Periodic homogenization of monotone multivalued operators
32
0
0
Texte intégral
(2) "!#$ % &(')* *+,%)' -. 0/ 132547658:9;4=<>2547<>65478 ?A@ 6CBEDF2547GIHKJMLN8N65JMO 478QPSR7JE478UT 4=8#VABXW 4=Y[Z\6C8N]FJM8 ^=_ Ya`MbAOc6CJEO ^7dFd7e ∗. †. ‡. hf gji kQlmg npoMqsr,tuvqswyxzo|{sr,}5~ so|,o|Mt{t xzrvXuvxoM{m s},tusojw;w }C~ t }zu, {Mu wyxzo|*{crq0,}z|,xz|{uvxzro 5pFE||E|| =oMqwycr t Eqsuxzo|{sr wy}z o|v 7\ M (x), d (x)) ∈ A (x) o|¡ A crauq{ o|{suxzo|{wyo|{E−div u}z r,d X}z=qf*r,r¢oM{Eu,(∇u wycr,o| 7t*Xuvq*r £jxzqy£>oM{ouvo|{cr¤ jo|q=rwy r%j¥j 7ous¦ r r"wy oMxzrvr|{su w oj*xz jx§uvt,rq A Arx¨}3rqxu wy ro| 7t*Xuvq*r X£yxz,|qy£,oM{ouvo|{ t }Xuvt r (T (A ))(x, y) oM{E Mv|q r¢ {sr w r©|* cr" | vr q{moM =t vuqsX£yx|},o|{o|uoM{ y) =oMq v rvMqs uoMqyu (x, y) ∈ Ω × Y Eqs{sw → 0 }zo|*r (u , B(x, o|{j | M ªxz«}z,{Euw|{sr¬q{"cr¢ =|¨wjo|«7o|}z |wytcEqsXu0 M*r¬q{ro|}zqyduvx)oM{ (u , d ) wy u (x), d (x)) ∈ A(x) oM¡ A rvXuxr¢ªxu }cr , cr j¥E 7−div ouv¦ r drE=q fA ¤7(∇u tcr¢q}uvux{s}qu}z"Mro|¡ r¢uq{%o|{=uxzo|{ ,o|{o|uo|{s%o|{Euxz{jq% =oMq v rvEquo|qu x w{sr ΩA¤ (x) ® ¯ i° ±X²³° ´ rx{suq{yoM}zwx{h,usojwoxzo|*{crq0=,},xz|{µ{=wxr¢o 5pFMM|M¶·r¢uq=wy¥,u·o|,oM|{sx¸cXuxzo|{,oMcEqsXuvxoM{sro|¬u·o|v N¶x§uv |{sw.¶v xr ;q{=¹ −div d = f uxzo|{¶oMr \|}qs r%v(∇u X(x), £yxzd}N(x)) ,o|{∈o|uo|A{s©(x)Mv| sr ¤´p{swy "A vo| sxXuv |voX¶u{swoj*xz jx§ua¥Mrrq, yuvxoM{sr¬xur Eq{=,oq{yo|}wycwµX£j¹ x|},o|{souoM{ Mv| sr oM{j |v|crx{µu©|* sxz }r{=r¢ y) o|}z,oMr¢u¨ | ¥ uo"X£yx|},o|{o|uo|{sp(T|*(A ))(x, |r → 0 Muv{ (u , d ) o|{j | M B(x, r ¶ |y)ºE}z¥x{3 rqx§u*«}z"jo|(x,«7o|}zy) ∈r sΩ|×pYuo ro|}zqyuxzo|{ (u , d ) o|¬u· vo|«}z −div d = f E¶x§uv (∇u (x), d (x)) ∈ {sw rvXuvxzr¢»scr¨uv"r||rvr¢qs uxzo|{sr|r ¤¨¼xrcr¢q}uxz{s}qsw r A(x) u |r;¶Av A (x) xr½,oM{ouvo|{ oM{Euxz{Eqso|Aqsr©q{suxzo|{¾o| |},oErau Mv¥ x x{ Ω ¤ á ÀÃaÅÖ¿7Î ÀâÁyã·Â äâÀÀ ÄaÌsÂcÈMÅÆÃaÇ Ç¨Ï,ÈsÉ ÀÊÃaËEËEÀ ÇpÌÆÍ|å Î[æcÇQÞ|â*ÇvĬީÈMØÇ ÃaÏ,ÑÄaÇ*ÀÅÖÄaÌFÐEØQÑ*ÒÇÈ|ÀÇvÄaçŧÓ×jÔMè|Ç*âάÀ ËEÊévÕEÇcÕEê7ÌÖëŧÓXÈMÔEÀ Ñ*Ò"Çv̧Î*À × Ò"ØQŧÙÀËyÚìÛÔEÜËMÏÅÖàÚµíªÕEÀÝcÃaÞ\ÅÖßÎ*Þ|âãX× Üê îÃËMï ŧàcÇvÃaÎ[ÅÆÄaÑ á á á ÀÃaÅÖÎ*ØQ×yÑvèMÃaÑ*âÒÀËEÀ é*ÈMÇ Çcê7× ëôÜÒ"ËMÇ*ÅÖÔMà\ËEÇvÅÖÃaÇvÎ[ÃÅÆÄaìÑ é*ÇÃaÀÇ*ÃaÒÅÖÎAÀcðÈ|À ÇcÔMê ÈEÕEÀ ÐMÔMŧÕEËMÐMÇcŧê ËMÇcÌê ÀîzÃé*ï ÇÈMÔ%Ï,ÀÃaÑ*é¢ÐjÀÌEÈ|Ǩ¿7ÀÄ[Ä[ÃaÇNÈMǨñFÀÎ[Î[ŧò ËXÍ"ó ßcócó ß ðÑ*ÕEÀÃ[ÄaÇ*Ò"ÇvËÄÈMÇÏ,ÀÄaÐEÑ*ÒÀÄaŧÓÔMÇc×sÜËMŧàcÇvÃaÎ[ÅÆÄaÑéÀÄaÐEÂcÌÖŧÓXÔEÇ"È|Ç"¿= ÔEà À ÅÖË=×7ãéÐMÇ*Ò"ÅÖË3È|Ô;ØÍí é*ÌÖ Ä[ÃaÂcËs×7â*õ æ\Ý"¿sÂcÔMàcÀŧË|íªÌÀíªÙÇ*ÔMà\Çc×|öQÇ*ÌÖò ÅÓXÔMÇ ëôà À ËMÎ[éÐEÀîÄaŧËMòcÇ*ËjìÒÀÄaÐsê ÔMé*Ìôê À écê ÁyÇïê . . . . . . . . . . 0. 0. 0. 0. . . . . . . . 0. . . . . 0. 0. 0. . . ∗. †. ‡. ÷. 0. 0.
(3)
(4) !!"$#%&('%$)+*,%-'.*0/ %$ !*1'2"34*,50*,6+%$798: !*,7.;<*, =74*,7#9974%> (/ !!? #@'2A;CB ;<%$ %$7D!? #%-EDF !*,74@GH9!0*I"$9#9#? !976"-*1%KJL"$%$71>M@NPO/Q%* ; %-EDF !*,7050*'%$# )I !*,F4/1O"> #R/ *,S#%$50( !976097T50%-'2? UGH9!V4*,#%-=*, 4%$%$ *,6+%$74%-*,F4(5L WB %$ !? #=GH9!T)I !*,F4"-*,5U/Q%$!976L#%$76I!2BP"> #%->M XP7ZY >[ \P] F4976!4%F72;^*,#'2976 50%$!4*'ZY`_ \P] !4%/Q%$ !*'2"=4*,50*,6+%$798: !*,7aG , "-*,74'%$ %-'b;^*, >c ( 97 d M :e −div (a (x, ∇u )) = f D 0 (Ω),. ÷. u ∈. GH4%$ %. ÷ ÷. W01,p (Ω),. fg9/4"h9!83*,/ %$7aSQ*,F74'%-'a%$H* ; N ] ] ] Ω R 1 < p < ∞ p−1 + q −1 = 1 74' F4"V! (* ;ji h !4k-'*1'*, !O f ∈ W −1,q (Ω) a : Ω × R N → RN a lO1/ % ] 50*,74*,*,74% ] 74'm ,09U6I *>GH!n"-*,7D! *,#9#%-'gMoXP;!4 %bF72;<*,#'%-'m;^F74"$!*,74 "-*,71)+%$ !6+% #950*Ip%$)+%$ !OGH4%$ % ] !4%$7q!4%%-EDF4%$74"-%(* ;R*,#9F!*,74 "-*,72B T (a ) (u ) * ;= b/ *,S#%$5rGH"mh !ts%-3!4%0h 50% )+%$ !6+%- ] , ] *T!4%L*,#9F!*,7 →0 u 1O1/ *,!4%-%-= 74'TGH4%$ % *,7#9Ob'%$/ %$740 '=*,7a!4% #995U9=* ; M a0 T (a ) d u 7 !F h #R6+%$74%$ h #998: !*,7v* ;@/ *,S#%$5 M :e H!4% / *,S#%$5. ÷ ÷. 97 D 0 (Ω), −div d = f (∇u (x), d (x)) ∈ A (x), u ∈ W01,p (Ω).. ÷. d M`w e. x(*>G 3 y50%> ,!F h S#%L5L /zh {976T9 )I #9F4%U97|!4% A : Ω → M(RN × RN ) * ; L 5 W } 9 L 5 N # 0 5 *,74*,*,74%L6I h /4 ;^ *,5 * M~XP;j!4%$ % h%$ M(RN × RN ) RN RN %K}2! 74' !F4"hb! ;<*, #950*I%$)+%$ !O ] 74'L;^*, %$)+%$ !O α>0 m ∈ L1 (Ω) x∈Ω ] (ξ, η) ∈ A (x) d M` e p q. ÷. α(kξk + kηk ) ≤ hη, ξi + m(x),. ÷. ÷. !4%$7Z/ *,S#%$5 d M`w e , =#%> ,! *,74%3h*,#9F!*,7Y ] \ M(*,#9F!*,7.74%-%-'.74*, S % F7E1F4%IM z%3"-*,74!'%$ 4%$ %&!4%&4*,50*,6+%$798: !*,7|* ; d M`w e MHfR%$ (u , d ) ∈ W01,p (Ω) × S %&*,#9F!*,74=* ;@!4% / *,S#%$5 d M`w e M XP;!4% h !t;^Oa ,!F5U/!*,7 p N L R ) A d M`(Ω; e F7A;^*, !5U#9O 74'A; !4%a%-EDF4%$74"-%V* ;(F72;<*,#'%-'6I h /4 "-*,7D)+%$ !6+%- T Y (A ) #950*I%$)+%$ !OGH4%$ % ] %$)+%$ !Oz"$#9F4!%$ 3/ *,97D * ;!4%y%-EDF4 %$74"-% ;<*, (u0 , d0 ) (u , d ) !4%&Gj%> {b*,/Q*,#*,6IOy97 1,p p N ] = U*,#9F!*,7.* ;. ÷. ÷. ÷. W0 (Ω) × L (Ω, R ). 97 D 0 (Ω), −div d0 = f (∇u0 (x), d0 (x)) ∈ A0 (x), u0 ∈ W01,p (Ω).. ÷. GH4%$ % h !!s%- d M` e 74'.='%Ks474%-'.97V%$ !50(* ;N!4% A : Ω → M(RN × RN ) #995U9j* ; 0 Y M@XP7aY >[ \ !4%1O1/ *,!4%-%-NGp%$ %!! *,76+%$ j 74'U95U/#9%-'0!4%H!! *,76 T (A ). ÷. w.
