfR1d'dmRTVVgyVgUWVhpdfb'ukfV dfkmlouhyzVgb?lh± Ticd+dmRTlob+lob+ubmsyzicdfVyz`hTsud+VMbmbmVhpdmlouyP¨ *2$G' !$M@# '"&!' 8Ý¡ H = H + H ¬+RTVgkmV H 3 kfVgbm{¨ H 6 lzb?wshaV| 3 kfVgbm{¨GwshwV 6 dfRTVh¬?V!bVd 3 I¨ 476 H (p , · · · , p ) = inf max sup [p · (y − q) − H (y) − H (q)] ¬+RVkfV Ω l = 1, · · · , k kmV°dfRTVuhTiTyouk.bmVgw;dfskvb+cV£hTVga`dfRTVBdmkflzhTyzVgb T , · · · , T 1d.hTsccV h ukfV!dmRTV VVhTkmV!dfkfhb¡¢skfUWb.su¡ H uh H ¨ VRVcVhTsdmVMp` x · y dmRTV M H Tsud+{TkfsaTiw;d.sH¡ x uh y ¨ 8Ý¡ lob(dfRTVbU¯yyzVgbd.kfclzib+su¡

(117) dmRVwlzkfwyVMb+su¡GwVghdfVk wshpdvulzhTVglzh Tl}¡ uh kmV lz{hbmwvRl}d KZwshbedvuhpdfbr¡¢sk H uh H pdmRTVgh¯dmRTV°bmwvRVUWMVZlobrUWshsudmshTV.{Tkf∪s1alzTTVg>dmR1LddfRTV.dflUWLV bdmVg{bf1dflzb£VMb H. H. . H. . . 1. G h. i. ∗ 1. l. 1. ki. 2. 1. 2. ∗ 1. i. q∈R2 0≤l≤ki y∈−Ωl +q. 1. ∗ 2. ∗ 2. 1. 1. ki i=1 j. i. 1. ki. 2. 1. 2. 2. ∆t 1 (L1 + L2 ) ≤ , h 2. shTV!whwshbmiTyd B 4 D ¡¢sk+UWskfV!cVdvulzyzbg¨ 8ÖhUWsbd?s¡¤dfRTVBhaiTUWVkflzwguyKV|TuUW{TyzVgb?OVyzs1¬! pdmRTV!¥ZUWlydmshTlzh5lob+wshpV|3 abs 3 I¨ 4-6 OVgwsUWVgb bmlzU>{yVgk 3 I¨Æ 6 H (p , · · · , p ) = max sup [p · y − H (y).] V±cVghTsudfV®p` H dmRTV±dmVkfU sup [p · y − H (y)] ¬+RTVkfVdmRTV¤uhTiTyzkbVMw;dfsk Ω lzbdmRTV bmVgwdmsk?s¡ T bVgVh¡¢kfsU0dfRTV!VgkdfV| M ¨ G h. T.

(118)

(119) # . 1. ki. y∈−Ωl. sup. Ú$P"Q;å;æ;æ. i. . pi · y −. ∗ 1. i. ∗ 1. '"# !$#%&$' 5 # J !$M@# '"&!' . y∈−Ωl. NON. 0≤l≤ki y∈−Ωl. H1∗ (y). . . . ,. l. ! , # M ! @"!$#%&$' 5.

(120) . 4I. H!$. + # ' 520# 0' # &!@7 G # ! !$ ' !$#% !@" + G h H#i" := # ' !*0 C ! 7 # ' #& M<M H H 7(σ i ,!u@"!i $, # u h)$ ! '"## H! 2$& '"# < # & ) . @' !$ "' # % !$' # 7. !$@"!# >$ !E # C5 # M '"## M"!$'"#%# , uh. .O" 7. #" . 7. # . σi Ω l. HiT. := sup y∈−Ωl. . pi · y −. H1∗ (y). . #

(121) # $

(122) 5

(123) #" '"# !

(124) ' GH. . σi. θli. Ωl. K.

