The topological asymptotic analysis for the Maxwell equations and applications

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(1)The topological asymptotic analysis for the Maxwell equations and applications Bessem Samet. To cite this version: Bessem Samet. The topological asymptotic analysis for the Maxwell equations and applications. Mathematics [math]. Université Paul Sabatier - Toulouse III, 2004. English. �tel-00005827�. HAL Id: tel-00005827 https://tel.archives-ouvertes.fr/tel-00005827 Submitted on 8 Apr 2004. 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..

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(25) 132*546 8792:4;=< D E3GF3 IHJ=K$L6'MONQP[. >?A@B?C>. ;È 8]@_5 Ω <*3y:*8]VcN]@_3*Qceg80693*}d:*Q R N = 2 8]< 3DQ?5:*Qd¾¤6A803)5A@Bw!69Q Γ N]bAb9Q!Y 69}!0<=UB@Bw?6AQP/J 5`N]3)5D:*803*3*} h ∈ H (Γ) p3*80<=bDb9<*n*ng80bA803=bDKM<hi@BULQ¿ =@_b95AQx<*3[<*3*@_KM<*Q 0}?6A@ ON]3)5 u ∈ H (Ω) :*N]3*b Ω, .]/~ ∆u + k u = 0 A b = < 6 ∂ u =h Γ, 8 ) ∂ :*}!b9@B03=QDUBNX:*}?6A@10}!QE3*8069VcNPUBQ]/*N$>!8]3*:*@_5A@B803N]< UB@BVX@_5AQ!b|b9<*6UBQFe8]6A:Qa 57}¿Ã 69@BQ?<*6º3hi@B3)57Q?69@BQ?3M5nONPb:*N]3*bU h~N]3ON]U1bAQFN]b;VXn=5A8]57@_K<=Q]/ È;8]@_5 ω <*3j:*80VxN]@_3*Qeg806A3=}{:*Q R >!803)5AQ!3ON]3)5U hi8069@B0@_3*QTQa5 x ∈ Ω /Otu8]<*6|5780<;5 NPbAbAQ?YDngQ?5A@_5!p*3*80<*bng80bA803=b ω = x + εω Qa5 Ω = Ω\ω /oNXng80bAb9@Be*@_UB@157}:=Q ε>0 ¾¨NP@B6AQ$<*3[>`WON]3=0Q!VXQ!3)5:=Qx>?8;8069:*803*3=}!Q!bE3*80<=b¸ngQ!6AVXQ?5E:*QXbA<=n*n8]bAQ!6DKM<*Q x = 0 / Â80<*bng80bA803=b u ∈ H (Ω ) UNbA80U_<=57@_803v:*<n*6A8]e*UBw?VxQ{ngQ!657<*69e}  :ON]3=b Ω , =0  ∆u + k u bA<=6 ∂ω , .]/ * u =0 bA<*6 Γ.  ∂ u =h t80<*6ºb9@BVXn*UB@ OQ!6©UNDn*6A}?bAQ!3)57NP57@_803j:ON]3*b|>?Q>`WON]n*@15769Q{@B3)57698;:=<*>?5A@_¾ pMUBQ?b|Q!b9nON]>!Q?bº¾¤803*>aÃ 5A@B803=3*Q!U_bº>!803*b9@B:*}?6A}?bb9803)55780<*b:*Q?b RÃ4Q?bAnONP>!Q!b0Q?>?5A806A@_Q!UBb?/ N. − 21. 1. Ω. 2. Ω. Ω. n Ω. n. N. 0. ε. 0. ε. ε. 0. Ωε. 1. ε. Ωε. Ωε. n Ωε. 2. Ωε. ε. ε.

