Error estimates for stochastic differential games: the adverse stopping case
32
0
0
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Error estimates for stochastic differential games: the adverse stopping case J. Frédéric BONNANS — Stefania MAROSO — Housnaa ZIDANI. N° 5441 Décembre 2004. ISSN 0249-6399. ISRN INRIA/RR--5441--FR+ENG. Thème NUM. apport de recherche.
(3)
(4)
(5)
(6)
(7) !"# $%
(8) '&)(' +* , ,-#. //
(9) $,(0+1 23547698;:<8=6?>A@CBDFE,E,GHEI J I1KMLONQPSR<>TP)UVGWDCI<D J5XZY\[<] R<P\P_^+` aGE,` b#cedfhgikjml nporqtsvuwdfhg?s xryefhz?{}|~=yeg?s 5{wvgumo= OeSO{}u eg{}g9ceg{cegxS<On kz9 gf%e{wgOOOnO'Og9s ∗. †. ‡. Z? ¡¢e£O O¤¦¥ g§Oeuw|¨x¦g{w{wO{SOyex©©sª«O{%fhOxeOu}Ox©g§ee{wM¬t|fu}|¨Ox®s}ceg?fhg?sOªH©{}u}| ye¯{ °vswO;s g9~=y©uw|;x ±b#ce|sk|skOxCg ¬ru}g?x©s}|;xFª<u}ceg'u}ceg?O{wqZª«;{kg9svuw|fuw|x©Zg{w{wO{s+ª«O{uwceg%²mfh|¨¯³uwOxt´ µ=OOe|´·¶¸g?¯¯¨fx§g9~=y©uw|;x ± ¹©O{Otu|¨xe|¨xeZye©g?{g?{}{wO{m;yex© º »+g%Ox©s}|tg{mu}ceg¼}½{}qr¯¨¾,{wg;ye¯¨O{}|¨¿?u}|¨OxrÀª7uwcegh°vs};O?s g9~=y©uw|;x¦u}"eye|¨¯¨_x_e©{}M¬t|¨fhu}gsQyee´Ts}O¯¨ytu}|¨Ox)ªuwcegFs}ceg?fhgO±ÁbÂ;g uh¯»+g{%g{w{};{%SOyex »+gg ¬ru}g?x©Zu}cegfhg uwcetª¶#O{}¯¨g?s#xµ=Ã;O©s}gxÅÄ Æ » ce|¨cFOx©s}|¨sQuws+|¨xH|x=u}{wtty©|xehhsQ» |uwc©|xe s}qtsvuwgf» c©;s}gs}O¯¨ytuw|;x©s {wgm¯¨tO¯ sQyeSg{}´·sQ;¯ytuw|;x©s+ªu}ceg°vswO;s+g?~=y©u}|¨Ox ± ÇFÈ=É\ÊwË'Ì ¡ÍO¤ °vswO;sg9~;yu}|¨Ox1º<²mfh|¨¯³uwOxt´·µ;; ;e|³´·¶+g¯¨¯fOxÅg?~=y©u}|¨Ox º7sQu}tc©;svuw|¨t|Î\g{wgx=u}|¯ =fhg?s?º=Ï©xe|u}gt|Î\g{wgx©g?s?ºeg{w{};{¸g9svuw|fu}g?s?±. ý þ äQÿâêÐMìQÑç ÒèÓÕÒÔQÖ9ÓâìÔQêMÓÕ×ÙØvØ äÚ ¨ÒÛ<ÓÕÜävÖMÜävÝ<Óâ×ÙØ!ÜÑ Ú\ÞOßá!êMà©êMÓâìáêMãÕä·ÿåæÞ×ÙêMç?ÓÕÖM×íáì9ØQÑ áMÓTè;Ñ éëêMÓÕ×íìwî«ï á?ØvðñMävêMØvá!ñMÓÕåQèeòá!óì×Ùê9ä7ÖMäæôá!õÙñMØväQìñ=è=Ú àÅöQ÷!ø9èrùú9öQø!ûü=ä ý þ äQÿâêÞ ìQåAç ä è¨ìÒêMÓâ×íìì êMØvä Ý<ëÞ ï åAÛ7ä ¨ÞMìêMÛ¸×íì9ß\Ñ àÓÕìáÓâãâá!ä·åÿâ á Þç9×ÙêMÖ9ÓÕá?×íØvì9áMÑ ¨è< Ó ·éëÑ ê9Óâ×íìwî«ï áØQðñMävêMØvá!ñMÓ«å}è¸òá!óì×Ùê9äÖMäôSþ á!õÙñMØväQìñ=è<Ú à öQ÷!ø9è<ùú9öQø!û§ü=ä ÞtݸèM û á!#ñMÚ\ÿâêá!ì!ñMì õÙä
(10) !ìé«ò1ÓÕÖÝô<Ü×ÙéTØ·ß=åâá!àÓQÓÕèáùãÕävø å¸ùû Þ ç9ÖMà©áìØvÓÕáM×Ùÿ èeý éëêMävÓÕÖM×íìwä 'î«ï1öQáø9ØvèMð?Òñ9ÓâäQìê9êMØQØvá! ä ñ9ÓÕ å#v×ÙÖìêMìêMÖ × ävêMêM× åâÿÕÔ#åTì9ÖMÑ äÓ TÑ ìwå Ôvóìwåâ×Ùð?ñMävÿ7ÝMõÙ×ÙðñMÔväQÿQèSÜ î ∗. † ‡. Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France) Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30.
(11)
(12)
(13) $, Q#<Z/Â,
(14) ! #< $%
(15) & <
(16)
(17) * & + &v+ Z-#<1 ¤ ikOy©sæOtuwgxe;x©sætg9s5g9svuw|fuw|;x©s5 g{w{wgye{s7SOye{+tg?s¸s}cez?fOs5 ©e{}M¬t|¨fu}|¨Oxhfh´ x©u};xeg?s# yexegz9~=y©uw|;x, °vs};O?s¸{}u}| ye¯¨|d?{}g;±æ¸g? | Ox©sQu}|u}y©gy©xegg ¬ru}g?x©sQ|¨OxHtg¯'uwcez;{}|¨gtg?s g9svuw|fu}|¨Ox©s g{w{wgye{s+;ye{ ¯ z?~=y©u}|¨OxHtg²kOfh|¯u};xt´Tµ=OOe|´·¶¸g?¯¯¨fx ± 1gfv;{wOx;ukeg¯ g{w{wgye{g9svuOtuwgxryeg{k¯fhz uwcettg%egm¼v{wz;ye¯{w|¨swuw|;xtg%½{}qr¯¨¾OÀ!º©~=ye|g?{Q´ fhgu¯ Oxsvuw{}y© u}|¨Ox" y©xegsQ;y©sv´·s}O¯¨ytu}|¨OxÅOee{wtcezghty®s}cez?fe ± 1g fh|¨xeO{x=ug?sQu;tu}g?xryÂg?x zu}g?x©ex=uæ¯fhz uwcettgktgm¶+O{}¯¨g?sæx©µ=Ã;O©s}gxÄ ÆAºr~=ye|Ox©s}|svuwggx ¯ |¨x=u}{wtty©!uw|;x§ yex§sQqtsQu}d?f g Ofhf'ytuuw|;xt;x=u#¯¨g?sksQ;¯ytuw|;x©s#sQ;x=u eg?ss}ye{sQ;¯ytuw|;x©s+¯t?¯¨g?s#tg ¯ z9~;yu}|¨OxH °vs};O?s± ¤ ! ~;yu}|¨Ox¦ °vswOO?sº z9~=y©uw|;x"tg§²mfh|¨¯³uwOxt´·µ;; ;e|³´·¶+g¯¨¯fOx º=vgyt¬®t|³Î\z{wgx=uw|g?¯¨s Ì 9 Ê £ "" sQu}tc©;svuw|¨~=yeg9sºtt|Î\z{wgx©g?s+Ï©xe|¨g?s?ºeg9svuw|fuw|;x©s g{w{}g?ye{9±.