(5) "-*,71)+%$ !6+%$74"-%* ;N!4%"-*, ! %-"$*, ] GH"h.=74*,=*,Sh 974%-'V4%$ %IM4% S %-!="-*,71)+%$ tB 6+%$74"-% 97v! '29 %-"$!*,7| d M _ e M&X ;j!4% h !!; O. ,''29!*,7 # ,h!F5U/!*,74 ] A *,74%& %-"-*>)+%$ !4%3!! *,76L"-*,7D)+%$ !6+%$74"-%* ;!4%3 "-*, ! %-"$*, d 4%-*, %$5 M [+e M T/ *,S#%$5 V97D!95L %$#9O "-*,774%-"$%-' * !4%z"-*,7D)+%$ !6+%$74"-%* ; 5L W}295L # 50*,74*,*,74%*,/ %$ h *, ] !4%!4%-*, !O * ;LGH"h Gp ,z ,"$!9)+%$#9O '%$)+%$#*,/Q%-' 97 !4% _+ ] 50*I=/ !!"$F#? !#9OV97Z!4 % =9#9S %$ !(/ ,"-%"> ,% d Y` \P] Y $\ 74'Y`w \ e M u (!4% !950% ] !4%~s4 ! F!4*, 3 ,'z!!F4'2%-'z9&97z!4%0"> ,%L* ;p % %K}29)+ % H 7 ,"!/ ,"-% ] %-/Q%-"$? #9#9O~97L)%$G * ;g!4%= %$74*, !5U976 %-!F#9j* ;g * ! 74{1 d {+%-%$/97697q5U974'0! 50*,74*,*,7"$9lOLj74*, !5 974'%$/ %$74'%$7D e Mj#%-'q* 3/ / %$ GH"T #*3974"$#9F4'%-' !4%"> ,%H* ;Q!FSQ'2
(6) g%$ %$7D!? #j* ;#*:Gj%$ %$5UABP"-*,71!97DF4*,F4j/ *,/Q%$ "-*,71)+%K};^F74"$!*,74 ] SF=Gp ,H74%$)+%$ =FS5U9!%-'a;<*, H/FS#9"> !*,7 Y \ M XP7v"-*,53S97 !*,7zGH9!Z!4% F72;^*,#'2976b50%$!4*' ] 05L 97v**,# 4%$ %U %~"-*,74'2AB !*,74F74'%$ GH"~!4% #*"> #2"-*,7D)+%$ !6+%$74"-%H* ;46I h /495U/#9%-6I#*,S #"-*,71)+%$ !6+%$74"-%IM Z*, %0/ %-"$h%$#9O ] '%$74*,!976.SDO !4%0"-*,7D)+%$ !6+%$74"-%y97 !4%L6I h /"> #ph%$74% ] Gp% #**,{T;<*, "-*,74'29!*,74&*,7 ! &h !t;^O A, An : Ω → M(X × X 0 ) An (t) A(t) F74'%$ =GH" H G 4 $ % % n. ÷. ÷. ÷. ÷. A A. A = {(u, v) ∈ Lp (Ω; X) × Lq (Ω; X 0 ) | (u(t), v(t)) ∈ A(t). ;<*, 2M %IM. t ∈ Ω}. 74'. ~!95U9#? ='%Ks479!*,7T;<*, n M A 4%*,F!#9974%* ;!4%/ /Q%$ ,g;<*,#9#*>G:M Xl7%-"$!*,7Uw ] Gp% %-"> #9#D!4%j'%Ks479!*,7 * ;Q5L W}95L #50*,74*,*,74%*,/Q%$ h *, p 74'U* ;!4%H74*,!*,70* ;g"-*,7D)+%$ !6+%$74"-%* ;Q5L W}295L # 50*,74*,*,74%6I h /4>M@Xl7L%-"$!*,7y ] F4!976qY \P] Gj%"-*,74!'%$ ph%-EDF4%$74"-%-j* ; 5L W}295L # 50*,74*,*,74%T), #9F4%-'50%> ,!F h S#%b;^F74"$!*,74 ] !4%$9 0"> 74*,7"> #=%K}%$74*,740 74'mGj% / *:)+% {+%$Ob %-!F#9= S *,F!4%$9 ="-*,71)+%$ !6+%$74"-%IM XP7T%-"$!*,7 Gj%3 %-"> #9#R!4%&'%Ks479!*,7v* ;!4% F72;<*,#'2976q*,/ %$ h *, ] >)+%$ h 6I976 *,/ %$ h *, & 74'V!4%~"-*, ! %-!/ *,74'2976a"-*,7D)+%$ !6+%$74"-%U/ *,/ %$ !!%- d "K;tM~Y`_ \P] Y $\ ,(Gj%$#9# ,3Y W\P] Y >[W\ e M XP7z%-"$!*,7 Gj%0"-*,74'%$ &!4%U4*,50*,6+%$798: !*,7/ *,S#%$5r 74'|Gj%q!h %U*,F 5L 97V %-!F#997T4%-*, %$5 M M *,50%z* ; !4%z %-!F#9T* ;~!a/ /Q%$ aGp%$ % 774*,F74"-%-' 97 Y >[ \ M 4%|"> ,% 3!4%q!FSQ'2
(7) g%$ %$7D!? #* ;="-*,7D)+%K} ; F74"$!*,7GH9! !F9h S#%q6I *>GH! GH4%$ % A (x) 74'V"-*1%$ "$9)9POa"-*,74'29!*,74GH9#9#SQ% / %-%$71%-'T97V ;<*, !!4"-*,5U976L/ / %$ 3Y w \ M x*,%q! !4%$ %a %q5L 7DOz/ / %$ 97 !4%q#99!%$ h !F %bF74'%$ !4%q!!F4'2O * ; BP"-*,71)+%$ !6+%$74"-%q 74'v4*,50*,6+%$798: !*,7 GH"z"-*,74"-%$ !7z!4% 74*,72B #9974%> 3"> ,%IM3z% G *,7#9Oy %K;<%$ *aY \ 74'a!4% S9S#9*,6I h /1OT!4%$ %$97RM. ÷. ÷. ÷y÷. ÷. ÷. ÷. ÷. . )+*-,. & . . =!#"H$&%'(. .0/214351768/:9<;. Xl7U!h%-"$!*,70Gj%H %-"> #9#*,50%HS ,!"-N74*,h !*,74 S *,F 50*,74*,*,74% 74'U5L W}295L # 50*,74*,*,74%.6I h /4b 74'm; F74"$!*,74q97n = 7 ,"hn/ ,"-%IM?>*, L50*, %V'%$h 9#y%-% Y` ] @ ] W\ M .
(8) fR%$ S %~ y %%K}29)+% H 7 ," / ,"-%0 74'v#%$ SQ%~9 '2F # M 4%U'2F #99PO X X0 u / *'2F4"$97 R ' $ % 4 7 , * % 3 ' D S O M " " , * 2 ' 9 7 6 * & ` Y w $\P] !4%$ % 73%-E1F9)I #%$7D X 0 ×X h·, ·i 74*, !5 *,7 !F4"q! S *,! 74' % # * > " 9 # 9 # ~ O F 7 A ^ ; *, !5U#9O0"-*,71)+%K} MNjGH9#9# X X X0 S %v ,!F50%-'o; *,5 74*>G *,7RM XPqy* S %V74*,%-' ! a*,7 #*"> #9#9O F7A;^*, !5U#9O "-*,71)+%K}y!/ ,"-% ] Gp%> {q"-*,71)+%$ !6+%$74"-% *,6+%$!4%$ GH9!y!4% "-*,71)+%$ !6+%$74"-%&* ;!4%(74*, !50 95U/#9%-!! *,76."-*,71)+%$ !6+%$74"-%IMb4%0'2F #99PO|/ 9 !976v ,*"$? %-'|*a!4%U74*, !5 * ; U'%$74*,%-' SDO 74'm5L /4 *v!4%yF7E1F4% F4"! X F ξ ∈ X F (ξ) ∈ X 0 74' M pOT!4%3#*"> #F7A;<*, !5 " , * D 7 )+%K}29POZ* ; kF (ξ)kX 0 = kξkX hF (ξ), ξi = kξk2X 74'|* ; 3 7 4*,50%-*,50*, !/!5TMqXl 97D)+%$ h%0 !4%0'2F #99lOv5L //976 ] X X0 F * q M l X | 7 !4%Lh%-EDF4%$# ] ;^*, &!95U/#9"$9lO ] GH9#9#j'%$74*,%0%$9!4%$ *, ;^ *,5 X0 X k.k k.kX !974"-% !4%&"-*,7D%K}2=GH9#9#g5L {+%&!4% 74*,h !*,7."$#%> >M k.kX 0 z%("-*,74'%$ h%$tB ), #9F4%-'b*,/ %$ h *, ] ! p5L /4pGH"ah {+% A : X → X0 %$)+%$ !O/Q*,971 *(h*,50%ph%$ M0 4%-%H //#9"> !*,74p %p95U/#9O"> #9#%-' ξ∈X Aξ ⊂ X M 195U9#? !#9O GH4%$774*( 53S96IF9!%-N !% ]
(9) GH4%$774*="-*,72;^F4!*,745L >O3 !%I+ GH9#9#( #h* '%$74*,%T!4 % * ; !4%V*,/ %$ h *, ] ! L0!4%.%$ A A {(ξ, η) ∈ M 4 % * ;. I 6 h / !4% %$ ]
(10)
(11) 0. ÷. X × X : η ∈ Aξ}. A. D(A) = {x ∈ X. F4"T! . Ax 6= ∅}.. 4%*,/ %$ h *,
(12) |*,7Z*,50% %$ ] A;@;^*, %$)+%$ !O ] A C⊂X ξ∈C -" *,71h 974= j50*Ip*,74% %$#%$50%$7D"!49jp74*,74%K}2/ 74!9)+%(A; Aξ kη − η k ≤ kξ − ξ k ;<*, %$)+%$ !O M$#N)+%$ !O~74*,74%K}2/ 74!9)+%=*,/ %$ h *1, j2"$#%> !#9O~1976I#2%KB (ξ , η ), (ξ , η ) ∈ A )I #9F4%-'|*,7 1 M 1>4*, 3*,2/ %$2 h *, GH4%$74%$)+%$ ;<*, ] Gj%UGH !9% X A, B A⊆B Aξ ⊆ Bξ %$)+%$ !O M ξ∈X .*,74*,*,74%3 74'T5L W}95L #R50*,74*,*,74%36I h /4="> 7T74*:G S % '%Ks474%-'gc %'&)($*,+-.+/0*2143653 4%Lh%$. . A ⊂ X × X0 *,/ %$ h *, e A;;^*, %$)+%$ !O ] (ξ1 , η1 ), (ξ2 , η2 ) ∈ A. 7 0
(13) 8 . d *, 50*,74*,*,74%. hη1 − η2 , ξ1 − ξ2 i ≥ 0.. 4%50*,74*,*,74%6I h / 7.9
(14)
(15) 0:
(16) 8 ;0 d *, N5L W}295L #50*,74* B A *,74%j*,/Q%$ h *, eK] A;;<*, %$)+%$ !O&50*,74*,*,74%j6I h / % 974"$#9F4!*,7 ] !4 B ⊂ X ×X 0 A⊆B 95U/#9%- M A=B. ÷)>. f %$ R S %q T976I#%KB ), #9F4%-'5L /GH9! MTXP; A : X → X0 D(A) = X ("-*,71!97DF4*,F4 74'v50*,74*,*,74%~!4%$7|9((5L W}295L #N50*,74*,*,74%IM3Xl7v/ !!"$F#? A !4% '2F #99lOq5L //976 '%Ks474%-= 5L W}95L #50*,74*,*,74% 6I h /R< M 1974"-% F : X → X0 0!! !"$!#9Om"-*,7D)+%K} ] U!! !"$!#9Om50*,74*,*,74%a97!4%T%$74h%b! k.kX F hF (x) − 95U/#9%- M 0 0 0 F (x ), x − x i = 0 x=x 4%3;<*,#9#*>GH976a %-F#9(/#? >O& L;^F74'4 50%$7Dh #N *,#% 97v!4% !4%-*, !O.* ;j5L W}295L # 50*,74*,*,74%*,/Q%$ h *, :M < 9:
(17) = . ?A@ &B/C &BDE1F3G1IH;JKC /BLM,&)CONQPSRUT"3=V W >7Y S A ⊂ X × X 0 X A7
(18) 0:
(19) 8 ;0 ;0 ^ > > ] A Z ["9
(20)
(21) 0 8\ ]\
(22)
(23) A + F Z \_ `ab" 6 K7cd > ]a
(24) A
(25) ^ bg h. 0 ce SfA. η∈X. ξ∈X. (ξ, η − F (ξ)) ∈ A.