(26) . !#";&%')(*'O+, Â80<*b>?803*bA@_:*}!69803*bF<*3*QXN]n=n*UB@_>NP5A@B803 J :*@_}!69Q!3)57@BN]e*UBQD:*Q H (O) :ON]3*b R p8*) ! Q b5<=3*QFnON]695A@BQ{]80@Bb9@B3*Q:*Q Γ /OÂ80<*bn8]bA803*b O .0/~ j(ε) = J(u ), ∀ε ≥ 0. tu80<*6bA@_Vxn=UB@ *Q!6!p!3*80<*bu}!>!69@_]803*b j(ε) = J(u ) /Lh~W)n8P57W*w?bAQ|¾¨N]@157Q©bA<*6UN¾¤803*>a57@B8]3 >!8*+=5 J b!hiQa ;n*U_@BKM<*Q$nON]6FU_Q¾¨N]@_5FKM<*QX:ON]3*bFU_Q!bFn*6A8]e*UBw?VxQ?b}!UBQ?>?5A6A80VxN]03*}a57@_K<=Q!bFUBQ?b >!6A@157w?6AQ!bD57Q!U_UBQxK<=QvUNSVxNP5769@B>!Q S bA803)5$:*} O3*@Bbb9<*6UBQ?bN]>?>!}!b:h~<*3 0<=@B:*Qd:h~803=:*Q 80< $U hi@B3"O3*@ng80<*6UBQ?b8]e*b957N]>!U_Q!be80693*}!b?/g^{N]3*bUBQF>`WON]n=@_5769QDXnON]6Q¿ =Q?Vxn=UBQ]po3*8]<*b N&]803*b}?5A<*:*@_}E<=3*QFUN]690Q>!UBN]bAb9QE:=QF¾¤8]3*>?5A@B803*bº>?*8 +=5Ab!/ Â8]5769Qde*<=5$Q?b95$:hi}?5A<*:*@_Q!6U_Qd>!80VXng80695AQ!VXQ!3)5N]bVxn=5A8]57@_KM<*Qd:*QvUBN :*@_}!69Q!3*>?Q U_806Ab9KM<*Q ε 5AQ!3*:d]Q!69b 0 /;ÂT80<*b\<=57@_UB@_bA803*b©:*Q?<; c80<=57@_UBb½VcN57W*}?VcNP5A@BKM<*Q?bZ j(ε) − j(0) UNVX}?57W=8;:*QFNP:µ980@B3)5AQ0}!3*}?67N]U_@Bb9}!QQ?5UBN¸576A8]3*>NP5A<*6AQF:*QF:*8]VcN]@_3*Q]/ K F3Q N EP F[ F3 K6 E >I?A@B?G@ ½'« ª )¡"f'O " !¡g¦ ¥

(27) M%'O£¨¦ M Â80<*b67N]n=nQ?UB803*bKM<*Q½>!Qa5A57Q½VX}?57W=8;:*Q½N}?5A}\@_3M5A6A8:*<*@157QLnON]6 «/ NPbAVX80<*:*@):ONP3*b k~q0r /=ÂT80<=b|UNE0}!3*}?67N]U_@Bb9803*bm:*N]3*bk~lmWON]noHp*lmWONPnOqPrNP<d>N]bm:h~<=3v>`WON]VXnv>!80VXn*UBQ¿ ;Q Q?5:hi<*3 n=6A80e*U_w!VXQ{3*803>!8Q!69>!@1¾ / È80@_5 V <*3 Q?bAnON]>?Qx:=Q$RT@BU_eQ?695 = ;Q]p ` <*3*Q¾¤806AVXQU_@B3*}!N]@B69QD>?803)57@B3M<*Q$b9<*6 V Q?5 Q!b5|<*3*Q¾¨N]VX@BU_UBQm:=Q¾¤8069VxQ?b\e=@BUB@_3*}NP@B6AQ?b >!8]3M5A@B3M<*Q?bmbA<=6 V /Â80<*b½bA<*n*ng80b9803*b (a ) K<=QEU h~W)ng8]57W*w?bAQEbA<*@1PN]3)57QFbA80@15ºb7N57@Bb¾¨N]@_5AQ]/ ®¢L¡O!¯ ε ≥ 0 "!$# % %'&)( %'*! a !,+-&.% "(-"!-&. 1. Ωε |O. Ωε. ε ε≥0. & ! %'&>= (@?.!-&>= &.@!4= ! ε @ ! !CB .! / 0!- / %'12!435% !768%9+-!:&)! / &;+- . ' α>0 < εA. D. +-. aε (u, u) ≥ αkuk2 ,. !768%9+-!. ∀u ∈ V.. &)! "!E# % %'&)( %'*! / &.F%'&..!C+- !-G&)! & / F%H& = (JIK&.%H! δa V f (ε) ! ! , 4 +. B. ! R∗+ '. .0/~G. lim+ f (ε) = 0,. ε→0. L. .0/I*. L2 (V). = (,+-%M2&)!. kaε − a0 − f (ε)δa kL2 (V) = o(f (ε)),. ON. !,+@?. .0/~. !C= !,+ -"!,+4# % %'&)( %',!,+ / &.F%'&..!,+P+- / VQ. È80@_5 u UBNbA80U_<=57@_803v:*<jn*6A8]e*UBw?VxQZ*5A6A80<=]Q!6. uε ∈ V. ε. aε (uε , v) = `(v),. ∀v ∈ V.. 57Q?UK<=Q. .0/1.&. ©J 3c<=57@_UB@BbAN]3)5½U hi@B3*}?)N]UB@157} .0/ ©ng80<*6 u = u − u p;ng80<*6½5780<=5 ε ≥ 0 UN¾¤803*>?5A@B803 u Q!b95UBN$b980UB<;57@B8]3v:*QFU hi}!KM<ONP5A@B803 .]/_.&*ap*U_Q!b}!)NPUB@_5A}!b .0/IG*mQ?5.0/~ mQa5TUBN>!803)57@_3M<*@_5A} :*QFUN¸¾¤8069VxQe*@_UB@_3*}N]@_6AQ δ pO3*80<*b80e;57Q!3=803*bUBQU_Q!VXVxQ{b9<*@_PN]3)5!/ ε. a. 0. ε.