(125)

(126) . O! & 52$&

(127). ¬+RVkfV. !$M@# $'"&!'. . . . . 3. . . K. . \¥.VgkmVB¬?VBbmVd. ¯) − HhLF (DuΩ1 , · · · , DuΩki ) = H(U. Ch. . kmVMb{ ¨ D 6 lzb+¯wlkvwyzV 3 c lobm³ 6 s¡

(128) wVhpdmVgk h. b= U. R. Dh. Mi. h. I. σi . [u(M ) − u(Mi )]dl,. ¨ 6. 3I. uhkvclzib h . Du dxdy. Ch. ,. h lob(yzkmVk?dmRhha` lz{bfwvRTl}dK!wshbedvuhpd+su¡ H clzalzTVga` 2π ¨ x cl²OVgkmVghd?Vgkfbmlsh¯su¡3dfRTV 1| =TkflVMckflzwvRb¥ZuUWlzy}dfshTlouh dfR1d?lzb%UWshTsdmshTV.iThTVk%dmRTVBbmUWV wshbdmkvulzhpdg alob(dmRTVB¡¢syyzs1¬+lzhT 1 HhLF (DuΩ1 , · · ·. , DuΩki ) =. Z. πh2. H(Du) Dh. πh2. − h. I. Ch. [u(M ) − u(Mi )]dl.. ¨ u 6. 3I. Ü< N ÜèÊ.

(129) 4.

(130)

(131) "!$#%&$'". . Q(Rlzb(VkvblzshwuhOV!kmVg¬+kmlddfVh®b HhLF (DuΩ1 , · · · , DuΩki ) =. X. X ~nil−1/2 + ~nil+1/2 θli tan θli H(DuΩil ) + ε · DuΩil . 2π 2. (Q RV'Vgw;dfsk ~n lob.dfRTV>iThTldBVgwdmsk\su¡rdfRTV>VgcVdfR1d!bVg{ukv1dfVgb.dmRTVWuhTiTyouk\bmVgwdmskfb Ω uh dmRTVhTyzV lob(dmRVuhTyVBs¡dfRTVuhiTyouk.bmVgwdmsk.ud σ bmVV

(132) =

(133) lziTkfV I¨Æa¨%Q(RTV!{ukvuUWVdfVk ε Ω lob?dmRVbmU>V!b(lhθ dmRV!{TkmVgalsib?Vkvbmlsh¨ 0≤l≤ki. 0≤l≤ki. l. l+1/2 i l. l+1. i. σi Ωi+1 ~ni+1/2 Ωi.

(134)

(135) . . . Ωi. . . . . . . . ~ ni+1/2. . . . x dfRTlzkfVgkfbmlsh a¬+RTlowvRlob?dmRV!shTV!¬?VBRV°ibmVglhdmRVblzUiTyo1dflshbg clzb HLF (DuΩ1 , · · · , DuΩki ) =. X. T 3Mi. |T |H(Du|T ) + α X. X. Mj ∈T. (ui − uj ). ß Þ;¾ . . ¨ w 6. 3I. |T |. h α ≥ h max ||D H|| ¬+RTVgkmV h lzb?dfRTV!youkfVgbd(VgcV°s¡ T ¨ Q(RV>U¯lh©cl²OVgkmVghwVWOVde¬?VVgh¤dfRTVgbmV¯cl²OVgkmVghd!¡¢skmU'iTyob°lzb!dmRud 3 I¨ 6 uh 3 I¨ 6 ukfV>lzhGF dfkmlzhbmlzwBlzhdmRV'bmVhbmV!lzVghlzh B 4 D ¬+RTlyzV 3 I¨ w 6 lob.hsudg¨ :?`5dfRTV¬(` cdfRTVbfuUWVlob(dmkfiTV!¡¢sk 3 I¨ 4 6 ¨ ¥ZVhwV ¡¢syzys1¬+lzhTBdmRVZbmUWV+kmV¡¢VkfVhwV 3 I¨ 6 uh 3 I¨ u 6 kmV.wshaVgkmVhpdruh'dmRV.VkfkmskGVgbdmlzU¯1dmV lob O(h ) ¨ =sk 3 I¨ w 6 bmiwvREh±VkfkmskZVMbedflU¯1dfVlobZhsud!lyouyV 3 1dByVMbd\¬+RTVgh®¡¢syzyzs1¬+lhT5dmRTV dfVgwvRThlznpiTVBsu¡ B 4 D TTicd.ld.lob+wshaVkfVhpd 1