(18)
(19)
(20)
(21) !" #$
(22) %'&( *)+ ,- . * /0102
(23) #(&3 *. 4. $% 1< ,æ
(24) $ b#c©gm|¨f ª uwce|¨s¸©Og?{¸|s5uw';|¾;gg{w{};{5SOyex©©s5ª«;{¸©e{}M¬t|¨fu}|¨Ox§swceg?f g9s5Oª1©{}u}| y©¯¨O{æxe;x Oxr¾Og¬§°vs};O?s¸g9~=y©uw|;x ±7lFO{wge{}g9 |sQg?¯q»+g ;x©s}|¨tg?{+u}cegª«;¯¯¨» |x©%g9~;yu}|¨Ox 7 8 min{sup L (x, Du(x)); u(x) − ψ(x)} = 0, x∈R , » c©g{wg 5. 6. α. N. α∈A. Lα (x, Du(x)) Lα (x, t, p, X). = Lα (x, u(x), Du(x), D2 u(x)), = − tr[aα (x)X] − bα (x)p + cα (x)t − f α (x).. ²kg{wg tg?xeuwg?s uwcegCsQgu§ª ;x;uw{};¯¨s?ºæ;s}s}yefhg?)u}¦SgHÏ©xe|u}-g 9 u}cegC?Os}gHª Ofh©;A!um=s}g u{α» |,¯¨¯. ..gh, αtg?O}¯³u» |³uwcÅ|¨xÅs}g?!uw|;xC©± 4e±b#ceg%rg :§ |¨gx=uws (a , b , c , f ) {wgOºª«O{g?;c º#;yex©tg9 x© |¨©swce|³uw¿Hª«yex© u}|¨Ox©s º¸» c©g{wg tg?xeuwg?s α ∈ A uwceg%s}g uª N × N sQqrfhfhg uw{}|fuw{}| g9s*9 ψ |sR§SO→yex©Seg? ×1|Rs}ce×|u}¿'Rª«ye×x© Ru}|¨OxCª«{wOf SR |¨x;uw R ± jkx©tg?{¯¨;s}s}|¨?¯<OswsQy©f eu}|¨Ox©s?º 8kc©OsZyexe|~;y©g%SOyextg?ž=|sw ;s}|uvqs}O¯¨ytu}|¨Ox1º\egxeOu}g? u ±b#ceg {wg;ye¯{w|³uvqhª u tg?g?x©es ;xZuwceg7 e{wOSg{}u}|¨g?s#ªu}cegr g :§ |¨gx=uws a, b, c, f ± b#c©|¨s<e{wO©¯g?f|¨s7©O{Quw|¨ye¯{7;sQg#ª\svuwtc©OsQu}|#t|³Î\g{wgx=uw|¨O¯t;fhg9sº?¯¨¯g9uwcegOt¾;g{sQg+svuwOe©|xe ?Os}gO±#°·xFªÕ;!u?ºS»+g'OxFxeuwgu}cuk|¨x 7 8 »+gc©M¾;guv»¸§©¯¨Mq;g{s º ; x©F¶±S5¯MqOg+{ ;c©;sZs}g ukOª Ox=u}{wO¯skOx©F»#x=uwsu}fh|xe|¨fh|¿?g'uwceg ;|¨x ±m5¯¨Mq;g{¶ xÅOxe¯¨qFtg?|¨tg'u}HsvuwOFuwceg ;fhg» |u}c uwcegOevg?!uw|¾;gmOªf¬r|¨fh|¿?|x©%uwceg;O|x1± b#c©gx»¸gOx©s}|¨eg{#fhOxeOu}Ox©gOee{wM¬t|fuw|;xZswcegfhg?s#Oª 7 8 ºeªu}cegª«;¯¯¨» |x©%ª«;{}3f < min{S(h, x, u (x), u ); u (x) − ψ(x)} = 0, x∈R , 7 18 » c©g{wg S |¨s# OxsQ|svuwgx=u?ºrfhOx©u};xe|mOx©§yex©|³ª«;{}fh¯¨qZ ;x;uw|xrye;y©s+ee{wM¬t|fu}|¨Ox§Oª sup L ± ¥ g'» |¨¯¨¯x©u}g u ∈ C (R ) u}c©g s}O¯¨ytu}|¨OxFOª 7 - 8!º\» ce|cC|suwceg%Oee{wM¬r|¨fuw|;xFª u ºSOx© h u}ceg fhg9sQcÅs}|¨¿gO±b#ce|smO©svuw{w;!uxeOuwu}|¨Ox©sm»+;s|¨x=u}{wrey© g9H=qF¶#{w¯¨g?sOx©Åor;ye;Oxe|¨e|¨s'Ä 4Æuw,t|sQ©¯¨Mq ¯g9{w¯quwceg#f ;xeuwOxe| |uvqªeuwceg s}c©gfh1g < x, r, v) |s<xeOx tg?{}g9Os}|x©|x r Ox©xe;x'|¨x© {wg?;sQ|¨xe |¨x v ±b¸qre|¯<©e{}M¬t|¨fu}|¨OxFswcegfhg9s u}cS(h, u»¸g%» |¯¨¯Ox©s}|tg{{wg ¸¯¨;s}s}|¯¹|xe|u}gh m|³Î\g{wgx©g?s Ĩ9MÆA$º =g?xeg{¯¨|¨¿g?,¹|xe|u}g% k|ÎSg?{}g?x© g9sÄ Æ1x©"Ä ÆTº©x©lCO{}Ã;¾,¸c©|¨>x ;kee{wM¬t|fuw|;x©sÄM9ÆT± jkx;uw|¯xe»ºk{}g9sQye¯uws,OxV Oxr¾;g{wOgx gH{wu}g9s§ª«;{ZfhOxeOu};xegÂOee{wM¬r|¨fuw|;x swceg?f g9s§Oªu}ceg g9~=y©uw|;x,» |u}cHOx©g©¯¨Mq;g{#c©M¾OgSggxH;tuwO|xeg9? 9t|A± g;±<ª«O{#u}c©gª«O¯¨¯» |¨xehg?~=y©u}|¨O@x < M. 1. α. N. N. α. α. N. α. N. N. h. h. b. h. N. h. α∈A. N. sup Lα (x, Du(x)) = 0,. x ∈ RN ,. α. 7 4-8. Ox©Zu}cegswcegfhg S(h, x, u (x), u ) = 0, x∈R . 7 (8 ! {}{wO{g?sQu}|¨fu}g9smª«O{u}ce|sg?~=y©u}|¨OxÂc©M¾Og g?gx"Otu|¨xeg?ÅrqŽ{}qr¯¨¾ÂĨA4ÆAº¨Ä?Æ5xCuwceg?s}gh{}g9sQye¯uws »+g{wgg ¬ru}g?x©tg?rqZ¶#{w¯¨g?s x,µ=Ã;O©s}gxÂÄ!ÆTº¨Ä ÆA± mye{w|x©Áu}c©gÅ{wg?e;!u}|¨Oxs§ª'u}ce|sZSg{»¸g¯g9{wxeg?_uwc©uF¯sQ µ;OÃOOsQg?x_»#OsZ»¸;{}Ãr|¨xeÁ;xu}ceg Oxr¾Og?{};gx©gk{uwgª«O{ fhOxeOu};xegee{wM¬t|fu}|¨Ox©s+ª 7 8 7 s}ggĨO!BÆ 8!±+²g;tuwO|x©s#g?{}{wO{#g9svuw|fuwg?s α∈A. h. ïï êC©øDDMö. h. N.