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(29) ©ª¯ª C + 12&;+ kuε − u0 k = O(f (ε)).. Â80<*bn8]bA803*b j(ε) = J(u ), ∀ε ≥ 0, 8 ) J Q!b95x<*3*Q¾¤803*>a57@B8]3 :*@1}?6AQ?3M5A@N]e=UBQv:*Q V :ON]3=b R /½tu80<=6x578]<=5 :=} O3*@_bAb9803*bUBQUNP067N]3=0@BQ?3dnON]6. .]/_.0.(. ε. ε ≥ 0. pL3=80<*b. .0/1.H* QE6A}?bA<*U15`NP5>!UBN]bAb9@BKM<*QEKM<*@:*@_5KM<*Q¸UBN:*}!69@_0}?Q¸:*QEUN¾¤803=>?57@_803>!8*+=5Q?b95{}?)N]U_Q$ UN Lε (u, v) = J(u) + aε (u, v) − `(v),. ∀u, v ∈ V.. :=}!6A@10}?QDnONP6957@_Q!U_UBQ:=<fUBN]067NP3*0@BQ?3dQ!b56AQ!VXn*UBN]>!}?QnON]6. J(uε ) − J(u0 ) = Lε (uε , v) − L0 (u0 , v),. .]/_.&q*. ∀v ∈ V.. LJ 3[<=57@_UB@_b7N]3)5 .0/1.&q apUNd:*@1}?6AQ?3M5A@N]e=@BUB@157}E:*QUNc¾¤8]3*>?5A@B803 J pUBQU_Q!VXVxQd.]pU hiW);ng8PÃ 5AW*w!b9Q .xQ?5DUNv>!803)5A@B3M<*@_5A}x:=QxUBNj¾¤8069VxQe=@BUB@_3*}NP@B6AQ δ p3=80<*bE80e=57Q?3*803*bDUBQ$6A}?bA<*U15`NP5 b9<*@_PN]3)5!/ D

(30) M¡o%ª ¯ 1 % F%H& j(ε) − j(0) =2"!- !P= (-12! ,? ?.!-"!-&. + ?> % B ! a. +- %'1 . & 3 . j(ε) − j(0) = f (ε)δa (u0 , p0 ) + o(f (ε)),. . . L !,+- +0 %H&"=2?>*2# "! 2= %'*! / @ ! ?. -! !,+- 0+ %H& u ε = 0 p0 < Q ,A =2 ?>0* 2# "! = %'&.G3 F* 12!- ! B .! p0 ∈ V . .

