Augmented Galerkin Schemes for the Numerical Solution of Scattering by Small Obstacles.
51
0
0
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Augmented Galerkin Schemes for the Numerical Solution of Scattering by Small Obstacles. Xavier Claeys — Francis Collino. N° 6195 Mai 2007. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6195--FR+ENG. Thème NUM.
(3)
(4)
(5) "!$#%& ('() *+,#+- #+ .#+! /% 0 1 325467+,892 0 +* /: =; <?>A@CBEDF5GH<IBKJ LNMPOPDQ<IR1ST@UL-F5VWGCGU@CR1V XY[Z]\5^+_a`cbedgf?hjilkmZ]\5^Qin?o[\qpsrutwvTo[^Qi xyrmzK{|^k}ix*~a^s\i K[IzKrmkcj^rm^Q}Y[^sr}}Y[^ nI ]TdbNt KK?ad?KK^]i 7n
(6) 1km]Y[s^ $s[zKnTEkm ^sTkPzK ¡^s¢w^]kmruzK\En[^kutw¤£*¦¥E^/rmzET$kmtwzKnA§Q£*^£ twimY
(7) kmz%Kjru^]iuiAkmY[^ iusklkm^]rmtwn[¨[ruzK©[¢w^s\ª «ruzK\ Iim\q^srmK «^]¢¢w¬^skur+¢h)kmYKzEn²njkmoY[^kmtw£*n[¦¥Eku^sY[¢w^stnn[K£ kmY³tru^]zi] ® km¯[Y[zK^-ratn?no[stw\q[^s^snTrutwk+]´¢A^]im¢wµt\
(8) §1o[tk
(9) ¢°$t°kmi+twzKn[n¡zKisk§IKIiuzEiliuo[il\qtw©[tw¢wn[^qkmz=kuY[k}^stwr ±K^7kuY[kmtwY[}^Q±?iln[^7^QimzEi ©kmilz7kuK©Is¢^+^Qi\
(10) tonT}kuYz K]zEo[nTk£ tkmYzKojk ^]nzEo[nTkm^srutwn[
(11) rmzE©[¢^]\iz n?o[\q^]rmt°sK¢µ¢zj}±?tn® kmXYY[twiq^=[¶ ruzEzK¢©¢°¢^]n¡\ ·§/\q©?zjh¸j\q^s¢Cz?§¤[£ t¹ «t°h?jtw^]n[¢h¸³o¡kmY[il^Q^=·n?o[twn·\q´¡^srun[t°t¹sku^=¢ jiut¹}ºWY[^s^sru\q^sn¡^)^=zKn,im}¥KY^]^srl\qkutw^]si]^]§yi5[¢wz?ruz$]¥?$kutwj^]¸^Qiqtwn,[kmY[r}^)En[\^]$tEkmY?t° ©¡zEilzEo[¢rmojYkuzTtzEzjn¸·kuzz kmY^ £ twru^]i]®·f?z »Kr5kmY[t°i5\5zjj^]¢ YKi5n[zKk5ru^]s^stw¥K^Q¸n?h¼ru^]¢/kmY^szKru^kutw]¢ {|oilkmt´¡skmtwzKn0§*n·tn?¥KzE¢¥E^]i
(12) ½³K^ ruKK\5rm^^skmt^]nTr]km§^]n¡rm^Q\qi|ku^]^]¼q¢wtwtn¸n^skut°Y[^)tnil[tw\qo¬[k}¢t´n^Qs¸^K§T[£ ruY[zKtw©[}¢wY^s\¾t°i1kuzKz+ c©¡^a©}twY[jzEtw\qim^s^sn5nzKimnqtwzKkunY[K^ ¢*©EEzKilt°o¡iPi|zkutw µ-Y[£*^]o[¦ru¥Etw^7ilkmt°[ ruzKzEnKilEt°$jku^]truzEn²kmtwtzKn,ni] ® \q^]jtwo[\twn¢wojtwn[il\K¢¢ zK©ilkuE¢w^+£ t¹kuYNY[zE\5zEK^]n[^szEoi ¿atrutw}Y¢^ska©IzKo[nruh szKnjtkmtwzKnA® ½À^
(13) [ru^]im^snTk n¡ÀKn¢whjil^5n?o[\q^]rmt°sK¢1iu}Y[^]\5^qimo[t¹k}©[¢w^5 «zKr¨´n[tkm^^]¢^]\q^snTkuiakmY¡$k+jz?^]in[zKk+ilojºW^sr¨ «ruzK\n?o[\q^]rmt°sK¢ ¢wz?}±?twn[§Wn¡=kuK±K^Qi kmY^
(14) [ru^]im^sn¡^zK PkmY[^qim\qK¢¢1zE©i|k}Ks¢^+twnTkmz KsszKo[nTkQ®aHktwi¨©Eil^Q=zKnÀ-zE\©[twnkmtwzKn ©Iimh?^ilk|km£*^]^s\^sn²$kmt°ku¨Y[^K^]´¡n[^skmr}tkmtw¢wzKtÁQo$i
(15) kutjzEn7zK\kmzqKtkun²Y[^\q¶c^kuzKY[¢wzj¢wK¼n-n\q²zj\j^]$¢Ak}tw}nY[^]¢w²ojEtwiln[h?7\qjn=kmzKKkmojt°qkmzE^\qjkmKt°niltwzKzKn¡\qis[® o[Xku$Y[kutwtizEn)rm^Qzil o¢¹kmk}Y[i%^%tnT¢wtkunzN^st° twnjokun¡^K®/~aor nK¢hjimtwi*¢w^]Ki*kmz5kmY[^´¡ruilkÂÃkuz5zEo[r ±?n[z$£ ¢w^]jE^QÄA{|oilkmt´¡skmtwzKn z kmY[t°ik|h?¡^zK \qzjj^s¢C® $Ë E \qÅ=^ÆTkuÇWY[ÈmzjÉµÊ §¶ ^]¢\qY[zE½Ì¢¹kutÁ%ru^K^]§0vTimo\kmtw¢wzK¢ n zE©ilkuKs¢^E§µim]$klku^srutn§¡´n[tkm^5^s¢w^s\q^snTk]§0¶czK¢w¢wKn=\qzjj^s¢C§0imtn[Eo[¢°ra «o[n¬kutzEn. 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.