(25) . @ *
(26)
(27)
(28)
(29) $ #
(30) , . #'-!#"$
(31) %". |¨x§uwcegOs}gkOª Ï©xe|u}gt|Î\g{wgx©g?s+s}ceg?fhgm» |u}c,fhu}{w|³¬ a |¨x©tg?g?x©tgx=u#ª x ºty©s}|xehSgx¯¨|¿9u}|¨Ox Oee{w;;c ± ¶+qy©sQ|¨xeuwceg'fhgu}cetesk|x=u}{wtty©g?Hrq¶#O{}¯¨g?sOx©Fµ=ÃO;©s}gx"Ä ÆAºS»¸g'O|¨¾Og ;xr¾Og{wOg?x© g{uwg ª«;{fhOxeOu};xe|¨Oee{wM¬t|fuw|;xÂswcegfhg9sª+u}ceg§uv»¸Fe¯MqOg?{g?~=y©u}|¨Ox ±b#ceg§yeeSg{%g?sQu}|¨fu}gOª |sg?Os}|¨g{u}HOeuw|¨xÂu}c©OxÅuwceg¯»+g{9±%b#c©ge{}rOªæ|xr¾;O¯¨¾Og?s¼}½{}qr¯¨¾{wg;ye¯{w|¿9u}|¨OxtÀOª u − u 8 e º â | ± O g < ± uwcegg?{Quwye{wg9,g9~=y©uw|;x 7 min{ sup L (x + e, Du (x)); u (x) − ψ(x)} = 0, x ∈ R , 7 18 Ox©"|³us¾r|¨sw =sQ|uvqÅs}O¯¨ytu}|¨Ox u ± ; {wg;ye¯{w|¿9u}|¨OxŪ u rqÂOxr¾O;¯ytuw|;xÂ;|¾;g?sOx®Oee{wM¬t|fuwg s}fhru}cs}yet´·sQ;¯yeu}|¨Oxhª 8!ºrtgx©u}g9 » ce|c|s5¯sQOx§e©{}M¬t|¨fhu}g s}yet´·sQ;¯ytuw|;xhª 1 8 ±5or©º rqÂy©sQ|¨xeu}c©gZ ;x©sQ|sQu}gx 7 qCe{wOSg{}uvqOº1»¸ug;tuwO|x"u}c©gyeeSg{SOyex ºªëu}g?{'cer;s}|x©Ox";7 tu}|¨f¯ {fhg uwg{'Oª{}g?Oye¯{w|¨¿?uw|;x ±Âjkxtª«O{}u}y©x©uwg¯¨q¦»¸gx u e{}tgg?Á|x_Ås}|fh|¨¯¨O{%»#Mq"u}Å©ye|¯) s}fhru}cFs}yeSg{}´Ts}O¯¨ytu}|¨OxHOª 7 8#» c©|¨cF»+Oye¯¯g9Ouwu}ceg'¯»+g{ g9svuw|fuwgOx u − u ±+°·xsvuwg?OHOª ,sQfhruwcÂs}yeSg{}´Ts}O¯¨ytu}|¨OxF»+g eye|¨¯¨ÂZs}g?~=yegx g ª5¯¨r?¯7s}f rOu}cÅs}yeSg{}´Ts}O¯¨ytu}|¨Ox1±m°·x©{}u}| ye¯{ »+g|x=uw{}tty©gu}cegª«;¯¯¨» |x© s}» |³uce|¨xe§sQqtsQu}g?f» ce|cOee{wM¬r|¨fuwg?s 7 8 h. α. ε. ε. N. α,|e|≤ε. ε. ε. ε. h. min{max{Lαi (x, Dvi (x)); vi (x) − min{vj (x) + k}}; vi (x) − ψ(x)} = 0,. 7'& 8. ª«;{ x ∈ R º Ox© i ∈ I = {1, . . . , M } ±¹©O{¯|u}g?{wu}ye{wg';xCuwcegsQ» |uwce|¨xeHsQqtsQu}g?fhs?º sQg?g,Ä & ÆTº7Ä ÆTº Ä (MÆ1Ox©CĨ?MÆT± ¥ gOx©s}|tg{¸u}ceg¾r|s};s}|³uvq§s}O¯¨ytu}|¨Ox ª1u}ce|s sQqtsQu}g?fºtOx©Z;|¾;gmOx g9svuw|fu}gªu}ceg{uwgªOxr¾Og?{};gx©gkOª v uw u ±7b#vceg=xH(v»+g, . O. x. ,sQv|tg{) Sg{}u}ye{wSg?HsQqtsQu}g?f j6=i. N. M. 1. min{max{ inf Lαi (x + e, Dviε (x)); viε (x) − min{vjε (x) + k}}; viε (x) − ψ(x)} = 0,. 7*) 8. ª«;{5¯¨¯ i ∈ I x© x ∈ R º=x© |uws7¾r|s};s}|³uvqs}O¯¨ytuw|;x tg?xeuwg? v = (v , . . . , v ) ± ¥ g {}g?Oye¯{w|¿?g =q Oxr¾;O¯¨ytu}|¨OxC;tuwO|x©|xe ºx©Åu}ce|sª«yex© u}|¨Ox¦¯¨¯¨»skuwHeye|¨¯¨®,¯tO¯5sQy©g?{Q´·sQ;¯yeu}|¨OxÅOª v 7 8 ±%b#ceg?xÂrqÅOee¯¨q=|¨xe,u}cegOvx©s}|¨sQu}g?x© qCOx©FuwceghfhOxeOu}Ox©|¨|³uvqFOªæu}c©ghswceg?f g »+g%;tuwO|xÂu}ceg eg?s}|{wg?SOy©x© ± ¥ |³uwc,;ye{+{}g9sQye¯u?º=»+g?x§e{w¾Ogmx§ye©g?{+SOy©x©§Oª h Ox©Z'¯»+g{¸;yex©§Oª ª«;{ ¯OswsQ|O¯SÏ©xe|u}gt|ÎSg?{}g?x© g9s s}c©gfhgx©Zª«O{;gxeg?{wO¯|¨¿g9Ï©x©|³uwgt|³Î\g{wgx g?s swceg?f g;± h b#ceg©Og?{ |¨s#;{}=xe|¨¿g9§Os¸ª«;¯¯¨»*s <<s}g?!uw|;xH'|¨x;uw{}tty g?s+uwcegOswsQy©f eu}|¨Ox©s+Oxg?~=y©u}|¨Ox 7 8 Ox©Zswceg?f g - 8!±æorg9!u}|¨O>x 4;tuwO|x©s¸uwcegm{uwgmOª ;xr¾Og{wOg?x© g Oª u}c©gs}O¯¨ytuw|;x ª 85u} ±7b#ceg Ï{wsQu+{}u¸ª<sQ7 g9!uw|;x§%;tuwO|xs+OxZye©g?{+SOy©x©ZOª u − u ºex©§u}cegs}g? ;x©§v{}u+O7+ª & u}ce|s#usQg9!uw|;x ysQg9s u}c©g'{u}g'Otu|¨xeg?H|¨xÅs}g?!uw|;x 4ª«;{kO|¨¾r|x©hu}ceg%¯»+g{m;yex©FOª u − u ±org9!uw|;xÅe©¯|¨g?s ;ye{+{}g9sQye¯uws¸uw%uwcegOgx©g{¯¨|¿?g?hÏ©x©|³uwge|³Î\g{wgx©gs}ceg?fhguwOÃOg?x§ª«{wOf ÄÙMÆTºex©,svuwy©t|¨g?s ;x©t|u}|¨Ox©s y©x©tg{u}ce=sQghOg?xeg{¯ lÅ{wÃO¾§c©O|xHe©{}M¬t|¨fhu}|¨Ox,O|¨¾OgSg uQuwg{g9svuw|fu}g?s u}c©Ox,uwce|ss}c©gfhgO± ¹|¨x©¯¨¯¨qu}ceg ;keSgx©t|¬HO|¨¾Og?s#s}Ofhgyt¬t|¨¯|{wqhu}c©gO{wgfs#» ce|cH{wgysQg9§uwce{};yeOc©Oytu#u}c©g©Sg{9± ¥ g%Ox©¯ytguwce|¨sm|¨x;uw{}tty!u}|¨OxÅ» |³uwcÅsQ;fhgxeuuw|;x©s±m°·xFu}c©g%s}g?~=yeg¯ C |smS;s}|u}|¨¾Og%Ox©sQuwOx=u |¨x©tg?g?x©tg?x;uOxF©O{wOfhg u}g?{ws ε Ox© h ±¶+q | · | »+gfhg?OxHu}ceg'sQuwOx©eO{w ! y ¯¨|¨tg9xFxe;{}f |¨xÅxrq uvqrgks}©; gO±°·x©O{Quw|¨ye¯¨O{?º;|³ª º;u}c©gx = tr(XX ) ºr» ceg{wg X |s7u}c©g u}{x©s}S;s}g R Oª X º=|â± gO± |X| |sæu}ceg¹e{wOSgxe|¨y©s¸xeOX{wfH∈±<S°Tª g |s¸'|X| ;yex©tg9ª«yex© u}|¨OxZª«{};f R |¨x=u}'g?|³uwceg{ R, R º ;{+u}cegs}©Ogª N × P fuw{}| g9sºt»+gsQgu j6=i. |e|≤ε. N. ε. ε. ε 1. ε M. ε. 1/2. 1/5. h. h. M. N. 2. >. >. N. M. |g|0 := sup |g(x)|. x∈RN. é«Üï éëÝ.
(32) .