(31). a0 (w, p0 ) = −DJ(u0 ).w,. .]/_. .0/I*DQa5SUN¾¤803*>?5A@B803 >!8*=+ 5. ∀w ∈ V.. 'Q?]Q!3*803=bXVxN]@_3M5AQ!3ONP3M5N]<yn*6980e*UBw?VxQcnQ?6957<=6Aeg}. g/ tu80<*6N080@_6T<*3 :*}?]Q!UB8]n*nQ?VxQ?3)5N]b;VXn=5A8]57@_K<=Q¸:*QE3*8]5A6AQ¸>?6A@157w!69Q]p 3=80<*bº3*Q{ng80<=]803*bºnONPbN]n*n*U_@BKM<*Q!6\:*@B69Q!>a57Q!VXQ!3)5ºU_Q57W=}!8069w!VXQ.>N]6 p Q?bAnONP>!Q¾¤803*>a57@B8]3*3*Q!UK<=@:*}!ngQ!3=:f:*<jnONP67N]VXw?5A6AQ ε /*t80<*6º>!Q?UN=p=3*80<*bu N08]∈3*Hbe(ΩQ?bA80@_)3 :=QE>?803*b9@B:*}?6AQ!6UBN5AQ!>`W*3=@BKM<*QFn*6A8]n80b9}!QE:ON]3*bU_QnON]67NP067N]n=W*QbA<*@1PN]3)5/ ©' &§Pu¦¨±+S" &%¡gu§0';+% " "¡gª 'O¦¨ lmQa5A57Q{5AQ!>`W*3=@BKM<*Q¾¤80<*693*@_5mUBQ>!N]:*69QVcNP5AW*}!VxNP5A@BKM<*Q3=}!>!Q?bAbAN]@B69Qn80<=6|U h~N]n*n*U_@B>!NÃ 5A@B803 :*Q$UBNjVX}?5AW*8:*Q$N]:µ980@_3)57Q0}?3*}!6AN]UB@_bA}!QP/ÂT80<=b ; =8]3*b R > 0 /Â80<*bE<=57@_UB@_bA803*b U_Q!bº3*8P5`NP5A@B803*bºb9<*@_PN]3)5AQ!bFZ 2 Ω = Ω\B(0, R) p 2 Γ = ∂B(0, R) p 2 D = B(0, R)\ω / t80<*6º578]<=5 ε ≥ 0 p=3*80<*b:*} O3*@Bb9bA803=bTU h~80ng}!6ANP57Q?<*6 %' % / !-@ 4!- &.& T nONP6 .0/1.* T : H (Γ ) −→ H (Γ ). j(ε) = J(uΩε ). Ωε. 1. R. R. ε. ε. ε. ε. 1/2. ϕ. R. −1/2. 7−→ Tε ϕ =. R ϕ ∇uε .n|ΓR ,. ε.

(32) $. !#";&%')(*'O+, 8*) n Q!b5TUBN3*806AVxN]U_QbA8]695`NP3M5AQDb9<*6 Γ Q?5 u Q?b95UNb980UB<=5A@B803v:=<fn*6980e*U_w!VXQ  :ONP3*b  ∆u + k u = 0 b9<*6 D , .]/_.&* u =ϕ Γ , A b * < 6  tu80<*6º5A80<=5 ε ≥ 0 p*3*80<*bng80ubA8]3*b u UBNbA80U_=<=57@_0803v:*<n*∂ω6A8]e*.UBw?VxQ{5A6A803*KM<*}  :ONP3*b  ∆u + k u = 0 b9<*6 Ω , .]/_. ∂ u +T u =0 Γ , b9<*6 Γ.  ∂ u =h ÂT80<*bN08]3*bN]UB8069bºUBQ69}!bA<=U_5`N5bA<*@1]NP3M5!/ %¡o¢L¡g¦¤&¦¨¡gC *!,+-F % / %H& Ω = ! u +0 %H& = ! < Q-A !,+- +0 %H& =2 ?>*2# "! u ÂT80<*bÆ67N]n*ngQ!U_803*buK< <=Q Q| 3*7A28]Q 5A6AQm¾¤803=>?57@_803¸>?*8 +=5LQ?b95L:=Q|UN¾¤806AVXQ j(ε) = J(u ) pP*8 ) O Q!b95\<*3*QnON]657@_Q]80@Bb9@B3*Q:*Q Γ /)´@B3=bA@ p)Q?3c<=5A@BU_@Bb7NP3M5©UBNn*6A80ng80b9@_57@_803j.0p)3*80<=bmng80<=08]3*b }!>!69@B69Q ), ∀ε ≥ 0. N¯¾¤8069V<*UBNP57@_803 PN]6A@BNP57@_803*3*j(ε) Q?UBU_Qf:*=Q J(u .0/_.&* cnQ?<=5fN]UB8]6Abvb!hi}!>?6A@_6AQ Z|576980<=0Q?6 u ∈ 57Q?UK<=Q H (Ω ) a (u , v) = `(v), ∀v ∈ H (Ω ), *8 )jUN¾¤806AVXQ{e*@_UB@_3*}N]@_6AQ a Q?5UN¸¾¤8069VxQU_@B3*}!N]@B69Q ` b9803)5:*803*3*}?Q!bnON]6 |ΓR. ϕ ε. R. ϕ ε. ϕ ε ϕ ε. 2 ϕ ε. ε. R. ε. ε. ε. n ε. 2. R. ε. R. ε ε. n ε. R. Ωε. ε. Ωε |O. ε. ε. 1. R. ε. aε (u, v) = l(v) =. ´@B3=bA@ p;UNPN]6A@BNP57@_803. Z. Z. 1. ε. ε. ΩR. ∇u.