(26) ÷)>. 1974"-% !! !"$!#9OV50*,74*,*,74% ] ;<*, %$)+%$ !O !4%$ % 50*I! F η ∈ X0 ! F4"T! M ξ∈X (ξ, η − F (ξ)) ∈ A X; ( ~5L W}295L #50*,74*,*,74%36I h / ] !4%$7T;<*, =%$)+%$ !O '%Ks474%-'v , ] A ξ ∈ X Aξ 3 #*T b5L W}295L # 50*,74*,*,74%L6I h /RM (%$74"-%~;^*, ] !4%$ % Aξ = A(. + ξ) λ>0 %K}2!& yF7E1F4% M ]ξ M %IM ] F4"z! (α, β) ∈ A (α − ξ, β) ∈ A 0 = F (α) + λβ + F4!!As%-!4%&'%Ks479!*,7Rc.
(27) . *,74%. %'&)($*,+-.+/0*. 143F3HfR%$. S %( 5L W}295L #Q50*,74*,*,74% 6I h /RM+>*, %$)+%$ !O. A ⊂ X × X0 74' '%$74*,%-!4% F7E1F4% / 9 H97 ! F4"T! ] A ξ∈X λ > 0 (Jλ ξ, Aλ ξ) A F (ξ − JλA ξ) = λAλ ξ.. w > 4%$7 3 =9#9S %$ !&!/ ,"-% ] 9=%-!8U*,50%$! !" X F : X → X0 h*,50*, !/!5TM Xl7a! "> ,% ] *0%$)+%$ !Oy5L W}95L #g50*,74*,*,74%36I h / ] A ⊂ X × X0 q74*,74%K}2/ 74!9)+%~5L / ,h*1"$? %-'v!F4"h|! A;p 74' φA : X → X (ξ, η) ∈ A *,7#9OyA;
(28) . F −1 (η) − ξ = φA (F −1 (η) + ξ).. ip*,71)+%$ %$#9O ] %$)+%$ !O 74*,74%K}2/ 74!9)+%b;^F74"$!*,7 '%Ks474%-L Z5L W}295L # φ : X 7→ X 50*,74*,*,74%36I h /T97a! G -O|Y \ MN4% 5L /4A 74' A % #997{+%-'. ,. ÷. φA. J1. φA (ζ) = ζ − 2J1A ζ. \C / d/ .+-.+/0*. 143
(29) 3 Y S 7 > Y b
(30) 6 S ;0. ÷. d w2M :e. :0 JλA : X → X
(31) Aλ : X → X 0
(32) f
(33) Z U
(34) 7.9
(35) 7 0
(36) 8 >. Aλ. 4% / *1* ;N* ;N *,/Q*I9!*,7.w2M %$#9%-*,7T!4% ;^*,#9#*:GH976LfR%$5U5L 2c e&BD. A. D7143ZP43 V W A ⊂ X×X 0 X "9
(37) 8
(38) 8 W;
(39) (ξn , ηn ) ∈ > ) S a S c c. n → +∞. ξn * ξ ηn * η. . ^. . ^. X,. . X 0,. d w2M`w e. lim inf hηn , ξn i ≤ hη, ξi, . (ξ, η) ∈ A. n→+∞.
(40) . lim inf hηn , ξn i = hη, ξi. ;"] >. 4*, =%$)+%$ !O >. n→+∞. (α, β) ∈ A. ] S1Oq!4%&50*,74*,*,7"$9POT* ;. A. ]. hβ − ηn , α − ξn i ≥ 0.. 4%$ %K;<*, %. 0 ≤ lim inf hβ − ηn , α − ξn i ≤ hβ − η, α − ξi. n→∞. 1974"-% H5L W}295L # 74' = !S9! h !O ] M A (α, β) ∈ A (ξ, η) ∈ A 4% ;< ,"$H! 74'T!4% 50*,74*,*,7"$9POV* ; 74*:G O%$#' . (ξ, η) ∈ A. A. lim inf hηn , ξn i ≥ lim hηn , ξi + lim hη, ξn i − hη, ξi = hη, ξi. n→∞. n→∞. . n→∞.
(41) >0>. ;"]A;] S 6
(42) . %$)+%$ !O. ξ∈X. ] Gp% -)+%. fR%$. (α, β) ∈ A. M. pO!4%Z50*,74*,*,7"$9POn* ; . A. ] ;^*, . hF (ξ − JλA ξ), α − JλA ξi ≤ λhβ, JλA ξ − αi,. 4%$74"-% ]. d w2M` e. kξ − JλA ξk2 ≤ λkβk kJλA ξ − αk + kα − ξk kJλA ξ − ξk,. !4%$ %K;<*, % ] A SQ*,F74'%-'.*,7aSQ*,F74'%-'Th%$>M u F50J% λ !! *,76I#9O.97 M 4%50*,74*,*,7"$9lOz* ; 74'|* ; 95U/#9O ξn → ξ X A F ! hF (ξn − JλA ξn ) − F (ξm − JλA ξm ), JλA ξn − JλA ξm i ≥ 0, hF (ξn − JλA ξn ) − F (ξm − JλA ξm ), (ξn − JλA ξn ) − (ξm − JλA ξm )i ≥ 0.. 1F5U5U976q!4% PGp*0/ %$)*,F4974%-E1F #99!%-6I9)+%- . hF (ξn − JλA ξn ) − F (ξm − JλA ξm ), ξn − ξm i ≥ 0.. pO d w2M` eK] !4% #%K;^tB 74'T!'% %$74'=* . lim. 0. ,. hF (ξn − JλA ξn ) − F (ξm − JλA ξm ), JλA ξn − JλA ξm i = 0.. m,n→+∞. f%$. (α, β). Mip*,74%-E1F4%$7D!#9O ]. m, n → ∞. S %& Gj%> {a"$#9F4!%$ =/Q*,971=* ;. (JλA ξn , Aλ ξn ). . M 74% ,. lim hAλ ξn − β, JλA ξn − αi = 0.. n→+∞. pO~fR%$5U5L w2M 2] 74' MN4%$ %K;<*, % ] (α, β) ∈ A λβ = F (ξ − α) α = JλA ξ β = Aλ ξ 74' H!4% F7E1F4% Gp%> {b"$#9F4%$ /Q*,971=* ; M pOyf%$5U5L 0w2M (α, β) (JλA ξn , Aλ ξn ) *,74% #*~ , . lim hAλ ξn , ξn − JλA ξn i = hAλ ξ, ξ − JλA ξi. n→∞. 1974"-% . kξ−JλA ξk2 = λ2 kAλ ξk2 = hAλ ξ, ξ−JλA ξi 97 M (α, β) X × X0 >4*, ] *,74% , ξ1 , ξ 2 ∈ X. ] *,74% "-*,74"$#9F4'%-! . (JλA ξn , Aλ ξn ) →. λhAλ ξ1 − Aλ ξ2 , ξ1 − ξ2 i = hF (ξ1 − JλA ξ1 ) − F (ξ2 − JλA ξ2 ), ξ1 − ξ2 i = hF (ξ1 − JλA ξ1 ) − F (ξ2 − JλA ξ2 ), (ξ1 − JλA ξ1 ) − (ξ2 − JλA ξ2 )i + hF (ξ1 − JλA ξ1 ) − F (ξ2 − JλA ξ2 ), JλA ξ1 − JλA ξ2 i ≥ 0. ! 7{2H*U!4% 50*,74*,*,7"$9lOV* ; 74' M A F 1974"-% !976I#%KB )I #9F4%-'V 74'T"-*,7D!971F4*,F4 ] 9=H5L W}95L # M Aλ. ÷. ÷. 9#9SQ%$ !@!/ ,"-% ] 74*,74%K}2/ 74!9)+% ] h%-%Y \ 74' d w2M :e
(43) > 4%$7 X JλA 97 %$5L !{Tw2M [.