(33)
(34)
(35) !" #$
(36) %'&( *)+ ,- . * /0102
(37) #(&3 *. °Tª g |s ¯sQ |¨©swce|u}¿ ;x;uw|xrye;y©s?ºt»¸gs}g u [g]1 := sup x,y∈RN x6=y. |g(x) − g(y)| , |x − y|. |g|1 := |g|0 + [g]1 .. ¥ gtgx©u}grq uwceg OfhSOxeg?x=u+» |sQg;{wtg?{}|¨xe%|x ºeOx©§rq u}cegO{tg?{}|¨xe'|¨x,u}cegs}gx©s}gmOª S;s}|u}|¨¾OgsQg?f |tg≤Ï©xe|u}gfuw{}| g9s¸|¨x S(N ) ±5b#cegsQO gRC (R ) 7 {wg?s} ± C (R )8+» |¨¯¯1egxeOu}gu}ceg s}©; g ª Ox=uw|xrye;y©sx©Â;yex©tg9ª«yex©!uw|;x©s 7 {}g9sQ ±hSOyextg?®x 1|©swce|u}¿ ª«yex© u}|¨Ox©s8mª«{wOf uw R ± R & & e/Â'1 $, ¦% , 6 1'¸)
(38) $ $ .% , b#c©{};yeOce;ytu+u}ce|s ©Og?{?ºt»+gsQye©=sQguwc©u#u}c©gª«O¯¨¯» |¨xeOswsQy©f eu}|¨Ox©s#{wgs}u}|svÏg??< b#ceg{wgg ¬t|¨sQu λ, K s}y©c¦u}cu?ºæª«O{h¯¨¯ x ∈ R x© α ∈ A, a (x) = σ (x)(σ (x)) º 7 ;' 8,» ceg{wg σ (x) |sºrª«;{ g?;c x º© N × P fhu}{w|³¬ ºeOx© N. b. N. N. b,l. N. . . N. α. 1 2. α. 7 ;18. cα ≥ λ > 0; min{1, λ} > supα {[σ α ]1 + [bα ]1 }. ÕÌ
(39) 2 #'*
(40) # *%
(41) #
(42) ' ,*. *. È . ±. α. α. T. |σ α |1 + |bα |1 + |cα |1 + |f α |1 ≤ K,.
(43) %% .
(44) *
(45) / %
(46) #/ 0 0 *. u ∈ C(RN ) x ∈ RN. min{sup Lα (x, u(x), Dϕ(x), D2 ϕ(x)); u(x) − ψ(x)} ≤ 0, α. (. 0 . ≥ 0),. 1 ϕ ∈ C 2 (RN ) B u − ϕ >% % 1
(47) ! / 0 % A%
(48) #
(49) ! x
(50) #2%%!#" ,%% .
(51)
(52) $ *%
(53) # %
(54)
(55)
(56) & B / *%
(57) # #' 0 */ *%
(58) # u ¥ g{wg ª«g{#uwHÄ;ºe¯¨gfhf '; ±Æ ª«O{#uwcege{wrªOª1uwcegª«O¯¨¯» |¨xeh{wg?s}ye¯³u9± ' ¡ Ì)(æÌ ÕÌ +* , '&% #2 '/. 2 *# B %% 0 "
(59) #(& *# % 1) /
(60) "B #2 u 0 *
(61) . *!% .
(62)
(63) $ / %
(64) # #' .
(65) *
(66) 0 */ %
(67) # 1 u1 2 #
(68) # RN u1 , u2 ∈ Cb (RN ) u1 ≤ u 2 /
(69)
(70) 2 2 * 3 -
(71) # # 5
(72) 4 .
(73) *
(74) *%
(75) # , %
(76) # 0 1 C (RN ) 2 u b,l ¸;x©sQ|tg?{kuwcegs}ceg?fhg 7 -!8 º » |u}c h > 0 x S : R+ × RN × R × Cb(RN ) → R ± ¥ ghfÃOg uwcegª«O¯¨¯¨» |xehOsws}yefhtu}|¨Ox©sA< 8 lFOx©u};xe| |uvq' < ª«;{¯¨¯ h > 0 º r ∈ RN º m ≥ 0 º x ∈ RN x©"SOyextg?®xÂOx=u}|¨xryeOy©s 7 o\,. ª«yex© u}|¨Ox©s. u, v. s}y©cu}c©u. u≤v. |x. RN. º. S(h, x, r + m, u + m) ≥ λm + S(h, x, r, v).. ïï êC©øDDMö.
(77) @ *
(78)
(79)
(80)
(81) $ #
(82) , &. #'-!#"$
(83) %". x φ ∈ C (R ) º x 7→ S(h, x, φ(x), φ) |sZSOy©x©tg? x© |¨s'yexe|ª«O{wfh¯q" Ox=uw|xrye;y©sª«O{';yex©tg9 r º<yex©|³ª«;{}fh¯¨qÅ» |u}c. ª«O{H¯¨¯ 7 oe18, ;gx;;uwye|xr¯ye{w;|³uvy©q2sA< 9. h > 0 r 7→ S(h, x, r, φ) x ∈ RN. b. N. }{ g9sQSg? u u} ± xÁC ;x©sQuwx=u K > 0 sQyc¦u}c©u'ª«;{%¯¨¯ i ∈ J ⊆ {1, . . . , n} 7 o 4-8,b#ceg{wgZgx¬t|¨sQu n,|¨x k >º\0,Ox© « ª O m { g ; ¾ g w { q } s h f = O } u c s}y©cHuwc©u |D φ| |¨smSOyex©eg? º h>0 φ ∈ C (R ) ª«O{ g?¾Og{wq i ∈x J ºrRu}c©gª«O¯¨¯» |¨xehceO¯esA< i. . c. N. i. 0. ±. α. Q(φ) :=. -. . r. n. |sup Lα (x, Dφ) − S(h, x, φ(x), φ)| ≤ Kc Q(φ),. » ceg{wg ,È Á¢t¡ ". N. * %. P. i∈J. |Di φ|0 hki. #' -* .
(84) 02%! B . v =u+m. . S.
(85) 1*
(86) #(&. ·° x_u}c©gHª«O¯¨¯» |¨xe©º¸»¸gFswMq¦u}cuZª«yex© u}|¨Ox s}O¯¨ytuw|;x$8æu}hu}c©gswceg?f g 7 18¸|ª. ". . v ∈ Cb (RN ). v. / / 1 . m=0. . . #2
(87) #'*
(88) # &. |s"s}yet´·s}O¯¨ytu}|¨Ox 7 {wg?s} ± s}yeg?{Q´. w{ g?s} ± ≥ 0), ª«;{¯¨¯ x ∈ R . jkx©tg?{H;s}s}yefhtuw|;x©s o\ 8,Ox© oe-8!º»¸g"c©M¾OgÂu}c©g"g¬t|¨sQu}g?x© g"ªh Ofh©{w|sQ;x e{w|x©|©¯gª«O{ SOy©x©tg?HOx=u}|¨xryeOy©s#s}7 O¯¨ytu}|¨Oxs#7 ª 7 18 9e|â± gO± / 0 v #' - #
(89) # / *%
(90) # / 0 0 / *%
(91) # ÈrÌ ¡ È
(92) . u min{S(h, x, v(x), v); v(x) − ψ(x)} ≤ 0,. .. ' #. ,. h.. ". . ,. ,. h. uh (x) ≤ vh (x). $ ,%%. N. (. x∈R. . N. . . . ¡ Ì Ìæb#ceg+e{w=Oª©|sOx'g9Os}qg¬ru}gxsQ|¨Ox%ª1ĨOº¯¨gfhft± 4ÆT± ¥ g+;s}s}yefhgæuwc©u m := sup (u (x)− Ox©®Oeuw|¨xÁCOx=u}{Ot|!uw|;x ± gu {x } SgZCs}g?~=yeg?x© gZ|x R sQyc®u}c©u δ := v (x)) > 0 ;xr¾Og{wOg9smu} Os ±Cb#ceg?x > 0 ª«O{%¯¨O{};ggxe;yeOc n ±C¶+q"y©sQ|¨xe u (x ) − v (x ) uwcegxeOu}|¨OxHª5s}yeCxFsQyeSg{}´·sQ;¯mytuw|;x º©nO→x©,∞u}cegªÕ;!uuwc©δu min(A, B) − min(C, D) ≥ min(A − ºeOgu C, B − D) '. h. h. ot|x©g. n. h. n. x. N. n n. h. n. n. 0 ≥ min{S(h, xn , uh (xn ), uh ) − S(h, xn , vh (xn ), vh ); uh (xn ) − vh (xn )}. uh (xn ) − vh (xn ) = δn > 0 0 ≥ ≥ ≥ ≥. ºt=qZy©sQ|¨xe 7 o 8!º©»¸gc©M¾;g. S(h, xn , uh (xn ), uh ) − S(h, xn , vh (xn ), vh ) S(h, xn , vh (xn ) + δn , vh + m) − S(h, xn , vh (xn ), vh ) S(h, xn , vh (xn ) + m, vh + m) + ω(m − δn ) − S(h, xn , vh (xn ), vh ) λm + ω(m − δn ),. » c©g{wg ω(t) → 0 » c©gx Ox=u}{Oe|¨ u}|¨Ox ± 2. t → 0+. |¨s;|¾;gx®rq 7 o©18!± guQu}|¨xe. n→∞. qr|g?¯¨es. m≤0. » ce|¨c®|s. é«Üï éëÝ.