(136) dmRTlob+lob.WblzUW{TyV{T{Tyzlzwg1dflsh5su¡ B D ¨ Q(RVEc1hdvuVs¡ 3 I¨ w 6 s1VkWdmRVEsudfRTVkde¬%s©VkvblzshbWlobldfbblzUW{Tyzlzwl}de`§lh wscclh¨»xZb 3 I¨ u 6 ¬%VWhTVgVg±dmsU¯u³V>ysas{¤s1VgkZdfRTVWVyzVUWVhpdgM ¨ =Tsk°VMwvR±VgyVgUWVhpdg ¬?VWwsUW{TiTdmV Du α T 3Mi. T. p. p. T. 1/2. |T. NON. Ú$P"Q;å;æ;æ.

(137) 4. . . H!$. hV1uyzi1dfV°¡¢sk+VgwvRTVkfVVBs¡ ¡¢kfVVgTsU lzhdmRTV!VgyVgU>Vghpdg |T |H(Du|T ) + α(ui − uj ).. (Q RV°haiTUWVgkmlowy¥ZuUWlzy}dfshTlouhlob(dmRTVkmldmRU>VdmlowBVkvuVZs¡ dmRVgbmVniuhpdmldmlzVgbg¨ Q(RVclzbfbml{udmlzsh©UWVgwvRuhTlobU¯bBlobU'iwvR©blzUW{TyVgkdfRuh¡¢sk 3 I¨ 6 ¨ /%shbmlocVkflh 3 I¨ 6

(138) dmRTV yzsas{Rb(dmsOVwgukfkmlzVgsicd\s1Vk(VMwvR®cVkmVgV!su¡

(139) ¡¢kmVgVgcsU¨?Q(RTVgh¡¢sk.VMwvRsu¡*dfRTVU ¬?V!hTVgVgdms U¯³V¯ysas{¤s1Vk!ldfb!hTVglRaskfbg¨ 8ÖhdmRTVwgbmV¯su¡ 3 I¨ w 6 dmRTVwscclh®lzb M" bmlUW{TyzVkM dfRTlzblzb ¬+Ra`¬?VB{TkfV¡¢Vk 3 I¨ w 6 ¨ O) ) , ¹ 1º µ , ¢ ¹ + ,Ô + ) Q(RTVkfVGukfVGU¯uha`.¡¢skmU¯uyzyz`.RTlzRBskvcVk3haiTUWVgkmlowy¥ZU>lzy F dfshTlouhb MTicd ld lob3UWskfVrcl ¯wiTyddms.wshbdmkfiw;d bdfTyzVruhBRTlzRBskfcVgk¥ZU>lzydmshlzhb9¨ =Tsk3V|TuUW{TyzV l¡rclobmwkmVdmV L ?pyzlz³Vbdfulyzl}de`{Tkfs{OVkmde`5lob.bsiTRpdM shV!Rb(dfskfVguyzLl KVBdmR1d.dfRTVkfVlob+hTshudmiTkvuy wsiThpdfVkf{ukmdsh©dfRTVwshpdflhaiTsibbmlzTV5OVgwibmV5su¡+dfRTVh1dfiTkmV5su¡.dfRTV5alobmwsbml}de`bsyiTdmlzshb 1¯dmRTV dfVgbd+¡¢iThwdmlzshb\ukfV°hsud.lzhdfVkfudmVga`5{kd.b(¡¢sk.bdfhTukvRpà{OVkfsylowB{TkfsTyzVU¯bg¨Q(RTV!hudmiTkvuy ?aylz³V.bdfTlyzlde`'{Tkfs{OVkmde`RbGdfsOVZ¬+kfl}dmdmVh5sh dfRTlobwibmVgbbmsUWV.cl wiTydmlzVgbdmR1d%U¯`>V L s1VkvwsU>V TbVgV!¡¢skZV|TuUW{TyzV B 4 I D uhUWskmV!kfVgwVhpDu dmyz` BÆu D ¨ 8Öh®dfRTV!¡¢syzyzs1¬+lhT ¬?VB£kvbed\wshbmlocVkZ ¡¢skmU¯yyz`RTlzRWskvcVgkrbmwvRTVgUWV(dmRudu{T{OVgkfb