(44) 9
(45) / ?3 6 ?3?/:9. )+* ). /:9. 73<; -,!. /214/:9. >*,#9#*>GH976 p %$8-&Y` \ 74' u !*,F4" Y`w \P] !4% "-*,7D)+%$ !6+%$74"-%3* ;@5L W}295L #R50*,74*,*,74% 6I h /4='%Ks474%-'V ,;<*,#9#*>G:c. SQ%~5L W}295L # 50*,74*,*,74%L6I h /4>M04% 143#"F3(fR%$ An , A ⊂ X × X 0 d h %-EDF4%$74"-% n "-*,71)+%$ !6+%-&* , ] eK] A;j;<*, %$)+%$ !O A A n → ∞ An A (ξ, η) ∈ A !4%$ %%K} 0h%-EDF4%$74"-% ! 4 F h " V ! !! *,76I#9Oa97 (ξn , ηn ) ∈ An (ξn , ηn ) → (ξ, η) M 0 , %'&)($*,+-.+/0*. X ×X. n→∞. < 9:
(46) = w > XP; 74' %3%$)+%$ !OGH4%$ % '%Ks474%-' An A d h%-%[@ # } 5U/#% :eK] 74'aA;;^*, %$)+%$ !O ] n. ÷. ] "-*,71!97DF4*,F4& 74'.50*,74*,*,74% M ] !4%$7 n. x ∈ X A (x) → A(x). 4%&"-*,7D)+%$ %&H! !F4% 97as479%KBP'2950%$74!*,7 #!/ ,"-%-. A A. 143%$83dV W > An ∈ C(RN ; RN )
(47) A ∈ C(RN ; RN ) X
(48) 8
(49) 8
(50) f ' & . \C / d/ .+-.+/0*. Y ]a . ( *). . An A c. ( +)_]W A ( +)_]W A . ^. ^. b
(51) =S:ba aa. ^
(52) K ⊂ RN c An → A ]W Kc. ξ ∈ RN c An (ξ) → A(ξ). >. fR%$qF40s4 !q/ *>)+%.! d e 95U/#9%- d 9 e M 1F// *I% 74'o#%$ An A S %q a"-*,5U/ ,"$~%$ * ; N MTij4*1*I% i ]. !F4" ! #9%-397 K R ξ i ∈ {0, ..., N } K !4% 971%$ !*, 3* ;j!4%~"-*,7D)+%K}|1F#9# * ; i M 1974"-% n ] !4%$ %0 %U%-E1F4%$74"-%- (ξn ) A A ! 4 F h " ! m M P X 7 / ! ! "$F#? ] #9%-~97!4% (ξni )n≥1 (ξni , An (ξni )) → (ξ i , A(ξ i )) K 971%$ !*, &* ;!4%~"-*,7D)+%K}Z DF#9#N* ; i ;<*, #? !6+% M fR%$ S % y%-E1F4%$74"-%397 (ξn ) n (ξ ) K F4"T! M pOy!4% 50 + *,74*,*,7"$9POV* ; n ] n ;"] >. ξn → ξ ∈ K. A. lim suphAn (ξn ), ξni − ξn i ≤ hA(ξ i ), ξ i − ξi. n→∞. 1974"-% !4%$ %& !F4"ha! ;^*, #? !6+% ] H"-*,7Dh 974%-'V97a!4% "-*,7D)+%K} δ>0 n B(0, δ) jS *,F74'%-'gM fR%$ 1F#9#g* ;R!4%=/ *,97D i ] !4%h%-EDF4%$74"-% (ξn − ξn )0≤i≤N (An (ξn ))n≥1 S % 3"$#9F4!%$ / *,97D* ; n M-,/0*~ &!FS4%-E1F4%$74"-% ] *,74% , η A (ξ ) hAn (ξn ), ξn i → M 1974"-% 3"-*,7D!97DF4*,F4n ] 9& 5L W}295L # ] 74'|!4%$ %K;^*, % ] SDO|fR%$5U 5L Tw2M ] hη, ξi A M η = A(ξ) 4%&*,!4%$ 95U/#9"> !*,74 % ! !9)1? # M . 97 N *,/Q*I9!*,7yw2M(; 9# A; !4%='2950%$74*,7q* ; 74' >
(53) (i) ⇒ (ii) X X0 972s479%IMHf%$ 4 7 ' H G 4 $ % % 0 S , ! = * ; ] X = l2 (N) An (ξ) = f (nhen , ξi)en (en ) 4 7 ' = 4 7 , * 4 7 ' % $ " > % , ! 9 7 a 6 " , * 1 7 ! 9 D 7 4 F , * 4 F 4 7 . ' ! 4 F h " . ! ] l2 (N) f :R→R f (0) = 0 M&z%U%-% ! n 74' SF d 99 e f (1) 6= 0 A (ξ − he , ξie ) = 0 he , ξie → 0 n n n n P 74*,! !F4%IM= {1976 ] Gp%3 >)+%3!4%3;<*,#9#*>GH976yGj%> {V"-*,71)+%$ !6+%$74"-% 1 ξ = m∈N m em S F ! 4. % ! , * 7 L 6 " , * D 7 + ) $ % ! + 6 %$74"-% '*1%-74*,H4*,#'gM ] An (ξ) * 0 4%y"-*,71)+%$ !6+%$74"-%a* ;=6I h /4~%$74!F %-~! Gj%> { #995U9U* ;=%$#%$50%$710* ; n A %&97 / *>)'%-'T!4%3'2F #99POb/ *1'2F4"$* ;!4% / 9 H/ %-%$ !)+%-'V !4%ϣU9>M A. .
(54) ?A@ &B/C &BD. 143F3 V W An , A ⊂ X × X 0 X A"9
(55) 8
(56) 8 : c
(57) > (ξn , ηn ) ∈ An
(58) (ξ, η) ∈ X × X 0 ] ce n → +∞ c. An A, ξn * ξ ηn * η. ^ X, f ^ 0. . X,. d w2M e. lim inf hηn , ξn i ≤ hη, ξi, . (ξ, η) ∈ A. W. n→+∞.
(59) . lim inf hηn , ξn i = hη, ξi. n→+∞. !! *,76I#9Oz97 ] !4%$7 d w2M e h !ts%-' ]
(60) > XP; (ξ , η ) → (ξ, η) X × X0 M %IM%$)+%$ !Oa!! *,76I#9nOb"-n*,7D)+%$ !6I976b%-E1F4%$74"-% * ;@%$#%$50%$71=* ; n H9 7 M4% h 50% A d A d %-!/RM " , * D 7 + ) $ % ! + 6 % ( ! ! , * 7 I 6 9 # b O 9 7 4*,#'GH4%$7V*,7#9O % R / 2 M 97 e 0e M (ξn ). (ηn ). 4% / *1* ;N* ;N4%-*, %$5. X. w2M !95U9#? =*~! =* ;@f%$5U5L Lw2M M. 1F3F3KV W ?A@ &B/C &BD An , A ⊂ X × X 0 X "9
(61) 7
(62) 0 8 >7Y Sd]W
(63) f K & W λ>0 . ( *). ( +) ( +). X. :.
(64) . ]a
(65) .
(66) . An A c n → ∞ c n ∀ζ ∈ X c JλA ζ → JλA ζ n → ∞ c. ( '). ∀ζ ∈ X c Anλ ζ → Aλ ζ n → ∞ c > Anλ Aλ c n → ∞. n f
(67) c b
(68) 8b
(69) JλA ζ → JλA ζ Anλ ζ → Aλ ζ
(70) f
(71) ^ > W 6; bf
(72) =S)ba aW \;] . X.
(73) . λ0 > 0. M 4%$ %K;<*, %A;!4%$ %H%K}2! [ > x(*,%! '*1%-@74*,N/#? >O~ 71O3 *,#%97 d e @ λ d d d d !F4"hT! 7DOa* ; 9 e B 9) e 4*,#' ] !4%$7 9 e B 9 ) e 4*,#'b;<*, %$)+%$ !O M λ>0. f %$qF4L/ *:)+%V! d e 95U/#9%-q! An R 74' Jλ ζ → JλA ζ d An ζ → A ζ F7A;<*, !5U#9O *,7 "-*,5U/ ,"$T!FS4%$a* ; M bGH9#9# 95U/#9O 9 e 74' λ d 99 e M fRλ%$ X S %F4"y! !! *,76I#9OU97 M (%Ks474% M (ζn )n≥1 ∈ X ζn → ζ X (ξ, η) = (JλA ζ, Aλ ζ) 1974"-% , ! 4 $ % . % y % 1 E 4 F $ % 4 7 " % ! 4 F " n ! ] An A n → ∞ (ξ , η ) ∈ An !! *,76I#9Oq97 M0 pOq!4% 50*,74*,*,7n"$9nPOT* ; n ] ;"] >. (ξn , ηn ) → (ξ, η). X ×X. A. n. λkAnλ ζn k2 = hAnλ ζn , ζn − JλA ζn i n. ≤ hηn , ξn − JλA ζn i + hAnλ ζn , ζn − ξn i ≤ kAnλ ζn k(λkηn k + kζn − ξn k).. (%$74"-%j!4%ph%-EDF4%$74"-%- n 74' An %pS *,F74'%-' 97 74' MfR%$ Aλ ζn Jλ ζn X X0 (α, β) ∈ ] M %IM ] F/T*0 0 S %& ~Gp%> {b"$#9F4%$ /Q*,971=* ;@!4%&%-E1F4%$74"-% An n . X ×X. . (Jλ ζn , Aλ ζn ).
(74) FS4%-EDF4%$74"-% ] An Gj%> {#9O 97 M j O !4%50*,74*,*,7"$9PO (Jλ ζn , Anλ ζn ) * (α, β) X ×X 0 * ; ] F. m. n. f%$!!976. A A hAnλ ζn − Am λ ζm , (ζn − Jλ ζn ) − (ζm − Jλ ζm )i ≥ 0.. m→∞. 6I9)+%-. n. m. A n hAnλ ζn , JλA ζn i + lim suphAm λ ζm , Jλ ζm i ≤ hAλ ζn , ζn − ζ + αi m→+∞. n. −hβ, ζn − ζ − JλA ζn i.. f%$!!976L74*>G. n→∞. ] Gj%&*,Sh 97 n. lim suphAnλ ζn , JλA ζn i ≤ hβ, αi. n→+∞. pOU4%-*, %$5 . w2M ]. (α, β) ∈ A. ] 74'. n. lim inf n→∞ hAnλ ζn , JλA ζn i ≥ hβ, αi. M %$74"-% ]. n. lim supkAnλ ζn k kζn − JλA ζk = n→∞. n. lim suphAnλ ζn , ζn − JλA ζn i ≤ hβ, ζ − αi ≤ kβk kζ − αk. n→∞. 4%$ %K;<*, % !! *,76I#9O 97 M pOo!4% n (ζ − JλA ζn , Anλ ζn ) → (ζ − α, β) X × X0 "-*,71!97DF9lO* ; ] M 74% ] 74' !4%$ %K;<*, % F λF (β) = ζ − α (α, β) = (JλA ζ, Aλ ζ) "-*,74"$#9F4'%-=! 9 7 , M n (J A ζ , An ζ ) → (J A ζ, A ζ) X × X 0 n→∞ 4%U%-EDF9)I #%$74λ"-%~SQn%$lGjλ%-%$n7 d 9 e 74λ' d 99 λe "-*,50%- ;^ *,5 !4%U"-*,7D!97DF9lO|* ;j!4% '2F #99lOb5L //976 M u h%$ !!*,7 d 99 e F95U/#9%- d 9) e d %-% #@}4 5U/#%3w e M > 97 #9#9O ] d 9) e 95U/#9%- d e cyf%$ M ] 74'm'%Ks474% (ξ, η) ∈ A ζ = ξ + F −1 (λη) pO!4%v ,!F5U/!*,7 ] !4%$ %.b %-EDF4%$74"-% !F4" ! ζn ∈ X (ζn , Anλ ζn ) → 97 M pOU!4%="-*,7D!971F9POb* ; −1 ] An 0 (ζ, Aλ ζ) X × X F Jλ ζn = ζn − F −1 (λAnλ ζn ) → M 4 $ % K % < ; , * % ] n (JλA ζn , Anλ ζn ) → (JλA ζ, Aλ ζ) = (ξ, η) ζ − F −1 (λAλ ζ) = JλA ζ ! *,76I#9Oq97 0M. . X ×X. U =9#9S %$ !U!/ ,"-% ] 74' %
(75) > XP; X φ n : X → X φA : X → X !4%~74*,74%K}2/ 74!9)+% ;^F74"$!*,743 ,*"$? %-'vA* n 74' 9 7 =%$5L !{|w ] !4$% 7|!4% A A / *,/ *I!9!*,74* ;!4% 4%-*, %$5 %&%-EDF9)I #%$7D=GH9!Rc >4*, =%$)+%$ !O , M ] ξ ∈ X φAn (ξ) → φA (ξ). )+* . n→∞. 73 $3 @32; 73 9. 1768/:9<;. 4 *,5 74*>G *,7 ] Gp%3! #9# ,h!F50% ! ] 74'a4%$74"-% ] %$/ h S#%IM4%3%$ X X0 * ;H5L W}295L #j50*,74*,*,74%y*,/ %$ h *, &;^ *,5 * ' $ % 4 7 *,%-' SDO M X X0 M(X × X 0 ) Xl7 !~%-"$!*,7 GH9#9#jS %q B<s479% BP"-*,5U/#%$%y50%> ,!F %KBP!/ ,"-%IMZ4% (Ω, T , µ) σ µ "h h ,"$%$ !!!" ;^F74"$!*,7. ,h*1"$? %-'a*~!4%3%$ ] ='%$74*,%-'aSDO M A χA 4% ;^*,#9#*:GH976q'%Ks479!*,7V"> 7VSQ%(;<*,F74'a97zY [ \ M >. _.