(93)
(94)
(95)
(96) !" #$
(97) %'&( *)+ ,- . * /0102
(98) #(&3 *. °·xF¯¨¯\u}c©gsQg9~;y©g¯ »+g» |¯¨¯ y©s}g s}g?~=yeg?x© gOªfhO¯¨¯¨|³Ï©g?{ws. (ρε )ε. ). eg Ï©xeg9HOs+ª«O¯¨¯¨»s*<. 7 -8. ρε (x) = ε−N ρ(x/ε),. » c©g{wg ρ ∈ C (R ), R ρ = 1, s}yee {ρ} ⊆ B(0, x© ±7°Tª |¨s Ox=u}|¨xryeOys檫y©x©!uw|;x Oª R wu R ºtu}ceg?xH»¸gtgÏ©xegu}c©gf ;¯¯¨|Ïu}|¨Ox¯ª g 1);s+ª«O¯¨¯ρ»≥sA< 0 g ∞. N. RN. N. gε (x) :=. Z. lCO{wg¾Og?{?º=|ª g |s |¨©swce|³uw¿ª«yex©!uw|;x ºtuwcegx °Tª. 7 {wg?s} ±. g ∈ Cbn (RN ). 7 (-8. g(x − e)ρε (e)de.. RN. 8 ºtu}ceg?x. 7 ?-8. |g(x) − gε (x)| ≤ ε[g]1 .. n Cb,l (RN ). |Di gε (x)| ≤ Cε−i |g|0 ,. (. {}g9sQ1±. Cε1−i |g|0 ),. 7 ;8. ∀ i = 1, . . . , n.. !
(99) <,
(100) $( ¸;x©s}|¨tg?{+u}cegª«;¯¯¨» |x©hs}» |uwce|¨xesQqtsQu}g?f e©{}M¬t|¨fhu}|¨Ox,ª 7 8 < . min. . . 7 M18 ±7°·x,{}u}| ye¯{+»¸gc©M¾;gOx,g9~=y©uw|;x§ª«O{ g?¾Og?{}q ;x=u}{wO¯A± ; x ∈ RN. max L (x, Dvi (x)); vi (x) − min{vj (x) + k} ; vi (x) − ψ(x) = 0, αi. ª«;{ i ∈ I = {1, . . . , M } x© ¾r|s};s}|³uvqHs}O¯¨ytu}|¨OxHuwceg;{}qª«O{ku}ceghs}» |³uce|x©,s}qrsQu}g?f ?xCSg%ª«Oyex©C|x Ä (MÆTº<Ä9MÆTºÄ¨ & ÆA± ¥ g {wg?O¯¯ c©g{wgmuwcegtgÏ©xe|u}|¨OxHª¾=|sw ;s}|uvq§sQ;¯ytuw|;xZª«;{k Og?xeg{¯ s}» |³uce|x©hs}qtsvuwgfOªuwcegª«O{wf3< F (x, v, Dv , D v ) = 0, i = 1, . . . , M, 7 A4(8 » c©g{wg F : R × R × R × S → R ± È ÕÌ 2 " #2
(101) # v = (v , . . . , v ) ∈ C(R )
(102) %% .
(103) *
(104) / *%
(105) # j6=i. i. i. . . N. M. N. i. 2. i. N. 1 / 0 " 0 */ *%
(106) # ,
(107) ' , *. *. . M. i∈I. #' . N M. x∈R. N. . . 0 ≥ 0 ,1 ϕ ∈ C 2 (RN ) v − ϕ % A% -
(108) ! / 0 % % ,
(109) #$
(110) x
(111) #2%%! " %% ,.A
(112) *
(113) $ *%
(114) i# 1
(115) "
(116)
(117) / > / *%
(118) # #' , 0 */ *%
(119) # v 1gfhf O)|¨fhe¯|¨g?sH Ofh©O{}|s}Ox e{w|¨x© |¨e¯¨gCª«O{ 7 91 8 ºmOx©uwcegx u}ceg®g ¬t|svuwgx©gÅOª%)y©xe|¨~=yeg ¾r|s};s}|³uvq§s}O¯¨ytuw|;x,Oª 7 91¸8 |¨x Cb,l(RN )M ºetgxeOu}g9 v = (v1, . . . , vM ) ± ¥ gg?{Quwye{w,uwcegs}qrsQu}g?f 7 M1+8 x,©ye|¯,u}cegª«;¯¯¨» |¨xe Oyt¬t|¯¨|{wq§sQqtsQu}gf 8 min{max( sup Lα (x + e, Dviε (x)); viε (x) − min{vjε (x) + k}); viε (x) − ψ(x)} = 0. 7 j6=i |e|≤ε 1gfhf O)©e¯|¨g?u} 7 !8 ºk|¨fhe¯¨|g9s§uwcegÂg¬t|¨sQu}g?x© gª%)yexe|~;y©g¾r|s};s}|³uvq sQ;¯yeu}|¨Ox Oª 7 !8 º Fi (x, v(x), Dϕ(x), D2 ϕ(x)) ≤ 0,. . i. egxeOu}g? eO±. ε v ε = (v1ε , . . . , vM ). ïï êC©øDDMö. º\|x. Cb,l (RN )M. ±kb#c©gª«O¯¨¯¨» |xe¯¨gfhf|smOx©s}g?~=yeg?x© gOª7u}ceg?O{wgf.
(120) . @ *
(121)
(122)
(123)
(124) $ #
(125) , . ¸È Á¢ . ". #2 1* 02
(126) # ,'. # %! -/0 #2 ,#. %. #' -'. . . ". #'-!#"$
(127) %". ., . 7 M18. |vi − viε | ≤ Cε,. #' . . ¹©O{g¾;g{wq º¸¯¨g u u}cegFfhO¯¨¯¨|³Ï?uw|;x)ª º+eg Ï©xeg9 ; sh|x (-8!± ot|x©g ª«y©x©!uw|;x ºeyexe|ª«Oi{wfh¯qZ»v ± {9± u ε > 0 sQy :|g?x=u}¯¨q,s}fhv O¯¯Aº 7 9-8¸|¨f © ¯|¨g?s 7. |s1|s}ce|u}¿ 7& 8 |v (x) − v (x)| ≤ max[v ] ε; ¯¨gfhf,Oh|fhe¯¨|¨g?smu}c©u max [v ] {wgf|¨x©sSOy©x©tg?C» cegx ε ↓ 0 7 u}ce|s{wOyefhgx=uk» |¨¯¨¯Sg'y©s}g? s}g¾;g{¯Su}|¨fhg?s#|¨xu}ceg©Og?{ 8 ± ¸È Á¢ 2 #'*
(128) # v − R
(129) ,%% i / *%
(130) #
(131) # ' * k , R := C k + ε + 7)8 C. K, λ. [ψ]1 2. ε i. iε. ε i. iε. viε. ε i 1. i. ε i 1. i. . %4. . iε. ". ε2. B # #$ '. ¡ Ì Ì 1g u. 10 #2 ,#$%! # K, λ #2 [ψ] C 1 ¥ w s w u ¨ | Q s « ª q 8 ± g © c M ; ¾ g } u h e w { O ¾ m g } u © c u?ºtª«;{ ¯¨O{};gmg?xeOye;c 7 ) R. min{sup Lα (x, D(viε (x) − R); viε (x) − R − ψ(x)} ≤ 0,. C. º. 7 ?-8. ∀ x ∈ RN ,. ª«;{O¯¯ i ∈ I ±'or|¨x© g v ∈ C (R ) º\u}c©g tgÏ©xe|u}|¨Ox"ªæ¾r|s};s}|³uvqHs}yet´·sQ;¯ytuw|;xC|smg9~=ye|¾O¯g?x;uu} uwcegxeOu}|¨Ox®ª ¯OswsQ|O¯7sQyee´Ts}O¯¨ytu}|¨Ox1±§b#ceg{wg ª«;{}gh»+g cM¾Og u}Fe{w¾Og%u}cuOx©gª+u}ceghª«;¯¯¨» |x© sQuwu}g?f g?x=uws#ce;¯¨es+ª«;{¯¨¯ x ∈ R < v (x) − R ≤ ψ(x), ∀ i ∈ I, 7 (;- 8 sup L (x, D(v (x) − R)) ≤ 0, ∀ i ∈ I. 7 (O$ 8 ¹©O{ g?¾Og{wq x ∈ R ºsQgu I (x) := {i ∈ I | v (x) = ψ(x)}. 7 O- 8 1g u ± kg?xeuwg'rq x, 2ε) u}ceg%©¯¨¯gx=u}g?{}g9HOx x˜ » |³uwcC{w;t|¨y©s 2ε ±kb#c©gxC»+g'c©M¾;gu}c©g uv»+ x˜ª«O∈¯¨¯R» |¨xe =s}s}|¨e|¯¨|u}|¨g?As B(˜ < ;o ! ; <®b#ceg{wgFg ¬t|svus s}y©c u}c©u ± ¥ g ¯|¨f u}cu 7 (O( 8 ceO¯es?± 6= ∅ ¥ gc©M¾Og v (y) = ψ(y) ºrª«y;{#∈s}OfhB(˜g xi, 2ε)∈ I (y) ± gu i I∈/ (y) 5 ± # b e c gkª«y©x©!uw|;x cOs?º ª«O{%sQy :§|g?x;uw¯q®s}fhO¯¯ ε > 0 º<H¯tO¯5f¬t|¨f'yef.I (y) u'HSO|¨x;u x s}y©c"vu}c©(x) u |x− |x−−y|y|≤/εε ± ot|x©g v |s+u}ceg¾r|s};s}|³uvqZs}O¯¨ytuw|;x,Oª 7 ?( 8 ºtOxegOª1u}c©gª«O¯¨¯» |¨xehsvuu}g?fhgx=uws c©O¯e*s < v (x ) ≤ ψ(x ), 7 e?( 8 α. ∞. iε. N. N. iε. α. iε. α. N. ε. ε i. N. ε. ε i0. 0. ε. ε. ε i. ε. 1. ε. ε i. max. . ε. 2 2I (xε − y), ); ε1 ε1 ε ε vi (xε ) − min{vj (xε ) + k} j6=i. 1. ε. ε. sup Lαi (xε + e, viε (xε ),. |e|≤ε. 2. ≤ 0.. 7 e$8. é«Üï éëÝ.