∇v dx − k. 2. Z. R. uv dx + ΩR. Z. Tε uv dσ(x), ΓR. hv dσ(x).. b!hi}!>!69@_5. Γ. (aε − a0 ) Z (aε − a0 )(u, v) = (Tε − T0 )uv dσ(x),. ∀u, v ∈ H 1 (ΩR ),. ?Q 5ºU_Qn*6980e*UBw?VxQTbAQ6AN]Vxw?3*Q ¸5A6A80<;0Q!6m<*3*QQa ;n*69Q!b9bA@B8]3fNPb9Vxn;578]5A@BKM<*Q{:*QUNEPN]69@NÃ 57@B8]3 T − T / L F F3BK6 Jd T − T >I?A@B? N K NSI E

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(41) >I? ?C> D E F3 H.J KLdi' MONQP[ È80@_5 Ω <=3«:*8]VcN]@_3*Qeg806A3*}$:*Q R Q?5E:*Q$¾¤6A803)57@_w!69Q Γ N]bAb9Q!Yx6A}!]<*UB@_w!6AQP/ÂT8]<*b bA<*n*ng80b9803*bdKM<*Q Ω >?803)57@_Q!3)5d<*3=QS@B3*W=80Vx8]0}!3*}?@_5A} ω = εω p©*8 ) ε Q!b95c<*3ngQ?5A@_5 nON]67NPVxwa576AQDQ?5 Q?b95<*3#:=80VcNP@B3*QDe80693*}:*Q >?803)57Q?3ON]3)5FU hi806A@_0@B3=Q]/gÈ;80@15 α UBN ¾¤803*>?5A@B803v>?803*b5`ωN]3)57QEnON]6ºVX8069>!QNP<; pO:*} O3=@BQFnORN]6 :ON]3=b Ω\ω , α :ON]3=b ω , α = α *8 ) α Q?5 α bA803)5j:*Q?b>?803*b5`N]3)57Q?b!/ÄT3=Q«NP<=5769QS¾¤803*>?5A@B803 β Q!b95j:=} O3*@_Q :h~<=3*Q VcN]3=@Bw!69QdN]3*N]UB80]<*Q]/tu8]<*6 ε ≥ 0 pu3=80<*bng80bA803=b u UNbA80U_<=57@_803¯:ON]3=b H (Ω) :*< n*6A80e=UBw!VXQ :ON]3=b Ω, .]/Iq0* ∇.(α ∇u ) + β u = 0 A b = < 6 Nd>?803*:*@157@_803 N]<; SUB@BVX@_∂5AQ!bu ∂ u = h 3hi@B3)57Q?69=@BhQ?3)5FnON]bΓ.:ON]3=bUBQ>NPUB>!<=U N]b;VXn=5A8PÃ 57@BKM<*QXQ?5DQ!U_UBQngQ!<=5 ?5A6AQx6AQ!VXn*UBN]>!}?QXn*N]6D<=3*QcNP<=5769Qx>?803*:*@157@B8]3/ÂT80<=b¸Â80<*bE@B3)57}¿Ã 6AQ!b9bA803=b ¸UNE:*}?5AQ!69Vx@_3ONP57@_803X:h~<=3d:*}?]Q!U_80n*ngQ!VXQ!3)5N]b;VXn=5A8]57@_K<=Q:*QTUNEPN]69@NP5A@B803 p*8 )fUBN¸¾¤803*>?5A@B803j>?*8 +=5 j Q?b95:*} *3*@BQFnON]6 j(ε) − j(0) .0/~q=. j(ε) = J (u ), ∀ε ≥ 0. >I? ?G@ * K=N F3 I H F3 H=H6 NQN t80<*65780<=5 ε ≥ 0 p u Q!b5LUNbA80U_<=57@_803E:ON]3*b H (Ω) :*Q\U hi}!KM<ONP5A@B803E]NP6A@N57@B8]3*3*Q!U_UBQ .0/Iq0*H a (u , v) = `(v), ∀v ∈ H (Ω), *8 )jUN¾¤806AVXQ{e*@_UB@_3*}N]@_6AQ a Q?5UN¸¾¤8069VxQU_@B3*}!N]@B69Q ` b9803)5:*803*3*}?Q!bnON]6 Z Z .0/Iq]q a (u, v) = α ∇u.∇v dx − β uv dx, Z .0/~q] `(v) = α hv dσ(x). N. ε. N. 0. ε. 0. ε. ε. 1. ε. 1. ε. 1. ε. ε. ε. ε ε. n ε. n ε. ε. ε. 1. ε. ε. 1. ε. ε. ε. ε. Ω. ε. Ω. 0. Γ.