(76) %'&)($*,+-.+/0* 14365 3 u ^; F74"$!*,7 A; 74'q*,7#9O A : d Ω → M(X × X 0 ) _ X A;Q;<*, %$)+%$ !OU*,/Q%$7Lh%$ % ! L / $ " # I * % 0 ' h $ % *, %$#%$ ] , * Q / $ % 70S #9# ] "$#*I%-' ] U ⊂ X × X0 S #9# eK]. 50%> ,!F h S#% 97. Ω. {t ∈ Ω | A(t) ∩ U 6= ∅}. M. ÷. 4%0;<*,#9#*>GH976Z/ *,/Q*I9!*,7G ,3/ *:)+%-'97 Y $\ ;^*, !4% 9#9SQ%$ ! "> ,% ] F4!976 %-F#9* ;Y [W\ M \C / d/ .+-.+/0*. 14365053=V a A : Ω → M(X × X 0 ) c W λ > 0
(77) K > Y Se]W
(78) f7 & W X ( *) Z _ X c. W. E X W a. A. ( +)_]W A ^. ( +)_]W A ^. ζ ∈ E c t 7→ Jλ. A(t). ζ _
(79) ; X c. > ζ ∈ E c t 7→ A(t)λ ζ [f ; X . 74*, /#? -Oy 7DO0 *,#%97 x(*,%! p74%$9!4%$ λ E %K}2! 74'z b'%$74%~%$ F4"|! &%$9!4%$ &* ; 0 E d 9 e 74' λ0d 9> 9 e 4*,#'b;<*, %$)+%$ !O 74'y;<*, =%$)+%$ !O
(80) . . >. de N M 4%$ %K;<*, %(A;R!4%$ % d 9 e ,* d 99 e 4*,#' ] !4%$7 M. ζ∈X. λ>0. 9!4*,F#*I* ;@6+%$74%$ h #99PO ] %$ n M λ = 1 u F50% d e M fR%$ SQ% "$#*Ih%-'gM >4*, =%$)+%$ !O. ;"] >. 4 *,5 >. ] %$. G⊂X ζ∈X . Cζ = (ξ, η) ∈ G × X 0 | ξ + F −1 (η) = ζ .. !4%&"-*,7D!971F9POV* ; n. F −1 A(t). t ∈ Ω | J1. ] 9H;<*,#9#*>G=! . Cζ. "$#*I%-'gM x(*>G Gp% -)+%. o ζ ∈ G = {t ∈ Ω | A(t) ∩ Cζ 6= ∅} ,. !4%&#? !%$ H50%> ,!F h S#% S1O d eK] *~! d 9 e !ts%-'gM *V4*>G ! d 99 e 95U/#9%- d eK] Gj%q"> 7 ,!F50%U! 3"-*,F71h S#%IM >4*, E 6I9)+%- ] %$!!976 ] !4%$ %K;<*, % A(t) ] (ξ, η) ∈ A(t) ζ = ξ + F −1 (η) ξ = J1 ζ η = A(t)1 ζ S1Oa!4%3"-*,71!97DF9POv* ; A(t) 74'.* ; GH9!. %-/Q%-"$* 74'V!4%'%$74!9PO J1 ζ A(t)1 ζ ζ * ; ] *,74% , E . t ∈ Ω | A(t) ∩ U 6= ∅ [. A(t) = t ∈ Ω | (J1 ζ, A(t)1 ζ) ∈ U ζ∈X. =. [. ζ∈E. !1F4 ] !4% #%K;^tB 74'T!'% 50%> ,!F h S#%IM. A(t). t ∈ Ω | (J1. ζ, A(t)1 ζ) ∈ U ,. 9#9S %$ ! !/ ,"-% ] !4%$7 N50%> ,!F h S#%HA;
(81) _ > XP; X A : Ω →d M(X ×X 0 ) 74'U*,7#9OA;!4% ,h*1"$? %-';^F74"$!*,7 SDO =%$5L !{ w e @50%> ,!F h S#% (ζ) d h%-% =%$5L !{aw2M`w2MN9t7 →Y φ\ A(t) ;<*, %$)+%$ !O eM. ÷. ζ∈X. ÷ .
(82) )+*. 3$9. /:9. 6. 3 1 . 9<;68/:9<;. ?3 6 ?3 ?/:9 /. 73<;. /214/:9. 9)+%$7 b;^F74"$!*,7 ' %Ks474%b a50*,74*,*,74%y6I h / ] *,74%q"> 7 A : Ω → M(X × X 0 ) * H G 4 $ % % , ;<*,#9#*>G>c ;^ *,5 ] ] p q 0. L (Ω; X). L (Ω; X ). 1/p + 1/q = 1. 14365S1F3 f%$. %'&)($*,+-.+/0*. ;^ *,5. *. ] !4%L"> 74*,7"> #j%K}2%$74!*,7* ;. A : Ω → M(X × X 0 ) ] GH4%$ % ] '%Ks474%-'aS1OQc q L (Ω; X 0 ) 1/p + 1/q = 1. A. Lp (Ω; X) . ;<*, = 2M %IM A = (u, v) ∈ Lp (Ω; X) × Lq (Ω; X 0 ) | (u(t), v(t)) ∈ A(t) t ∈ Ωd .. 74%& %> ,'29#9Ob"h4%-"{2! . A. w2M +e. H50*,74*,*,74%IM. \C / d/ .+-.+/0*. 14365 3=V W > A : Ω → M(X × X 0 ) X \_ X > [7.9
(83)
(84) 8
(85) 8. A. . ]. A 6= ∅ ce S. ÷. > 4%5L W}295L #99PO * ; ;<*, @ #950*I%$)+%$ !O 74*,@!F2JL"$%$7D@97
(86) A(t) t∈Ω ,N!4%=#? !%$ j"-*,F#'LSQ%%$5U/lOQc *, '%$ j*&%$74!F %=!4%H5L W}95L #99lO0* ; A Ω = (0, 1) 74' N N −1/q M. A(t) = {(ξ, η) ∈ R × R. ;"] >. }. :η=t. M 1F//Q*Ih% 1974"-% ] !4%$ %U%K}2! A 6= ∅ (α, β) ∈ A (u, v) ∈ Lp (Ω; X) × ] 0 !F4"V! H;^*, %$)+%$ !O 0 0 . Lq (Ω; X ). *, =%$)+%$ !O >. (u , v ) ∈ A. t∈Ω. ] %$. Z. hv 0 (t) − v(t), u0 (t) − u(t)i dµ ≥ 0. Ω. A(t). u(t) + F −1 (v(t)) , v∗ (t) = A(t)1 u(t) + F −1 (v(t)) .. u∗ (t) = J1. pOz"-*,74!! !F4"$!*,7 ] ;<*, #950*I~%$)+%$ !O M > *, ] (u∗ (t), v∗ (t)) ∈ A(t) t ∈ Ω k > 0 '%Ks474% . Ωk = {t ∈ Ω | ku∗ (t)k ≤ kku(t)k. 74'. kv∗ (t)k ≤ kkv(t)k} ,. 74'. 74%& ,. (uk , vk ) = χΩk (u∗ , v∗ ) + χΩ\Ωk (α, β). (uk , vk ) ∈ A Z. GH"hV %> ,' Z. ] *~! . hvk (t) − v(t), uk (t) − u(t)i dµ ≥ 0,. Ω. hv∗ (t) − v(t), u∗ (t) − u(t)i dµ ≥ −. Z. Ω\Ωk. Ωk. ÷y÷. hβ(t) − v(t), α(t) − u(t)i dµ.
(87) 1974"-% ] Gj%3 -)+% ;<*, 2M %IM ] SDOa!4% u (t) + F −1 (v (t)) = u(t) + F −1 (v(t)) t∈Ω 50*,74*,*,∗7"$9lOT* ; ] ∗. F. hv(t) − v∗ (t), u(t) − u∗ (t)i = hv(t) − v∗ (t), F −1 (v∗ (t)) − F −1 (v(t))i ≤ 0,. 74'. 0≥. Z. hv(t) − v∗ (t), F −1 (v∗ (t)) − F −1 (v(t))i dµ Ωk Z hβ(t) − v(t), α(t) − u(t)i dµ. ≥− Ω\Ωk. f%$SQ%-!6IF4% = 50*,74*,*,74%&"-*,7D)+%$ !6+%$74"-%!4%-*, %$5 0=. Z. hv(t) − v∗ (t), F −1 (v∗ (t)) − F −1 (v(t))i dµ Ω. h*~! SDOb'%Ks479!*,7.* ; 0=. Z Ω. 74*:G 6I9)+%-. F −1. − kv∗ (t)k2 − kv(t)k2 + hv∗ (t), F −1 (v(t))i + hv(t), F −1 (v∗ (t))i dµ Z −(kv(t)k − kv∗ (t)k)2 dµ. ≤. 95U/#9%-. ;<*, #950*I%$)+%$ !O. Ω. 74'. kv(t)k = kv∗ (t)k t∈Ω hv∗ (t), F −1 (v(t))i = M 4%$ %K;<*, % ] M 1974"-% kv(t)k kv∗ (t)k (u, v) = (u∗ , v∗ ) (u, v) ∈ Lp (Ω; X) × 74' 7 %L"-*,74"$#9F4'%-! ] ;^*, #950*I!%$)+%$ !O ] *,4 q 0 L (Ω; X ) (u (t), v∗ (t)) ∈ A(t) t S %$#*,76+ M ∗ (u, v) A. u. / !!"$F#? 95U/Q*, !h 71"$#? ,* ;@5L W}295L #R6I h /4H'%Ks474%-'VSDO 14365
(88) 3>*, . 50%> ,F h S#%v; *,5 * 74' ] ] m Ω R α ∈ R p > 1 !4%=%$ * ; 4 F " 0 ! N 0 5 > % , F h S # % M(Ω, X, p, α, m) A : Ω → M(X × X 0 ) A 74'V!F4"T! ;<*, #950*I!%$)+%$ !O ] ;^*, %$)+%$ !O ] %'&)($*,+-.+/0*. α. GH4%$ %. p−1 + q −1 = 1.