(140) dfsV.bdfTyzV.bGyzshT!bGhTsbmlhTiTyoukfl}de`s¡ Du u{T{OVgkg¨ 8Ýd¯lob>lhªdmRTV®bm{Tlzkmld>s¡ZdfRTV X ?pyzlz³VbmwvRTVgUWVs¡ B c 4D h B 4 D ¨§x2bmVgwsh@V|TuUW{TyzVlzbWuyobms wshbmlocVkfVg¡¢syzyzs1¬+lhT0 B 4 D ¨ZQ(RVbmwvRVUWVlobZUWskfVwsUW{wdg OTicd\shTyz`Tu{cdfVgdfsRTsUWsVhVsib ¥\uUWlzy}dfshTlouhbg¨G¥.s1¬?VVkM ¬?VB{Tkms{spbV°uh !2 7 V|admVhblzshdms>dmRV°lzhTRTsUWsVghTVsib?wbmV°lhdmRlzb {u{OVkM¨ $# !$#%&$'" 8ÖhWdmRTlob%{ukvukf{TR 1¬?VZwshblocVgkrVyzVUWVhpdvbGdmR1d%kmV.lzUW{Tyzlzwl}dfy`'dmRTsiTRpd%bGdfkmlouGh F iTyouk+VyzVUWVghdvb Ticd.dfRTlob.kfVgbdmkflzwdmlzshlzbZhTsud.VMbmbmVhpdmlouyP¨ V!£kvbdZ{TkfVgwlobmV\dfRTV!deà{OVsu¡GlzhdfVkf{syzhpd 3 kmVMwshbedfkmiw;dmlzsh 6 s¡ ¬%VwshbmlzTVk(lhdfRTV!V|TuUW{TyzVgbg¨ {u } Q¬?s®³plzhTbsu¡. kvuhTVdeà{OVlzhdfVkf{syzhpdfbkmVwshbmlzTVkfVg3¨®_alzhwV¯dfRTVkfV¯lobhTsEUTlzil}de` ¬+RVh¬%VwshbmlocVklVhlhpdfVkf{syzudmlzsh u kfV{kmVMbVghdvb°dmRTV5lhpdmVgkm{Osyo1dflsh¤s¡%dfRTV5wshpdmlzhaiTsib ¡¢ihw;dflsh u ¨ 8Öh dmRTV¤£kvbedwbV .lh VgwvR dmkflzhTyzV ZdmRTVbmsyzicdmlzsh lobu{T{Tkfs|clU¯udmVga` ª{syàhTsU>louy s¡ZcVkmVgV r GdfRTVlzk¯bVd'lob P (T ) ¨¥ZVhwV GTdfRTVbmsyzicdflsh«lzb>cVMbmwkmlzVM«a` cVkmVgVgbs¡ ¡¢kfVVMcsU¨ 8Ýd!lzb!³ahTs1¬+hEdmRud!dmRTV¯{Oslzhpdfb!s¡ T ¬+RTlowvR¤km`cwVhpdmkflzw>wsaskfclzhudmVgbBkmV ( , , ) ¬+ldmR {spbldmlzV°lzhdfVVkvb%h i + j + k = r kmV\iThTlobsy1hdM¨ |TuUW{TyzVgb%¡¢sk r = 1 2 kmV°clob{Tyo`VM i, j, k sCh =

(141) lziTkfV I¨ c¨rQ(RTVcVgkfVVMb%s¡ ¡¢kfVVMcsU σ lzhdfRTV!UWVgbmRlzb?dfRTVwsyyzVgwdmlzshsu¡

(142) dmRTVMbVBhTVg¬ {Oslzhpdfbg¨ ZdfRTVkWwvRTslzwVgbkmV{spbmbmlzTyVbiwvR@bdmRTVlzhdfVkf{syzhpdfb>cV£hVg«lzh BÆ D ¨©Q(RTVblzUW{TyzVgbd'VJ| F UW{TyVlzb\cVgbfwkflOVg lzhEuha`dfkmlouhyzV a`dmRTV'cVkmVgVgb+s¡