(42) ?u?%&¦¤¡g "1+u#¦ü¡gª ¡o¥> ¦¤0f"'Oux£¨f"¡ª«'O¦ 0¸'*¢ ¸'*¢ ² ^N]3*bmU_QnON]67NP067N]n=W*QTb9<*@_PN]3)5!p3=80<*bmn=6A}!b9Q!3)578]3*b|:*Q?<; xVx}a57W*8:*Q!b½:*@1}?6AQ!3)5AQ!bmng80<*6 >!N]UB>?<*UBQ?6ºU h~N]b9VXn=578P57@BKM<*QT5780ng80UB8]0@BKM<*Q{:*N]3*b>!Qa5A57QFb9@_57<*NP57@_803/ { H F3 E L E E M5Q N G F; E >? ? %ª¯¦¨M%¸¢L¡g!¦ u¦ £¨¦ 0 g+¦ £¨¦¨!'=¦¤¡g "x£¤'dª M&¡"'O " !¡g¦ g&¥ ; M%'O£ ¦¨ M ^{NP3*bF>!Q?5957Q$bA@157<ON57@B8]3p3*8]<*bFn80<;0803*bEN]<=bAbA@©NPn*n*UB@_KM<*Q!6UBNdVx}a57W*8:*Q$N]:µ98]@B3)57Q ]}!3*}?67N]U_@BbA}?Q]/ N]@Bb?p;@_U3=Q¾¨N]<=5ºb9<*695A80<=5ºn*N]bm578]VeQ?6m:ONP3*b|UBQn*@Bw?0Q]"/ m@BQ?3vKM<*Q{3*80<=b 5A67NPN]@BU_UB803=b:ON]3*bd<=3Q!b9nON]>!Qf¾¤803*>?5A@B803=3*Q!U = ;Q¯Z H (Ω) p½3*80<*bx3*QSng80<=]803*bcn*N]b NPn*n*UB@_KM<*Q!6:*@_6AQ?>?57Q?VxQ?3)5DUBNvVx}a57W*8:*QXN]:µ980@_3)57Q0}?3*}!6AN]UB@_bA}!QP/JL3 Q?Q?5!p:ON]3*bE>!Qa5A5AQ b9@_5A<ONP57@_803p;UN¸PN]6A@BNP57@_803d:*QFUBN¾¤8]6AVXQ{e*@BU_@B3*}!N]@B69QQ?b95:*803*3=}!QEnON]6 . 1. (aε −a0 )(u, v) = (α1 −α0 ). Z. ∇u.∇v dx−(β1 −β0 ). Z. ∀u, v ∈ H 1 (Ω).. uv dx,. È @03*80<*b}?>!6A@10803=bK<=Q δ (u , p ) = (α −α )∇u (0).∇p (0)−(β −β )u (0)p (0) p 8 ) u Q?5 p bA8]3M5Æ6AQ?bAngQ!>?5A@_]Q!VXQ!3)5LU hi}?57NP5u:*@B69Q!>?5ÆQ?5ÆU hi}?5`N5ÆNP:µ980@B3)5!p3=80<*b 3*Q\n8]<=0803=b n*N]bDQ?3 :*}?:*<*@_6AQ$UNdPN]6A@BNP57@_803#:*Q j(ε) − j(0) < KM<*QXU h~N]n*n=UB@B>!NP57@_803 δ 3hiQ!b5¸nONPb >?803)57@_3<=Q:ON]3*b H (Ω) /Q6A}?bA<*U15`NP5N]@B3*b9@u8]e=57Q?3< Q?b95¾¨N]<; /´@B3=bA@ pg>!8]VxVXQ¸:ONP3*b U_Q>NPbE:h~<*3=QXngQ!6¾¤8067N57@B8]3p3*8]<*b:*Q?]803*bE<=57@_UB@_bAQ!6TUNc5AQ!>`W*3*@_KM<*Q$:*Q576A8]3*>NP5A<*6AQ:=Q :=80VcNP@B3*QPp=VxN]UB069}{UBQ{¾¨NP@_5ºKM<*QFU hiQ!b9nON]>?QE¾¤803=>?57@_803*3*Q?UbA8]@_5 = ;Q]/ v+,¦¨ª Ç¢L¡g¦ Æ¦¨£¨¦¤0 +u ª )¡"'O " !