(89) . ÷y÷)>. XP;. M. kξkp p. +. t∈Ω kηkq q. (ξ, η) ∈ A(t). ≤ hη, ξi + m(t),. d w2M [+e. %$5U/PO c:pO*,F76 974%-E1F #99PO ]. ] !4%$7. α>1 M(Ω, X, p, α, m) !4%b6I h / !4*,F#'S %ySQ*,F74'%-'97 ;^*, 0 #950*I!U%$)+%$ ! O ] 97 A(t) X × X0 t ∈ Ω "-*,71! h ,'2"$!*,7.GH9!V!4% 5L W}95L #99lOT* ; M A(t). ÷w. X; 74*,%$5U/lOTA;j 74'V*,7#9O ] !4%$7 −1 ≤ α ≤ 1 M(Ω, X, p, α,m) A; MV4%06I h /2B )I #9F4%-'z;^F74"$!*,7 m≥0 A(t) = (ξ, kξkp−2 F (ξ)) | ξ ∈ X 97 ;<*, &%$)+%$ !O 74' Maip*,7D)+%$ h%$#9O ] F//Q*I% M(Ω, X, p, α, m) α≤1 m≥0 A∈ M 1974"-% 05L W}295L # #950*I!L%$)+%$ !OGH4%$ % ] SDO4%-*, %$5 M(Ω,d X, p, α, m) A(t) w2M %-%aS %$#*>G eK] !4%$ %a !F4" ! M?pO (ξ, η) ∈ A(t) η = −kξkp−2 F (ξ) d w2M [+e-] M
(90) . >. ÷. m(t) ≥ 0. ÷w.
(91) ÷. > jO *,F76 R974%-E1F #99PO ] A; 74' ] !4%$7
(92) −1 m≥0 M(Ω, X, p, α, m) 6I h /2B )I #9F4%-'Uα;^F≤ !4%%$ * ; 50%> ,!F h S#% 74"$!*,74>M@4%"-*,74'29!*,7 74%-"KB m≥0 %-hh !OQcX ; ] !4%$7 ] GH"z50%> 74 ! 97v! "> ,% ] 71O m(t) < 0 (0, 0) 6∈ A(t) 6I h /VGH"."-*,71h 974 H74*,97 M. ÷. (0, 0). M(Ω, X, p, α, m). X; 74' =74%$)+%$ %$5U/PO ] !4%$7 α < −1 X 6= {0} M(Ω, X, p, α, m) 974"-% 9N #9Gp -O2@"-*,7Dh 974@!4%j6I h /2B ), #9F4%-';^F74"$!*,7 ] A(t) = {(ξ, η¯(t)) | ξ ∈ X} GH4%$ % = ! 4 F " T ! M 0 q
(93) . . >. η¯(t) ∈ X. −(α + 1)k¯ η (t)k /q ≥ m(t). /0C / C. 1F365SP43=V W > α > 0
(94) m ∈ L1 (Ω) ] A ∈ M(Ω, X, p, α, m) c S > A [7.9
(95)
(96) 8
(97) 8
(98) f
(99) c
(100) Z `ab" 6 > p. ÷. A. D(A) = L (Ω; X). "> 7 SQ%UGp%> {+%$74%-' * ]
(101) > >4*, 3!4%05L W}95L #99lO * ; α>0 α > −1 d *,/Q*I!9!*,7 w2M`w4mSQ%$#*:G e M 4%z"-*,74A V*,/!95L # ;^*, *S % '29!*,7 α > 0 A %$)+%$ !OGH4%$ %('%Ks474%-'q 74'L*,71*MNXP74'%-%-' ] #%$ Mip*,74'29!*,7 A(t) = {(0, η) | η ∈ X} w2M [ h !ts%-' GH9! 74' M 195U9#? !#9O ] GH9! ] 74' m = 0 α = 0 D(A) = {0} H74*,=!F !%-"$!9)+%IM ]. A(t) = {(ξ, 0) | ξ ∈ X} A ;"] >. fR%$. u ∈ Lp (Ω; X). 74'a#%$;^*, . λ>0 A(t). pO0'%Ks479!*,7 ] . α. (uλ (t), vλ (t)) = (Jλ (uλ (t), vλ (t)) ∈ A(t). ]. u(t), A(t)λ v(t)).. ;<*, p #950*I%$)+%$ !O. t∈Ω. M p O d w2M [+e$] *,74%( ,. ku (t)kp kv (t)kq λ λ + p q ≤ hvλ (t), uλ (t)i + m(t) = −λkvλ k2 + hvλ (t), u(t)i + m(t) ≤ hvλ (t), u(t)i + m(t).. (%$74"-% ] !4%$ % . !F4"V! H;^*, = #950*I=%$)+%$ !O. C>0. t∈Ω. ]. α kuλ (t)kp kvλ (t)kq + ≤ Cku(t)kp + m(t). 2 p q. d w2M e. lX 7Z/ !!"$F#? ] M (74*, %$5U/POZ4%$74"-% (uλ , vλ ) ∈ Lp (Ω; X) × Lq (Ω; X 0 ) A 5L W}295L #RS1O *,/Q*I!9!*,7Zw2M 2M XP71%$6I h !976 d w2M e$] 6I9)+%-. ÷. 4%$ %K;<*, %. α 2q. Z. q. kvλ k dµ ≤ C. Ω. lim. Z. λ→0 Ω. Z. p. kuk dµ +. Z. m dµ.. Ω. Ω. kλvλ kq dµ = 0,. 74'TF/a*L ~!FS4%-E1F4%$74"-% ] ;<*, = #950*I!=%$)+%$ !O. t∈Ω. ]. uλ (t) − u(t) = λF −1 (vλ (t)) → 0,. ÷.
(102) , MN*,6+%$!4%$ GH9! d w2M eK] fR%$S %-!6IF4% ='*,5U97 %-'V"-*,7D)+%$ !6+%$74"-%34%-*, %$5 λ→0 95U/#9%- 97 M<1974"-%=!4%(%- E1F4%$74"-% jSQ*,F74'%-'y97 q ] uλ → u Lp (Ω; X) vλ L (Ω; X 0 ) 9pjGj%> {#9Oq"-*,5U/ ,"$ ] 74'y!4%$ %= 4 F " b ! j G > % { 9 # 0 O 9 7 v ∈ Lq (Ω; X 0 ) v *v M < 1974"-% L 5 W } 9 L 5 # R f $ % 5U5L 0w2M = //#9"> S#% ] 74λ' M ] q 0 L (Ω; X ) A (u, v) ∈ A x*,%3! −1 h !!s%-!4% h 50% ,F5U/!*,74( , A = {(v, u) | (u, v) ∈ A} d GH9! 74' %$)+%$ %-' e M pO !4%b/ ** ; S *>)+% ] ] GH"h A p q D(A−1 ) = X 0 %-E1F9)I #%$7DGH9!T!4%3!F t%-"$!9)9POa* ; M A. 9
(103) / . )+*. /:9. 329. /. 9. 3 1 6. 9. ;6-/:9. ;. 9)+%$7 ;^F74"$!*,74 n : Ω → M(X × X 0 ) 74' !4%$9 ~"> 74*,7"> #H%K}%$74*,74 A, A d S1O d w2M + eeK] Gj%q"-*,74!'%$ !4%LE1F4%-!!*,7 GH4%$!4%$ !4%0/ *,97DlGH%q"-*,71)+%$ tB A, An 6+%$74"-% n 95U/#9%-N!4%="-*,7D)+%$ !6+%$74"-%=* ; !4%974'2F4"-%-'U6I h /4 n M . A A. A (t) A(t). 1F365 "F3=V W. ?A@ &B/C &BD. ( *). ]W \ W
(104) ^. ( +). t ∈ Ω c An (t) A(t) n → ∞ c. A
(105) An
(106) f\"9
(107) 7
(108) 0
(109) 4c. ( +) . > A, An : Ω → M(X × X 0 ) X A_
(110) ; X . (αn , βn ) ∈ An
(111) (α, β) ∈ Lp (Ω; X) × Lq (Ω; X 0 ) bW_ h.
(112) ^ (αn , βn ) → (α, β) 6
(113) Lp (Ω; X) × Lq (Ω; X 0 )
(114) n → ∞ c Sf 9
(115) . An A. ÷>[. >. . u !F5U/!*,7 d 99 e "> 774*, SQ%q'2 *,//Q%-'gM 74%L"> 7 -)+% n A (t) ;^*, Z%$)+%$ !O GH9#% M >4*, Z%K} 5U/#% ] #%$ ] 74' n A(t) t ∈ Ω A 6 A X = R. #%$ M (%Ks474% 4 7 ' Ω = (0, 1) An (t) = (x, y) | y = n1/q χ(0,1/n) (t) A(t) = M 74% , n ;^*, %$)+%$ !O M 7T!4%&*,!4%$ 74' {(x, y) | y = 0} A (t) A(t) t∈Ω ;<*, %$)+%$ !O 74' 74' ] *,74% , (un , vn ) ∈ An (u, v) ∈ A kvn kLq = 1 kvkLq = 0 h*b! "> 7z74*,3"-*,71)+%$ !6+%U* 97 , H G 4 $ % 4 7 "-% ] vn v Lq (Ω) n→∞ An A 95U/ *I!9S#%IM
(116) . >. . ;"] >. . fR%$. M 4 > *, . ] #%$. (u, v) ∈ A t ∈ Ω w(t) = u(t) + F −1 (v(t)) M %Ks474% A(t) (u(t), v(t)) = (J1 (w(t)), A(t)1 (w(t)) An (t). Ω0 = {t ∈ Ω | kJ1. 74' ] ;^*, . n≥1. w(t)k ≤ 2ku(t)k. 74'. ] *z! . kAn (t)1 w(t)k ≤ 2kv(t)k},. ]. (un (t), vn (t)) =. (. An (t). (J1 (w(t)), An (t)1 (w(t))) (αn (t), βn (t)). A; t ∈ Ω0 , A ; 0M t 6∈ Ω. pOZ"-*,74!! !F4"$!*,7 ] 97 MpOv'*,5U97 %-'z"-*,7D)+%$ !6+%$74"-% ] (u , v ) An (un , vn ) → !! *,76I#9Oy97 pn n q 0 M . (u, v). L (Ω; X) × L (Ω; X ). ÷.