(143) ¡¢kfVVMcsU wshbmlzbdmlzhT¯su¡*dfRTsbmVsu¡*dfRTV P lzhpdmVgkm{Osyo1dflsh5{Tyzib?dmRVBwVhpdmkfslos¡ dfRTV°dmkflzhTyzV cbmVV =

(144) liTkmV I¨ IT¨r\VhTsdmlzhTWa`¯Vgw dmRVBbmVd+s¡ ¡¢ihw;dflshbZs¡GdmRTVdeà{V λb ¬+RTVkfV λ ∈ R dfRTV'lzhpdmVgkm{Osyo1dflsh±b{wVlob Pe (T ) = P (T(b)) L Vgwd (b) ¨ Q(RV°iTTTyzV!¡¢iThwdmlzshlob Tl¡ {Λ } cVhTsdmVMb?dmRTV!km`cwVhpdmkflzw°wsaskfclzhudmVgb(lzh T b = Λ Λ Λ ¨ Q(Rlzb!lzhpdmVgkm{Osyo1dflshbm{wV¯lob!ibmVglzh¤dfRTV¯de¬?s±clzU>VghbmlshuyrV|TuUW{TyzVgbg¨ 8ÝdàlzVyoTbdfRTlkvskvcVk wwiTkfudmV°lhpdfVkf{syzudmlzsh Th5Vgh1tes1`cb?dmRTVB¡¢syyzs1¬+lzhT>npickv1dmikmV°kmVgyzudmlzsh Z X 3 I¨ I 6 ω f (x ) f (x)dx = |T | 2. 2.

(145). i i=1,··· ,nΣ. h. (r+1)(r+2) 2 i j r r. k. k r. 2. 2. j j=1,3. 1. j. T. 2. 2. 3. j. j. Ü< N ÜèÊ.

(146) 4.

(147)

(148) "!$#%&$'". 1. 1. 6. 4. 2. 3. 2. ¬+ldmR ω = ¡¢sk j = 1, . . . , 3 ω hsudmRVk+V|TU>{yVMb¨ 1 20. 3. 5. r=1. r=2. . j. 9. . . j. =. . .

(149) . 2 15. . r = 1 r = 2 . ¡¢sk j = 4, . . . , 6 uh. ω7 =. 9 20. ¨G_cVV ÆB D ¡¢skrUWskfV+cVdvulzyzb. 3. 6. 5 7. . . . 2. 4. 1. . .

(150). . . . . . . . . . V\TVgbfwkflOV.dfRTVZblobG¡¢ihw;dflshb%¡¢sk P (T ) P (T ) uh Pe (T ) ¨ V.¡¢syyzs1¬§dmRTV°hTsudv1dflshbrs¡

(151) lziTkfV I¨ 'hCI¨ IT¨ T µ P (T ) ¨ V!RV N = Λ ¨ • T µ (T ) ¨ V+RV N = Λ (2Λ −1) uh'bmlUWlzyzk*¡¢skfUiTyoZ¡¢sk i = 2, 3 N = 4Λ Λ • hbP lzUWlyouk(¡¢skmU'iTyz¡¢sk i = 5, 6 ¨ T µ ¹ Pe (T ) ¨GQ(RTV°bmlobr¡¢iThw;dmlzshb%kmV N = Λ (2Λ ) + 3b hbmlUWlzyzk¡¢skmU'iTyz!¡¢sk • uhbmlUWlzyzk(¡¢skfUiTyo'¡¢sk i = 5, 6 N = 27b ¨ i = 2, 3 N = 4Λ Λ − 12b 1. =. 1. i. 2. 1. 2. NON. Ú$P"Q;å;æ;æ. 4. 2. 2. i. 1. 1. 1. 1. 3. 1. 1. 7. 4. 1. 4.