¡g¦ ¢L¡g+%¯£¨#¦ü¡gª ¡o>¥ ! ¦¤ ; Â80<*bbA<*n=n80b9803*bKM<*QDU_Q!bºW)n8P57W*w?bAQ!bTbA<*@1PN]3)57Q!bb9803)5TbANP57@_b9¾¨N]@157Q?b!/ ®j¢L¡O;? a. 0. 0. ωε. 0. 1. 0. 0. ωε. 0. 1. 0. 0. 0. 0. a. 1. L. Jε (uε ) = Jε (u0 ) + DJε (u0 ).(uε − u0 ) + f (ε)δJ1 + o(f (ε)), Jε (u0 ) = J0 (u0 ) + f (ε)δJ2 + o(f (ε)),. !-. ®j¢L¡O;? L. δJ1. pε. !,+-. δJ2. . +0&. = !- 6 &) # *!,+:*(,! + B % = (@?.!-&>= !-&. = ! & F%H& / / Q D !768%9+-@! ! B .! δa ∈ R (aε − a0 )(u0 , pε ) = f (ε)δa + o(f (ε)),. +0 %H&$= ! . ∀w ∈ H 1 (Ω).. Â80<*b|3=8]57803=bºK<=Q{U hiW);ng8]5AW*w!b9QEH¸Q!b95º]}!69@ O}?Qn80<=6m<*3*QUBN]6A]Q>!UBN]bAb9Q{:*Q¾¤8]3*>?5A@B803*b >?8*+=5Ab!/oÂT8]<*b80e=57Q?3*803*bTN]U_806Ab|U_Q{57W*}?806Aw?VxQFb9<*@_PN]3)5/ D

(43) M¡o%ª 1 % %H& J (u ) − J (u ) =2"!- != (-12! ,? ?.!-"!-&. +? aε (w, pε ) = −DJε (u0 ).w,. @ % B !:+- %'1 . & 3 L. δj = δa + δJ. ε. !-. ε. 0. 0. Jε (uε ) − J0 (u0 ) = f (ε)δj + o(f (ε)),. δJ = δJ1 + δJ2 Q.

(44) !#";&%')(*'O+,. #. O803=>?57@_803v>!8 +=5. j(ε). δJ 0. J(u Z ε|Γ ) αε |uε − ud |2 dx ΩZ 1 αε |∇(uε − ud )|2 dx 2 Ω. (α1 − α0 )|ω||u0(0) − ud (0)|2. 1 (α1 − α0 )(∇u0 (0)T Mω ∇u0 (0) − |ω||∇u0(0)|2 2 +|ω||∇u0(0) − ∇ud (0)|2 ). 0. /1. 2 O803=>?57@_803*bº>?8*+=57bnONP6957@_>!<*U_@Bw?6AQ!b ¨@B3=W*80VX800}!3=}!@_5A}(¿/ ¸¡gª ¢'*%'O¦¨!¡g";Xª)¡"; Â80<*bN08]3*bn*6A}?bAQ!3)5A}E:=Q!<; dVx}a57W*8:*Q!bº:=@_}!6AQ?3)57Q!bºng80<*6\5767NP@_57Q?6|<*3vn*6980e*U_w!VXQ :h~@_3*W*80VX800}?3*}!@157}P/ºN n*69Q!VX@Bw!69Q Qa ;@B0Q[<*3*Q[5769803*>N57<*69Q«:*< :*80VxN]@B3*Q#KM<*@E6AQ?3*: U hi}?57<=:*Qv:hi<*3*Qv¾¤803*>a57@B8]3y>!*8 +=5$]}!3*}?67N]U_Qv<*3 ngQ!< >?80Vxn=UB@BKM<*}?Q ¤]80@B6n*N]6Qa ;Q!VXn*U_Q kIH]r ¿/Nd:*Q!< =@_w!VXQ}?