(117) . 1F35 0 $ 3. /0C / C. _ α > 0 c mn ∈ L1 (Ω) c An ∈ M(Ω, X, p, α, mn )
(118) ^ > A : Ω → M(X ×X 0 ) f ; X ] mn bf
(119) W 6
(120) m L1 (Ω)
(121) ^ Q ] ]W
(122) W t Ω c An (t) A(t) h c SW A ∈ M(Ω, X, p, α, m) > n . A A. ÷. Uh !!s%-'g= M pOmip*, *,#9#? !O z%a //#9O 4%-*, %$5 w2M >[ M =O/Q*,!4%- (i) w2M 2] n 35L W}295L #50*,74*,*,74%IM ,=/ *Z .!FS4%-E1F4%$74"-% d 74*, %$#? S %$#%-' eK] A mn "-*,71)+%$ !6+% * 2M %IM 74'b %$5L 974='*,5U97 %-'a97 1 M+%$74"-% ] n ] m L (Ω) A (t) A(t) ;<*, = 2M %IM 95U/#9%-H! * H5L W}295L # M t∈Ω A ∈ M(Ω, X, p, α, m) A z% 74*>G "h4%-"{a1O1/ *,!4%-! M (%Ks474% ;"] >. ÷. (iii). An (t). (un (t), vn (t)) = (J1. 0, An (t)1 0).. ij#%> !#9O ] ;<*, 2M %IM 74' (u (t), v (t)) ∈ An (t) t∈Ω hvn (t), un (t)i = −kun (t)k2 ≤ Mjip*,5S974n%-'TGH9!n d w2M [+e-] !HO1%$#'. 0. α. ku (t)kp kv (t)kq n n ≤ mn (t). + p q. 4%$ %K;<*, % ] ;<*, %$)+%$ !O 74' (un , vn ) ∈ Lp (Ω; X) × Lq (Ω; X 0 ) n≥0 (un , vn ) ∈ M pOb4%-*, %$5 2 w ` M 0 _ / / 9 # % ' < ; , * 9 # 0 5 I * ! % $ + ) $ % ! O * ] ] ] n n. A. t∈Ω. (un (t), vn (t)) → (u(t), v(t)),. !! *,76I#9Ob97. A (t). X × X 0,. GH4%$ % M x(*>G fR%$SQ%-6IF4% y'*,5U97 %-' "-*,71)+%$ tB A(t) 0, A(t)1 0) (u(t), v(t)) = (J 6+%$74"-%4%-*, %$5 95U/#9%-!1 ;<*, 7DO0 %-E1F4%$74" ] ! 4%$ %j !FS4%-E1F4%$74"-% nk → ∞ !F4"hT! . n kj. (unkj , vnkj ) → (u, v). !! *,76I#9Ob97. Lp (Ω; X) × Lq (Ω; X 0 ).. 4%& !6IF50%$71 S *>)+% 95U/#9%-H!4% "-*,71)+%$ !6+%$74"-%&* ; !4% GH4*,#% %-E1F4%$74"-% (un , vn ) * M (u, v). )+*. . :6 9?1. 329. . . 6 1 ?3 6-9 3. 4%-*, %$5 w2M`wq 0/ !!"$F#? &"> ,h%* ;p L50*, % 6+%$74%$ h #@ %-!F#9 Y ,\ c q50*,74* B F *,74%a"-*,71!97DF4*,F4y"-*1%$ "$9)+%b5L / ] 74'm!4%$ %bUF7E1F4%$74%-~GH4%$7 U!! !"$!#9O F 50*,74*,*,74% ] M %IM GH4%$7 MXP7v)%$G * ; hF (ξ ) − F (ξ ), ξ − ξ i > 0 ξ 6= ξ ∈ X !4%&/ *1* ;N* ; *,/Q*I!91!*,7 M`w ] 2Gj%&1"-*,742'%$ '%Ks474%-'T1;^*, 2 S1O ξ∈X. Fp. Fp (ξ) = F (ξ)kξkp−2 , ∀ξ ∈ X.. Xlh !ts%- 4%="> ,%H* ; 4%-*, %$5 %. hFp (ξ), ξi = kξkp = kFp (ξ)kq .. w2M`w "-*, ! %-!/ *,74' *. ÷. . p=2. M p *>G'%$ %-!F#9 97U! "> ,h% .
(123) \C / d/ .+-.+/0*. 14365 H;J C /hLM &BCON
(124) R T"3=V W > A ⊂ X × X 0 X K 7 0
(125) 8 ^ Y ] `"ba 6 [0c A Z =7.9
(126)
(127) 0 8= ]
(128) A + Fp A > > > ]W A ^ bg h: 0 ce \. η∈X. ξ∈X. 195U9#? !#9Oy*0!4% "> ,% . p=2. ] !4% ;<*,#9#*>GH976y'%Ks479!*,7V5L {+%-%$74h%Ic. 14365 F 3 fR%$. %'&)($*,+-.+/0*. (ξ, η − Fp (ξ)) ∈ A. S %H (5L W}95L #50*,74*,*,74%H6I h /RM 4 > *, %$)+%$ !O. A ⊂ X ×X 0 74' '%$74*,%-!4% F7E1F4% / 9 97 ! F4"T! ] A ξ∈X λ > 0 (Jλ,p ξ, Aλ,p ξ) A A Fp (ξ − Jλ,p ξ) = λAλ,p ξ.. ÷. 9!b!4%(/ %$)1*,F4'%Ks479!*,7V 74'b4%-*, %$5 w2M ] #9#Q!4% %-F#9H* ; %-"$!*,7 w 97Z!4%"> ,%* ;!4% L '2F #99POT5L //976b4*,#'ZGH9!v*,SD)*,F450*'2As"> !*,74 97 Fp !4%&/ *1* ;<>M 4% F4% * ; O%$#'H95U/ *>)+%$50%$71 * ; ip*, *,#9#? !%- w2M 74'Vw2M c. ÷. Fp. 1F3Z1 F3 Y Se W SS
(129) W =;]
(130) ^ ( W ; ] \ S[ ^ S. . \C / d/ .+-.+/0*. . α > −1. > c,
(131) )>. . ÷. >
(132) f.
(133)
(134) ^. α>0. HHS ,h%-'T*,7a!4% ; ,"$H! A;. ;"] >. A(t). (u(t), v(t)) = (J1,p 0, A(t)1,p 0),. *,74%& ,. v(t) = −Fp (u(t)),. 74'TF74'%$ DO/Q*,!4%- d w2M [+eK] !H6I9)+%-. ku(t)kp kv(t)kq ku(t)kp kv(t)kq + = −hv(t), u(t)i ≤ m(t) − α + . p q p q. ÷>. jO *,F76 R974%-E1F #99PO ] $% )+%$ !O 6I h / h !ts%- d w2M [+e GH9! M A α = −1 *,6+%$!4%$ GH9! =%$L 5 !{2 q 74' >[ !50%> 74(! =!4%"-*,74'29!*,7 α > −1 *,/!95L # M
(135) .
(136). '. %. . ÷. @Qh . ÷.
(137) - h ". 4%&/Q%$ !*'2" F72;^*,#'2976q*,/Q%$ h *, G ,H97D! *'2F4"-%-'TS1OVij*, h 74%-"$F ] 5U#? 5U? 7 74' !h*yY`_ \ M@z% %-"> #9# !4% '%Ks479!*,74= 74'y/ *,/ %$ !!%-* ;!H*,/ %$ h *, >M 4% / ** ;<"> 7aSQ%(;<*,F74'a97zY`_ ] I] >[W\ M. ÷y÷ ÷. +*-,. 9<6 1768/:9 / 1 4 176 ;. 68/. . <6. . /<6-9 / 73514/ 9. 329 . 7/ . lX 7 SQ%& ~ %K;<%$ %$74"-%&"-%$#9# d %IM 64M ] #%$ ] *, H50*, % 6+%$74%$ h #9#9OT ~%$ -)976 RN Y ]0, 1[N !4%=/ -)976/ *,/ %$ !POLGH9!q %-!/ %-"$* S ,! '%Ks47976 !4%/Q%$ !*' e M (b1 , ..., bN ) >*, ' $ % 4 7 , * %. ! 4 ~ % F 7 1 E 4 F U % 9 D 7 % $ + 6 $ % 3 " , * 5 S 9 7 !*,7 PN ] ] GH9! y ∈ RN [y]Y kj bj j=1 S %$#*,76+H* 74'a'%Ks474% ] * ;@!4% /Q%$ !*'!F4"ha! . kj ∈ Z. y − [y]Y. {y}Y = y − [y]Y ∈ Y.. ÷>[. Y.
(138) 3503HfR%$. %'&)($*,+-.+/0*. 5L /. u : RN → S. S % 3 %K;<%$ %$74"-%&"-%$#9# ] / *I!9!9)+%(7DF5SQ%$ ] ~%$ 74' Y S M 4%K .]W Q
(139) Y '%Ks474%-'TSDO T. TY (u) : RN × RN → S (x, y) 7→ TY (u)(x, y) = u(. 74% %> ,'29#9Oa%-%-H! ;<*, =%$)+%$ !O. x ∈ RN. ]. x + y). Y. TY (u)(x, {x/}) = u(x).. Z*, %-*>)+%$ ] . TY (u). 971), !? 71=F74'%$ H!4%(;<*,#9#*>GH976q ,"$!*,7T* ;. ZN. c;<*, . ]. k ∈ ZN. TY (u)(x + k, y − k) = TY (u)(x, y).. XP;. u : RN → S. 74'. f : S → S0. ] !4%$7. TY (f ◦ u) = f ◦ TY (u).. Xl7~/ !!"$F#? NA; 74' ] !4%/ %-"-%-'2976&/ *,/ %$ !PO~ //#9%-' u : RN → S v : RN → T *~!4%&/ * t%-"$!*,74 74' O%$#' P : (u, v) 7→ u. Q : (u, v) 7→ v. TY ((u, v)) = (TY (u), TY (v)).. 4%$ %K;<*, % ] A;. F :S×T →R. ]. ÷. d 2M :e. TY (F (u, v)) = F (TY (u), TY (v)).. ,(%K;^F#/ !!"$F#? U"> ,%-U %qGH4%$7 74' 74' ] S = R T = R F : (s, t) → st GH4%$7 4 7 ' H ! 4. % ' , * / * 2 ' 4 F $ " >M ] N N S=R. \C / d/ .+-.+/0*. T =R. F3Z143. Z. RN. . F. u ∈ L1 (RN ) c SW TY (u) ∈ L1 (Rn × Y )
(140) Z 1 u(x) dx = T Y (u)(x, y) dx dy. |Y | RN ×Y . ].
(141) S
(142) W 6 b B _ ce ] 1 ≤ p < +∞
(143) u ∈ Lp (RN ) c TY (u) ∈ Lp (RN × Y ) c . ÷. kTY (u)kLp (RN ×Y ) = |Y |1/p kukLp (RN ) .. lX 7!4%yh%-EDF4%$# ] V;^F74"$!*,7GH"hm '%Ks474%-'*,7 Z%$ * ; N ] A R "> 7ZS %3)%$Gj%-'| , U;^F74"$!*,7|'%Ks474%-'Z*,7 ] A; Gj%~"-*,74!'%$ (9 %K}%$74!*,7ZSDO RN *,F!'%3* ; M 0 A 4%&"-*,53S97 !*,7v* ; *,/Q*I!9!*,7Z2M`w~*,6+%$!4%$ GH9! d 2M :e O%$#'>c
(144) . . >. ÷. ÷.