(152) 4. ;. . H!$. Q(RVGUWlh!cl²KVkfVhwVrVde¬%VgVh°dmRVrbmlobO¡¢iThw;dflshbs¡ P (T ) h\dmRTspbVGs¡ Pe (T ) lzb3dmRud Z N dx = ¡¢sk σ = 1, 2, 3 lh P (T ) ¬+RTlzyzV Z N dx > 0 ¡¢skhp` σ lzh Pe (T ) ¨Q(RTlobBVgRalskB{Tyz`cbBuh 0 lzUW{skdvuhpd+kmsyVBlzh5dfRTV!hTV|pd.{ukvukvu{R¨ ZdfRTVk(deà{OVgb(s¡ ukvuhVZlzhpdmVgkm{Osyouhpdfb(wsiTyoRV°VgVh®wshbmlzcVgkmVM3¨ O) )&() &, µ M µ µ ) /%shblocVk%uha`'cVgkfVV+s¡O¡¢kfVVMcsU ¨ 8Ýd%OVyzshTpb*dms bmVVkvuyOdmkflouhTyVMb T c{spbmbmlzTy`5shTyz`5shTVBl}¡ r ≥ 3 ¨%ZVghTsudfV!p` V dmRTV!yzlzbd.su¡

(153) dmRVgbmσVBdmkflzhTyzVgbg¨ V TV£hTVBdfRTV!RTlzRskvcVk+¥\uUWlydmshTlouhb 3 I¨Æ 6 H := H (Du , T ∈ V ) ¬+RVkfV H lzb¯hp`©s¡ZdfRTVys1¬-skvcVkW¥ZU>lzydmshlzhb¯cV£hTVg§uOs1Va` 3 I¨ 4 6 3 I¨Æ 6 3 I¨ 6 sk 3 I¨ u 6 ¨ ZdfRTVk.RlR®skvcVk.¥\uUWlydmshTlouhbg bmiwvR®b(dfRTVRTlzRskvcVgkZwVghdfkfy3shTV!s¡ B c 4 D wsiTyo RV°VgVh®wshblocVgkmVM3¨ 8Öh«dmRTVVgkm`Vgkmyz`Tkfu¡èd'su¡+dfRTlob'¬?skf³K 3 I¨Æ 6 Rb'VgVhªlUW{TyzVUWVhpdfVg«h©dfVgbdmVg¨ Vcs hsudB{TkmVMbVghpd\dfRTVWkmVMbiTydfb°RTVgkmV 3U¯ulzhTy`®OVgwguibmV>¬%V>¬?hpd\dfsbdmkfVgbfb\sh¤bmlzU>{ylowldeÈbm{OVgw;dBsu¡?siTk TVkfl11dflsh¨ 8Öh!dfRTV%wgbmVrsu¡ X

(154) X @deà{V?bmwvRTVgUWV MdmRTV%kmVMwshbedfkmiw;dmlzsh!bdmVg{lzb Vgkm`°wsUW{TyzV|3 VMb{OVgwlzyyz`W¡¢sk+ihbedfkmiw;dmikmVM5UWVgbmRTVMb¨ 8ÖhdmRTV!hV|ad.bVMw;dmlzsh c¬%VB{TkfVgbmVhpd.>UiwvRblzUW{TyzVk(UWVdmRsa ¡¢sk.wshbedfkmiw;dmlzhT>RTlzRskvcVk+bfwvRTVUWVB¬+RTlowvRRb(yzVgib(dfsWuhcshdmRTV X shTV¨ O) ) ) ¹ TuWg µ µ + µ ,Ä , ¢ ¹ + ,Ô + ) 8Öh5dfRTlzbZbVMw;dmlzsh a¡¢sk+uha`¯dfkmlouGh F yV admRTV!lzhpdmVgkm{Osyouhpd u OVyzshb?dms P (T ) sk P^ ¨ !# &' V¡¢syzys1¬ B 4 D 1?lh®dfRTV'wgbmVsu¡r(Th±) RTsUWsVhVsib+¥ZuUWlzy}dfshTlouh®s¡cVkmVgV k lh c¡¢sk+V|TuUW{TyzV p 2. 2. σ. T. 2. 2. σ. T. σ. H σ. M. h Tk. k. σ. M. h. 1. 2. . ¬+ldmR. H(x, u, p) ≡ H(p) − f (x) H(p) =. ¬?V!whyzsas³5ud. 1 Dp H(p) · p, k. b+WwshaVgw;dflsh5{TkfsyVgU ¬+l}dfR®bsiTkfwVZdfVkfU . H(x, Du) = 0. ¨ 6 ¬+RVkfV ~λ = D H(Du) ¨ Q(RVh>¬%V+wshbmlzcVgk*dmRTV(iThbedfVgT`{TkfsTyzVUbmbmscwlo1dfVg!dfsBdmRTlob*Vgnpiudmlzsh 3 ¬+ldmRTsicdGdmRV+siThJF km`wshTl}dflshb 6 ~λ · Du = f (x). 1 k. 3I. p. ∂u 1 + Hp (Du) · Du = f (x) ∂t p. Ü< N ÜèÊ.