(132)
(133)
(134)
(135) !" #$
(136) %'&( *)+ ,- . * /0102
(137) #(&3 *. °Tª. viε (xε ) ≤ ψ(xε ). º©s}|¨x© g. viε. (. Ox© ψ {wg 1|s}ce|u}¿;ºt»¸g;tuwO|x viε (y) ≤ ψ(y) + ([ψ]1 + max[viε ]1 )ε.. 7 ;18. viε (y) ≤ max[viε ]1 ε + viε0 (xε ) + k ≤ 2max[viε ]1 ε + viε0 (y) + k.. 7 14-8. kuwceg{w» |sQg;º=» |u}c 7 t?$8!ºt»+gc©M¾Og. i. ot|x©gg|u}ceg?{ 7 ;18¸O{ 7 4-8¸ce;¯¨©sºt»+gtg?ty gu}cu i. i. viε (y) − Cε − k ≤ ψ(y),. 7 (8. ∀i ∈ I,. » c©g{wg tgSgxesOx [ψ] x© max [v ] ±ot|x©g y ∈ B(˜x, 2ε) º1x ψ x v O{}g |¨©swce|³uw¿Oº uwce|s |fhCe¯¨|¨g?s v (˜x) ≤ ψ(˜ ª«O{kO¯¯ i ∈ I ± ;©e¯qr|¨xe 7 & 8 ºeOtu|¨x v (˜x) ≤ ψ(˜x) + R º ª«;{¯¨¯ i ∈ I ºe» ce|cH|fhex¯¨|¨)g?+s 7 k9-+8 ± Cε, ;o ! ¶ <+¹©O{mO¯¯ y ∈ B(˜x, 2ε) º©»+gc©M¾Oguwc©u I (y) = ∅ ± ¥ g'¯¨O|fu}c©u?ºª«O{¯¨¯ e ∈ B(0, ε) º |s %¾r|s};s}|³uvq,sQyee´Ts}O¯¨ytu}|¨Ox,ªu}cegª«;¯¯¨» |¨xe s}qtsvuwgf (v (· − e), . . . , v (· − e)) . 7 O- 8 max L (x, Dw (x)); w (x) − min{w (x) + k} = 0, x ∈ B(˜ x, ε). ¹|¬ Ox© i ∈ I ± gu ϕ ∈ C (R ) gsQyc¦u}c©u v (· − e ) − ϕ(·) cOs C¯¨r?¯ ε) f¬re|¨f'y©∈f B(0, | H x w u ceg©¯¨¯ x, ε) ±¸b#ceg?x v (·) − ϕ(· + e ) cOsh¯¨t¯ f¬r|¨f'y©fu x − e ± ot|x©g x −x e ∈ B(˜x, 2ε)B(˜ º5»+gZc©M¾;g§u}cu v (x − e ) > ψ(x − e ) ºæOx©)s}|x g v |s%u}ceg ¾r|s};s}|³uvq§s}O¯¨ytuw|;x,Oª 7 ( 8 ºt»¸g;tuwO|x i. 1. ε i. ε i 1. ε i. iε. ε. ε 1. ε M. αi. i. i. ε i. e1. 1. max. bÃr|¨xe. N. 2. 1. e1. e = e1. . j. j6=i. ε i. 1. 1. ε i. e1. e1. 1. 1. e1 ε. 1. sup Lαi (xe1 − e1 + e, viε (xe1 − e1 ), Dϕ(xe1 ), D2 ϕ(xe1 ));. |e|≤ε. ºtOtu|¨x (. viε (xe1 − e1 ) − min{vjε (xe1 − e1 ) + k} j6=i. . ≤ 0.. Lαi (xe1 , viε (xe1 − e1 ), Dϕ(xe1 ), D2 ϕ(xe1 )) ≤ 0, viε (xe1 − e1 ) − min{vjε (xe1 − e1 ) + k} ≤ 0.. #b c©|¨s%g?|x©Åuw{}yegZª«O{hx)O{}©|³uw{wO{}q e ∈ B(0, ε) x© i ∈ I º5»+gZ;tuwO|xÁu}c©u?º<ª«O{h¯¨¯ |e| ≤ ε º |sH¾r|s};s}|³uvqÅs}yet´·s}O¯¨ytu}|¨Ox®ª ;18 ±>;kee¯¨q=|¨xeÁĨº¯g?fhfh ;± 4Fx© (v (· − e), . . . , v (· − e)) ¯¨gfhfÂe± ) ÆAº7s}|x g v (·) |s'¯¨|fh|u Oªk ;x=¾;g ¬® Of%e|¨x©uw|;xÁ7 ª v (· − e) º<ª«;{ e ∈ B(0, ε) º<u}ceg?x |¨s ¾r|s};s}|³uvqsQyee´Ts}O¯¨ytu}|¨OxOª O- 8!±lFO{wg¾;g{9º?s}|x©g5|u|ss}fhru}cª«y©x©!uw|;x º (v (·), . . . , v (·)) |u|s s}yet´·s}O¯¨ytu}|¨Oxª 7 ;1 8¸|xuwceg ¯OswsQ|O¯ sQg?x©sQg;7 ºex©,»+gcM¾Og 7 & 8 L (˜ x, Dv (˜ x)) ≤ 0, ∀ i ∈ I. j6=i. 1. ε 1. ε M. ε i. iε. 1ε. Mε. αi. ïï êC©øDDMö. iε.