@157QUBQ?b{n*6980e*UBw?VxQ?b{:*QUBNx5769803*>N57<*69QQ?5FQ!U_UBQD3*80<*bngQ!69VxQa5 :h~}a57<*:=@BQ!6cn*UB<*bx¾¨N]>!@_UBQ!VXQ!3)5x<*3*Q#UBN]6A0Qf>!UBN]bAb9QS:*Q ¾¤803=>?57@_803*bd>?*8 +=5Ab!/ºt80<*6c5A80<=57Q?b >!Q!bº6AN]@Bb9803*b?p=3*80<=bº<=57@_UB@Bb9803*b\UN¸:*Q?<; ;@Bw?VxQ{VX}?5AW*8:*Q{ng80<*6º80e=5AQ!3*@_6mU h~Q¿ =n=6AQ!b9bA@_803j:*Q U h~N]b9VXn=578]5A@BKM<*Q5A80ng80UB80]@BKM<*Q]/ E LdiQ H . L5 K ; E K N F3 F E >I? ? ^N]3*b{U_Q¸>`WON]n=@_5769Q¸*pg3*80<=b{}?5A<*:*@B8]3*bU h~}?K<*NP57@_803 :*QRTQ!U_VxW=80U_5AYEQ?3#:*@BVXQ!3=bA@B8]3 png80b9}!Q$:ON]3*bF<*3 :*8]VcN]@_3*Q>?803)57Q!3*N]3)5D<=3*Q@B3*W=80Vx8]0}!3*}?@_5A} ω = εω /8069bAKM<*Q N Q?b95x:*Qv¾¤806AVXQvKM<*Q!U_>!803*KM<*QPp©U hiQa ;n*6AQ?bAb9@B803 :*QjU h~N]b9VXn=578]5A@BKM<*Qd5A80n8]UB800@_KM<*QdQ!b5 ω :*803*3*}?QFnON]6 . . . . ε. n o j(ε) − j(0) = εN < (α1 − α0 )∇u0 (0)T Mω ∇p0 − (β1 − β0 )|ω|u0(0)p0 (0) + δJ. 8*). 8*). +o(εN ),. ?Q b95<*3*QVxNP5A6A@B>?QKM<*@:*}?nQ?3*::*QUND¾¤8]6AVXQ{:*Q ω /*^{NP3*bºUBQ{>!N]bnON]657@B>?<*UB@_Q!6 Q?b95UNeg80<*UBQ<*3*@_5A}]pO3*8]<*b80e=57Q?3*803*b. Mω ω. . N α0 (α1 − α0 ) |ω|∇u0(0).∇p0 (0) − (β1 − β0 )|ω|u0(0)p0 (0) (N − 1)α0 + α1 +δJ} + o(εN ). N. j(ε) − j(0) = ε <. ÂT8]5A803*b©KM<*QU_Q5AQ!69VxQ KM<*@gN]n*n*N]67-N ,_5©:ON]3*b½>!Q?b\¾¤8069V<*U_Q!b©:*}!ngQ!3=:c:*QUN¾¤803*>a57@B8]3 !> 8*+=5©}a57<*:=@B}!Q 080@_6L¹ÆNPδJe*UBQ!N]<j.0/_.(a/0Â80<*b©80e=5AQ!3*8]3*b½N]<*bAb9@OU hiQa ;n*69Q!bAb9@B803$:*QU hIN]bVxn;Ã 578]5A@BKM<*Qm5A80n8]UB800@_KM<*Qm:ONP3*b©UBQ|>!N]bL8*) ω Q!b95½<*3=QQ?UBU_@Bn*b9Qm:*Q06AN]3*:$N ;Q a Qa5©:*QºngQ?57@15 N ;Q b /o^{NP3*b>!Q?5957QEbA@157<ONP5A@B803p=UNVxNP5A6A@B>?Q M Q?b95:*803=3*}!QEnON]6 ω. . α0 (1 + a) + α1 (b − 1)  α0 a + α 1 b Mω = πab  0. 0. .  . α0 (1 + b) + α1 (a − 1)  α0 b + α 1 a.