(155) 4.

(156)

(157) "!$#%&$'". . lP¨ V¨rWwshpVgwdmlzshJ?aylz³V\{TkmsTyzVU ¬+RTlowvRlzb.{T{Tkfs|clU¯1dfVg3 aalodfRTV_cYZq U>VdmRTsc ÆB $I D Tb . Z. ∂u ω + H(x, Du) dx + ∂t Ω×[tn ,tn+1 ]. Z. ∂ω ∂u + DH · Dω τ + H(x, Du) = 0. ∂t ∂t. ¬+RVkfV ω lobBuha`®yzlhTVMukBwsUTlzh1dflshEs¡rdmRTVWblob\¡¢iThwdmlzshb Q(RV°{OsbmldmlzVBkmVMuy3haiTU'Vgk τ lob?deà{Tlowuyzyz`5wvRTsbmVhb τ=. XZsudflzwlhT>dfR1d. . Ω×[tn ,tn+1 ]. 2 ∆t. 2. +. . 2|Dp H| h. 3 I:¨ 9 6 + ¬ } l f d R z y z l T h g V ° k z l ± h f d lUWV¨ N (x)ϕ(t) ϕ σ. 2 −1/2. .. ω(x, tn+1 ) − ω(x, tn ) ∂ω = , ∂t ∆t. iblzhT 3 I¨ I 6 ¬%V{T{TyzÙ¯bfb(yiTUW{TlzhT¯uhVd. un+1 − unσ σ + Hσ (σ, uσ , uξ ξ∈Vσ ) = 0 ∆t. ¬+ldmR Hσ (σ, uσ , uξ ξ∈Vσ ) := Z. H(σ, Duh )Nσ dx Ω. +h. Z . !−1 ! Z Dp H(σ, Duh ) h

(158)

(159)

(160)

(161) Nσ dx

(162)

(163) Dp H(σ, Duh )

(164)

(165) · DNσ H(σ, Du )dx Ω. ¨ ; 6 8Öh 3 I¨ ; 6 lob

(166) dmRTV(U¯1|clzUiTUTlzU>VdmVgkGsu¡TdfRTV(dmkflzhTyzVs¡dfRTV(U>VMbR ¨ Q(RTV+¥\uUWlydmshTlouh 3 I¨ ; 6 ¬+lzyy OV°ibVM¯¡¢hsk 3<4 ¨ 4-6 c¬+RVh H lzb?RTsUWsVghTVsibrlzh p = Du ¨G[®sccl£wg1dmlzshb¡¢skdfRTVBlhRTsUWsVhTVgsib wgbmV\lob.wshbmlzTVkfVgyo1dfVkM¨ VrRV*dmRTV%blzUW{TyV VUWU¯(¬+RTlowvRwh°VMbmlyz`ZOVrVhTVgkfylLKVMZdfs+sudfRTVk3deà{OVgb su¡clhpdfVkf{syzhdvb¨ Ω. 3I. . .O H.

(167) # !# G C 1 !$'2 $'" %2E #" ' # % !$' #' P1(T ) $ P^ 2 A' !$ G' #%&$'" !$H H 52 .!-,H$ # $G'32E!$& 579H !# &' 3 I ; 6 2 ' !M(T ') # !'"# !$M@# $'"&!' ' #"

(168) !$'"' ' , 2E '"#%&$' ! H-< 8 = h. ¨ ¨ Ý¡ u lzb(yzlhVgukM u = u ¨ Tsk.ha`5cVkmVgV\s¡¡¢kfVVMcsU σ T¬%VBRV Z. NON. Ú$P"Q;å;æ;æ. H(σ, Duh )Nσ dx Ω. ! Z. Ω. Nσ dx. !−1. = H(σ, Du(σ)).. ¨ 6. 3I. .