(138) 9. @ *
(139)
(140)
(141)
(142) $ #
(143) , . ¥ gÃrxe»uwc©u. |viε (y) − vjε (y)| ≤ k. ª«O{O¯¯ i, j ∈ I Ox©. Dn viε (˜ x) − Dn vjε (˜ x) ≤. y ∈ B(˜ x, ε). Ck , εn. » c©g{wg C tg?g?x©es ;xe¯qZOx ρ tg Ï©x©g?|x 7 (8!±7°Tu ª«;¯¯¨»s+u}c©u Lαi (˜ x, Dvjε (˜ x)) − Lαi (˜ x, Dviε (˜ x)) ≤. #'-!#"$
(144) %". ±¸¸;x©sQg9~=yegx=u}¯¨q. ∀n ≥ 1,. Ck , ε2. ∀ i, j ∈ I.. ¸;f'e|¨xe|¨xe» |u}c 8!º=Ogu º;ª«;{+O¯¯ i x© j |¨x I, ª«O{#s}Ofhg C tg?g?x©r´ |¨xeOx ρ Ox© K ºO7 x©'& cegx gOLº sup(˜x,LDv(˜x,(˜xDv)) ≤(˜xCk/ε ª«;{O¯¯ i |¨x I ±7or|¨x© g L |syexe|ª«O{wf ¯¨q Ck/ε 1|s}ce|u}¿;º»+g'cM¾Ogu}cuª«O{mO¯¯ i |x I º sup L))(˜x≤, Dv ±mb#c©g{wg ª«O{wg 7 9-8 (˜ x) − Ck/(λε )) ≤ 0 O¯¨s} ceO¯es#|¨x,uwce|s;sQg;± 2 #' " ,%% x ∈ R . ÈrÌ ¡ È *. * i∈I 7 ) 8 0 ≤ v − u ≤ Ck , αi. α. α. 2. jε. iε. α. . ". . ¡. α. 2. α. 2. iε. N ". 1/3. i. - /0 #2 ,# %! # λ #' K ' Ì ÌÅ-8 ¥ g§e{w¾Og uwcegÏ©{svu|¨xeg9~;y¯¨|³uvqª 7 ) 8!±. gu w = (u, . . . , u) Sguwceg¾Og9!u};{» ce=sQg ¥ O h f S O e x ? g ; x u % s O } { Z g ? g = ~ © y ¸ ¯ } u ± , g ¯ ¨ | . f } u © c u |¨s C¾r|s};s}|³uvq®sQy©t´Ts}O¯¨ytuw|;xÁª 7 918 ±1g u M u c©Os,¯t?¯<f¬tw|f%yef u S g Q s y Å c w u © c u ± or|¨x©g u |sZ¾r|s};s}|³uvq ϕ ∈ C (R ) u−ϕ x ∈R s}yet´·s}O¯¨ytu}|¨Oxª 7 8 º g|u}ceg?{ u(x 1 º O { ± °Tª ) ≤ ψ(x ) sup L (x , u(x ), Dϕ(x ), D ϕ(x )) ≤ 0 uwceg¯uQuwg{ ce;¯¨©sºruwcegx '. C. 2. N. 0. 0. α. 0. Lαi (x0 , u(x0 ), Dϕ(x0 ), D2 ϕ(x0 )) ≤ 0,. ¸;f'e|¨xe|¨xehOu}cF;sQg9sºr;tuwO|x min. N. 0. α. . 0. 0. 2. 0. ∀ i ∈ I.. max Lαi (x0 , u(x0 ), Dϕ(x0 ), D2 ϕ(x0 )); u(x0 ) − min{u(x0 ) + k} ; j6=i. u(x0 ) − ψ(x0 ). . ≤ 0,. ∀ i ∈ I.. b#c©g{wg ª«O{wg |sæ¾r|¨sw =sQ|uvq's}yet´·sQ;¯yeu}|¨Oxhª 7 M18 ±¶+q%u}cegkOfh©O{}|sQ;x%©{}|¨x© |¨e¯¨g 7 ¯¨gfhf'O-8!ºOu}ceg Ï{wsQu |xeg9~=y©w¯¨|³uvqZª 7 ) 8¸ceO¯es?± ' 8 ¥ g,xe» e{}¾;g u}cegsQg9 OxÂ|¨xeg9~;y¯¨|³uvq"|x 8!±F¶+q¯¨gfhf x®e{};=sQ|u}|¨OxÁeº»+g§c©M¾;g uwc©u v − R ≤ u º=ª«O{ ¯¨¯ i ∈ I ºex© x ∈ R ºt7 » ce) g{wg R s}u}|svÏ©g9s ) 7 ) 8!± ;©e¯qr|¨xe 7 & 8¸x© 7 9-8!º ;tuwO|x N. iε. vi − u ≤ |vi −. viε |. +. |viε. . k − viε | + |viε − u| ≤ C 2 + ε + k , ε. ∀ i ∈ I,. ∀ x ∈ RN ,. » c©g{wg C tgSgxesOx K, λ, max [v ] , [ψ] ± lC|xe|¨fh|¿?|x©¦» |³uwc_{wg?s}Sg?!uhuw ε º+»¸gH;tuwO|x u}ceg eg?s}|{wg?,yeeSg{ SOyex ºe» |u}c ε = k ± 2 ε i i 1 1/3. 1. é«Üï éëÝ.
(145) ;.
(146)
(147)
(148) !" #$
(149) %'&( *)+ ,- . * /0102
(150) #(&3 *. ,È Á¢t¡ . . $. .
(151) # *
(152) / @
(153) # B + * # %! #2 0'% 01* * .@ B 1 *#'
(154) # &B * B . . ,$ , $ u − u h ¥ gsQuwuwg|xu}c©|¨ss}g? u}|¨OxOye{ fO|x{}g9sQy©¯³us*<u}c©gyeeSg{kOx©¯»+g{+;yex©esOx.
(155)
(156)
(157) u − uh <g?{Quwye{},u}c©gg?~=y©uw|;x 7 8#s}Os¸u};tuwO|x . min. . α. ε. ε. sup L (x + e, Du (x)); u (x) − ψ(x). . u − uh. ± 7 -8. x ∈ RN .. = 0,. jkx©tg?{h;s}s}yefhtuw|;x©s 7 ; 8%Ox© 7 ;m18 º5rq¦e{wOS;s}|³uw|;x tº 7 (8c©;shCyexe|~=yegH¾r|¨sw =sQ|uvq¦sQ;¯ytuw|;x ±æ°·xH¾r|g?»VOª<b#c©gO{wgf t» |u}c ºe»¸gc©M¾;gu}c©u ºtª«;{ksQ;fhg C u egS∈gx©Ct|¨xe(R;x ) λ º K x© [ψ] ±5 mg Ï©x©gu}cegOx=uwk;!=u0tOfO|xª u Os |u − u | ≤ Cε ε. α,|e|≤ε. N. b,l. ε. ε. 1. X(uε ) := {x ∈ RN |uε (x) = ψ(x)}.. 1g u g+uwceg#f ;¯¯¨|Ïu}|¨Ox'Oª ºOtgÏ©xeg? Os|¨x (-8!±7or|¨x© g u |s< 1|©swce|u}¿+ª«yex© u}|¨Ox ºOyexe|ª«O{wf ¨¯ q »± {9± uu ε > 0 sQy :§|g?x;uw¯q,sQf¯¨¯Aº u7 9-8¸|fhe¯¨|¨g?s 7 |u (x) − u (x)| ≤ [u ] ε, 7 (-8 » c©g{wg [u ] {wgf|¨x©s#SOyextg? ± ' + * % #' % B >0102 -
(158)
(159) # * ÈrÌ ¡ È ' . #
(160) *%
(161) # u
(162) # C (R ) 2 # $ !
(163) *# % %% h > 0 . u − u ≤ Ch , ∀x∈R , 7 4O- 8 ε. ε. ε. ε. ε. . . /. . h. *
(164) #(&> 1`#":=#2min
(165) # i∈J* {k i /i} '. . b. . . !. `. h. C. %0. N . 54. . 1. 1. .. ". ε. ε. 10 *#' #$%! #. ". N. . λ K, [ψ]1. #' . Kc. . # # . ki. ¡ Ì Ì ¥ g ¯|¨fuwc©u min{S(h, x, uε (x) − R1 , uε − R1 ); uε (x) − R1 − ψ(x)} ≤ 0,. ∀ x ∈ RN ,. #2 . KC. 7 4e8. ª«;{'s}Ofhg R > 0 ª#u}ceg§ª«O{wf R := λ Q(u ) + Cε º<» ceg?{}g Q(·) »#OstgÏ©xeg9®|x 7 o 4-8x© egSgx©esmOx©¯qOx [ψ] Ox© [u ] ±k°·x©tgg9 ºS»¸g%» |¯¨¯©{}¾;g§sQ¯¨|¨Oc=u}¯¨qsQu}{wOx©Og{k{}g9sQye¯uA<+ª«;{mxrq C ºtOxegu ¯g9OsQu#Oªuwcegª«O¯¨¯¨» |xe%uv»+hsQuwu}g?f g?x=uws#ce;¯¨esA< x∈R u (x) − Cε ≤ ψ(x), 7 4;O- 8 7 4=$ 8 S(h, x, u (x) − K λ Q(u ), u − K λ Q(u )) ≤ 0. 1. 1. ε. 1. −1. ε. 1. N. ε. ε. ïï êC©øDDMö. C. −1. ε. ε. C. −1. ε.