(145) \C / d/ .+-.+/0*. . d8 . 3F3,V W. . >. A ⊂ RN X _
(146) ; X ] u ∈ L1 (A) cF S TY (χA )TY (u) RN × RN c TY (χA )TY (u) ∈ L1 (RN × Y ) c
(147) Z Z 1 u(x) dx = TY (χA )TY (u) dx dy. |Y | N R ×Y A . f
(148) c7 ]
(149) 1 ≤ p < +∞ u ∈ Lp (A) cK S TY (χA )TY (u) Z d8
(150) RN × RN c TY (χA )TY (u) ∈ Lp (RN × Y )
(151) . . . kTY (χA )TY (u)kLp (RN ×Y ) = |Y |1/p kukLp (A) . 1974"-%&!4%&F72;<*,#'2976q*,/ %$ h *, ( , U#*"> # ,"$!*,7 ] 9==7 !F h #*L%K}4 5U974% 9=%
(152) %-"$=*,7a#*"> #9#9Ob!F5U5L S#% ;^F74"$!*,74>M. \C / d/ .+-.+/0*.
(153) e]
(154) .
(155) O ^ 1 ≤ p < ∞ c TY > p p N Lloc (R ) Lloc (RN × RN ). z% !F !7T74*>G ?A@ &B/C &BD . . F3
(156) 3. *~!4%. 3GPF3 V W. Lploc (RN ) S. Lploc. #*,S #"-*,7D)+%$ !6+%$74"-%-=;<*,#9#*>G. ?A@ &B/C &BD ] . u → u. 8
(157) g
(158) . "-*,7D)+%$ !6+%$74"-%/ *,/Q%$ !!%-;^*, . > (u ) , u Lploc (RN ) c 1 ≤ p < +∞. TY (u ) → u ⊗ 1 . . W 6;
(159) ^. . bf
(160) 6 . 1 ≤ p < +∞ . ]. M.
(161) ^ u → u 6
(162) . > Lploc (RN × RN )
(163) → 0. %> ,!9#9O+M. 3#"F3dV W A ⊂ RN X _
(164) ; X
(165) ^ W 6;
(166) Lp (RN ) ce . > (u ) , u Lp (RN ) c 1 ≤ p < +∞.
(167) ^ TY (χA )TY (u ) → (χA u) ⊗ 1 6
(168) Lp (RN × Y ) → 0,
(169) ^ TY (u )|A×Y → u ⊗ 1 6
(170) Lp (A × Y ) → 0.. 4%T;<*,#9#*>GH976 %-!F#9b!h %-L! y!4%T#995U9 d A;&9b%K}! e * ; 7 F72;<*,#'%-' h%-EDF4%$74"-% / %$ !*'2"IM e&BD. ( . D. 3%$03 V W u p N ) > ]. Lloc (R ). ( ∈ L1loc (RN ) Lploc (RN )) u ˆ ∈ L1loc (RN × RN ). ^ ^ TY (u ) * u ˆ ∗ M(RN × RN ), ( f f Lploc (RN )). . . M(RN ×RN ) 0: S. . _ f= S:b c$ S. > u ˆ Z Y Q b. x %K}2 ] Gj%L %-"> #9# !4%U/ *,/ %$ !!%-3* ;!4%~F72;^*,#'2976Z*,/Q%$ h *, //#9%-'z*,7z!4% 6I h ,'2%$71Z* ;0*,50%|; F74"$!*,74>M XP; 1,p N !4%$7 SDO *,/Q*I!9!*,7 2M ] u ∈ Wloc (R ). ÷ .
(171) 74'. M Z *, %-*>)+%$ ] ;<*, . TY (u) ∈ Lploc (RN × RN ) TY (∇u) ∈ Lploc (RN × RN ) %$)+%$ !Oy%-!H;^F74"$!*,7 ϕ ∈ D(RN × RN ) Z Z Y ∇y ϕ T (u) dx dy = ∇y ϕ(x, y) u([x/]Y + y) dx dy RN ×RN RN ×RN Z =− ϕ(x, y) ∇u([x/]Y + y) dx dy RN ×RN Z ϕ TY (∇u) dx dy, =− RN ×RN. 4%$ %K;<*, %. TY (u). HGp%> {1#9Oa'2
(172) g%$ %$7D!? S#%GH9!T %-!/ %-"$H*. y. ] 74'. TY (∇u) = ∇y (TY (u)).. d 2M`w e. 4%b;^*,#9#*:GH976 %-!F#9L6I9)+%-L v %$#? !*,7oS %$PGp%-%$7o!4%a#995U9q* ; 7 F72;<*,#'%-' h%-EDF4%$74"-%3 74'b!4%ϣU9=* ;!4%3%-EDF4%$74"-%Ic F3F3eV W a (u ) X " & 0b ] Lploc (RN )
(173) u ∈ Lploc (RN ) c ^ > _ h. u ˆ ∈ Lploc (RN ×RN ) u * u Lploc (RN )
(174) TY u * u ˆ ^ f Lploc (RN × RN ) ce \C / d/ .+-.+/0*. 1 u(x) = |Y |. Z. u ˆ(x, y) dy.. Y. 4% ;^*,#9#*:GH976L/ *,/ *I!9!*,7T= 7a95U/ *, !h 7D*1*,# ;<*, H!4%&%-E1F4%$# M 3 F3,V W = W > 1,p (u ) X = a'& W0b ] Wloc (RN )
(175) u ˆ ∈ Lploc (RN ; RN ) ] (u ) X Lploc (RN ) c (∇u ) X (Lploc (RN ))N
(176) ^ TY (u ) * u ˆ Lploc (RN × RN ) → 0,. \C / d/ .+-.+/0* . ^ > TY (∇u ) * ∇y u ˆ f Lploc (RN × RN ) → 0. > u ˆ Y G6 6b y.
(177) f
(178) . 4% ;^*,#9#*:GH976L!4%-*, %$5 365F3AV W. ?A@ &B/C &BD. !4% 5L 97T %-!F#9>M. 1,p (u ) X a'& W0b ;] Wloc (RN ) Wb I h c u * u. >7Y Scd SfK9
(179) \ X a & 1,p 8b ( 0 df X )'
(180) Wloc (RN ) 1,p ] 0ba 6
(181) F]W bf 0b=h Q u ˆ Lploc (RN ; Wloc (RN )) b S S . . f ^. f ^. . TY (∇u ) * ∇u ⊗ 1 + ∇y u ˆ,. . > Lploc (RN × RN ) → 0. 4%$7. . ]. : 6 66 0 ^ c. Ω ( RN A : Ω → M(X × X 0 ). > u ˆ Y G6 6b. HF72;<*,#'%-'V ,;<*,#9#*>G>c. 74' %K}2%$74'%-' ] s4 A : Ω → M(X × X 0 ) A S1O M N%K}2%$74!*,7~N!!9#9#'%$74*,%-'~SDO M@4%pF72;<*,#'%-'~6I h / kξkp−2 ξ A TY (A) 74*>G '%Ks474%-'.*,7 M N N %'&)($*,+-.+/0*. 36553pfR%$. d 2M` e. Ω ( RN. R ×R. ÷_.
(182) 3 6-9 / 73514/ . +* ). %'&)($*,+-.+/0* F365143fR%$ S % 3 %K;<%$ %$74"-% "-%$#9#g 74' 97 ] ] >0 Y u L1loc (RN × RN ) !4% -)+%$ h 6I976q*,/ %$ h *, Y '%Ks474%-'V ,;<*,#9#*>G. U. UY (u)(x) = \C / d/ .+-.+/0*. F365 3. \C / d/ .+-.+/0*. F365
(183) 3. W 6;
(184) ^. 1 |Y |. .
(185) A
(186) ^. ]. Z. Y. x x + z, dy. u Y Y. u ∈ L1 (RN ) c
(187) 8 h UY TY (u) (x) = u(x).. 1 ≤ p < ∞ w ∈ Lploc (RN × RN ) TY UY (w) → w. > Lploc (RN × RN ) → 0. . z%T!F !7 4 7 *:G M. *|!4%. Lp. #*"> #9#9O"-*,7D)+%$ !6+%$74"-%v%-EDF9)I #%$74"-%T/ *,/ %$ !!%-L;^*, . 1 ≤ p < +∞. 63 5SP43[V W ?A@ &B/C &BD (u ) Lploc (RN )
(188) u ˜ ∈ Lploc (RN × RN ) c 1 ≤ p < > Y ]a 6
(189) bf
(190) 0b 7
(191) fK & W +∞ . +). TY (u ) → u ˜. +) . u − UY (˜ u) → 0. . " N H . W 6;
(192)
(193) ^. . 6
(194) ^. . $+(
(195) t. Lploc (RN × RN ) → 0,. > Lploc (RN ) → 0. (. Xl7v! %-"$!*,7|Gj%Uh %U*,F 5L 97|4*,50*,6+%$798: !*,7 %-!F#9>M>9 ! Gj%~6I9)+%~!4% %-F#9~ 74' !4%L/ ** ;* ;!4%b"-*,7D)+%$ !6+%$74"-%IM|4%$7Gj%b!!F4'2O|!4%y/ *,/ %$ !!%-~* ; !4%T4*,50*,6+%$798>%-'o*,/ %$ h *, >M >97 #9#9O Gp%a6I9)+%T*,50%T%$74%$ !6IOm"-*,7D)+%$ !6+%$74"-%v 74' "-*, ! %-"$*, %-!F#9>M. 9
(196) ; 817;. *-,. /:9. ?A@ &B/C &BD W X . Ω > >0.
(197) 3653. _ 1 < p < ∞ c p−1 + q −1 = 1 c m L1 (Ω) c α > 0
(198) ^ > V a ;] X aW RN A ∈ M(Ω, RN , p, α, m ) ]a
(199) . . _g h [ Sf9
(200) g 1 N N
(201) . . M(R × R ). b . m ∈ L (Ω × Y ). . Y ⊂ RN cA _ X B : Ω × Y → b h h]a
(202) 7
(203) ^ c. (x, y) ∈ Ω × Y. TY (A )(x, y) B(x, y) .
(204) ^ > T (m ) fb
Documents relatifs
Pardoux : Homogenization of a linear and semilinear Second Order Parabolic PDEs with Periodic Coefficients: A Probabilistic Approach, Journal of Functional Analysis, 167, 1999,
In this paper we study homogenization for a class of monotone systems of first-order time-dependent periodic Hamilton-Jacobi equations.. We characterize the Hamiltonians of the
Browder has constructed in [3] a degree function for maps of class (S)+, demicontinuous from el(G) to X*, which is normalized by J and reduces to the Leray-Schauder
If we assume that the initial data u 0 is concentrating at the bottom of a higher level Bloch band (see Section 7 for a precise statement), we obtain a convergence result similar
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des
— We establish homogenization results for both linear and semilinear partial differential equations of parabolic type, when the linear second order PDE operator satisfies
Any contraction in Hilbert space can be interpolated at a .finite number of given points by means of a piecewise unitary
To prove the comparison principle we first prove comparison in the case where either the subsolution or the supersolution is bounded Lipschitz continuous and then we proceed