(166) M. @ *
(167)
(168)
(169)
(170) $ #
(171) , . #'-!#"$
(172) %". ¹|¬,x x˜ ∈ R ± ¥ gc©M¾;gkuwcegª«O¯¨¯¨» |xeh¯u}g?{}xu}|¨¾Og-< ;o ! ; <Fb#ceg?{}gHg ¬t|svus s}y©c)uwc©u ºæ|A± g;± ± or|x g Ox© ψ O{}gyexe|ª«O{wfh¯q 1|ys}∈ce|B(˜ u}¿;º5xª«,;2ε) {hs}Ofhg C tg?g?yx©∈t|¨xeX(u ®Oxe¯¨)qÁOx [ψ]u (y)x©= [uψ(y) º] æ»+gOtuu|¨x 7 4;O- 8¸uSO|¨x;u x = x˜ ± ;o ! ¶ <mx©g'c©;s ) ∩ B(˜x, 2ε) = ∅ ± ¥ g ¯|¨f u}c©u u (· − e) |sºª«O{¯¨¯ e ∈ B(0, ε) º\ ¾r|s};s}|³uvq§s}yet´·s}O¯¨ytu}|¨OX(u xª N. ε. ε. ε. 1. ε. ε. 1. ε. sup Lα (x, Dw(x)) = 0,. 7 414-8. x ∈ B(˜ x, ε).. ¹|¬ e ∈ B(˜x, ε) º\x©C¯gu ϕ ∈ C (R ) Sg%s}y©cFu}cu u (· − e ) − ϕ(·) c©Os¯¨r?¯fh¬t|f%yef u,SO|¨x=u x ∈ B(˜x, ε) ±hb#c©gx u (·) − ϕ(· + e ) cOs¯tO¯7f¬t|f%yef u x − e ±§or|¨x© g ºeOx©cegx©gOº x − e ∈/ X(u ) ºe»+gc©M¾Og;ºt» cegxeg?¾Og?{ |e| ≤ ε º x − e ∈ B(˜ x, 2ε) α. ε. ε. e1. e1. N. 2. 1. e1. 1. 1. e1. 1 ε. 1. 1. sup Lα (xe1 − e1 + e, uε (xe1 − e1 ), Dϕ(xe1 ), D2 ϕ(xe1 )) ≤ 0.. bÃr|¨xe. ºt»¸gcM¾Og α,|e|≤ε. e = e1. sup Lα (xe1 , uε (xe1 − e1 ), Dϕ(xe1 ), D2 ϕ(xe1 )) ≤ 0.. b#c©|¨s©{}¾;g?s%Oye{Z¯¨O|f u}c©u u (· − e ) |¨s§"¾r|s};s}|³uvq)s}yet´·s}O¯¨ytu}|¨Ox_ª 7 414(8!±Vot|x©g e |¨sOx O{}©|³uw{wO{}q§SO|¨x=ukª B(0, ε) º u (· − e) |¨sm¾r|¨sw =sQ|uvq,s}yet´·sQ;¯yeu}|¨OxHª 7 4-4-8 ºeª«O{m¯¨¯ |e| ≤ ε ± ot|x©g |¨s# ª«yex© u}|¨Ox ºeOx©§|u |¨s+¯¨|fh|u ªOxr¾Og¬hOf'©|x©u}|¨Ox,ª sQg?g%Ĩºt¯¨gfhf,;'± 4 u (·) Ox©¯¨gfhfC%e± ) Æ 8 º;ceg?x© g;ºtee¯¨qr|x© 7 ĨOºr¯¨gfhf%t± ) BÆ 8!ºr»+gkOxZswMq'uwc©uu (·u−(·)e) |7 s¸%sQyee´Ts}O¯¨ytu}|¨OxOª 7 414(8æ|¨xu}ceg¯¨;s}s}|¯ s}gx©s}gO±7b#ce|s |¨fhe¯|¨g?s 7 4O( 8 sup L (˜ x, Du (˜ x)) ≤ 0. ¶+q OxsQ|svuwgx©qOº S(h, x˜, u (˜x), u ) ≤ K Q(u ). ;e©¯qr|¨xe 7 o 8S» |u}c u = v = u −K λ Q(u ) º º1x º Otu|¨x 4;' 8muSO|¨x=u ±h¸;f'e|¨xt´ r = u (˜ |¨xe®;sQg9xs ) ; − KOx©)λ¶Q(u »¸gHOe) uw|¨x 7 m4e 8!=±Kor λu −Q(uR )|s"sQyee´Ts}7 O¯¨ytu}|¨Ox ª 7 1 8 ±_x ¶+=q¦x˜uwceg;{}g?f ©º ºt|â± gO±¨º u − u ≤ K λ Q(u ) + Cε, » c©g{wg C tgSgxes Ox [u ] Ox© [ψ] ±5jks}|xe u − R ≤Pu º x© O 8!º\Otu|¨x Q(u ) ≤ C P ε h ±b#ceg%{wg?s}ye¯ukª«;¯¯¨»srq Q(u ) = s}g u}u}|¨xe ε = max|D uh|h ± 2 7 α. ε. 1. 1. ε. ∞. ε. ε. ε. α. ε. α. ε. ε. C. h. 1. ε. i∈J. i. i∈J.
(173) . ε. −1. ε. ε. C. ε. ki. h. C. C. −1. ε −1. ε. C. . ε. ε. ε. 1. ε. ε. ε. ki /i. i∈J. 1. 1. 1−i ki.
(174) u − uh ¥ gSg{}u}ye{wZuwcegs}» |³uce|x©hs}qtsvuwgf 7 9-8+Os+ª«O¯¨¯¨»s . −1. . min{max( inf Lαi (x + e, Dviε (x)); viε (x) − min{vjε (x) + k}); viε (x) − ψ(x)} = 0, |e|≤ε. j6=i. 7 4;-8. é«Üï éëÝ.
(175) 4.
(176)
(177)
(178) !" #$
(179) %'&( *)+ ,- . * /0102
(180) #(&3 *. ª«;{,O¯¯ i ∈ I Ox© x ∈ R ± ¶+q_¯g?fhfh_©º#u}ce|sHsQqtsQu}gfc©;sZ)yexe|~;y©gžr|¨sw =sQ|uvq sQ;¯ytuw|;x |¨x ±¸;x©sQ|tg?{ º¸u}c©gfh;¯¯¨|³ÏSuw|;x Oª º+eg Ï©xeg9 ;s |x ((8!± v = (v , . . . , v ) ot|x©g v |s1|s}ce|u}¿ Cª«yex(R!u}|¨O)x1º yexe|ª«O{wf ¯¨qC»v± {?± u?± ε > 0 s}y :§ |¨gx=u}¯¨qÅs}fv ¯¨¯âº ©e¯qr|¨xe¯g?f 7 f 7 t 8¸Ox© 7 ?( 8+»¸gc©M¾;g N. ε. ε 1 ε. ε M. N M. b,l. |vi − viε | ≤ Cε,. » c©g{wg C tg?g?x©es ;x λ, K x© [ψ] ºx© ¸È Á¢
(181) * x ∈ R i ∈ arg min 1. N. 0. ε. ε. 0. 74 & 8. |viε − viε | ≤ max[viε ]1 ε,. {wgf|¨x©s#SOyextg? ± i. maxi [viε ]1 vjε (x0 ). j∈I. . #' . 74 ) 8. ε ≤ (12sup[viε ]1 )−1 k. i. 2 # %% 0"
Documents relatifs
À cet effet, un nombre considérable de recherches démontre cliniquement que les enfants de parents souffrant de problème de santé mentale sont plus à risque d’un retard dans leur
La question de l’extension des limites de Paris jusqu’à cette ligne avait été discutée mais ne fut tranchée que par un décret impérial du 9 janvier 1859 qui décidait de
C'est aussi le résidant pas chez leurs parents fréquentent deux fois parativement à la province, les étudiants de Nanterre ne lement ce moindre attachement à la ville
In this paper, we study the performance of an enhanced channel estimation technique combining estimation using an autocorrelation based method and the Expectation-
Selon Laeven et Valencia (2012) , le coût budgétaire de la crise bancaire s’est respectivement élevé à 44 et 41 points de PIB en Islande et en Irlande, ce qui place ces deux
Dans ce travail, nous comparerons les entreprises coopératives, détenues par leurs membres, dont le produit risqué est la ressource critique apportée et dont l’objectif est
L’indice EMBI a augmenté de 77 points de base entre le point bas de mi-septembre 2007 et fin février, en partie conduit par la hausse de 300 points de base du spread de l’Argentine