(16)
(17) "!$#%& ('() *+,#+- #+ .#+! /% 0 1 325467+,892 0 +* /: ¿¨ni¨¢w^5szKnTkm^sTku^
(18) j^5¢°-[ruzK¡EkmtwzKn=j^QiazEnj^Qic^]¢^Q¬kmruzK\KKn[pskmt°vEo^]i]§¡n[zEoi¨n[zKo¡ictwnTkmpsru^]iuimzKni¨oNrmz ©[^]il¢wZskc\q©[^¤tw^sjn=^y[¢t¹oºWicruE¡¬^skukmttzEkmn¨^5¡vEorA^+j¢w^Qqiµ¢w´zK¢°niµK\qo[^]twno[rc^Q i0 zKKnrl[ »^Kt¹kus^sK\qruE^s¬nTkmk0p]rmtzEkmt°nvTjo[o^%kmj^s^+o¢
(19) ru i]zE® nf?jtK^% ¢ twzKnn+t°imjo[^][nEIku^KzE§Iim^yt¢ vTn o[ ^P^]il¢^]kco[r0Kp]i ¡zTt°imilimtw^s©[o¢w^r j^%[ru^sn¡jrm^¨^sn=szK\qjkm^+[^]i*´¢°i \qtn¡^]i iun¡i* »twrm^¨ »Ks^
(20) on rmzE©[¢Z]\q^[^%¥K^sruruzKo[tw¢¢°E^cn?o[\qpsrut°vEo^K® 0^a\qz?[Zs¢w^¨j^¶ zE¢¢°n¡µ§?¢wKrmE^s\q^snTk¤ojkut¢wt°ilp[n¡i¤¢w^]izjj^Qi/[t¹ºWpsru^sns^]i¤´n[t°is§T «zEo[rmnt¹ko[n[^ilzE¢o[kmtwzKn7[r} \$kutwvTo[^ ^ [ruzK©¢Z]\5^E§K^]n\5zjjt´¡KnEk/¢^¨im}Y[p]\nTo\5p]rmt°vTo[^cimo[r¤vTo[^]¢wvTo[^Qiyn[z?^so¡[i¤jo\Kt¢w¢wKK^ ¦¥EzKt°ilt nKnEky¢w^]iy´¢°is® Ko¡imvT o [rup]im^snTk¤^ \qzjjZs¢w^ n EiPrm^ oj^P{|o¡i|kut¹´¡]$kutzEn
(21) kuY[pszErmt°vTo[^ imzK¢wtwj^E§K^sk¤t¢¡t\q[¢wt°vEo^ kmo[t°nNvTo[¡^]i]r}® \qZkurm^+K[¡^]¢p
(22) ¢ twnjokun¡^¢wtwn[pst°vTo[^K§WvT o tw¢1jzKtk skmru^}Y[zKt°imt imo[t¥$KnEk¨j^]i¨zEnilt°jp]rlkutzEni Y[^]o[rmt°i _ ¦¥KzE^Qoian[zEzKnojiatkmtwtwnEzKkun·psruj^]^ iuil¿czEntwrui tw}t°Y[t1¢w^kqo[o¼rmzE©I©[zK¢wr}Zsµ\q§P^^sn \qzjjj^sojZs¢w¸^qjjtw^\q¢°^s-njimtwtzKºWnruEi
(23) ¬[kmtwzKnnÀiKo[n·zEo[\qrmkmtwt°¢vEtw^soo·^+Y[zK\qr¨zKo[EnÀZsn[¡^E^s®Nkmtk¨_ zEzE©oi|ik}Ksrm¢p^E § im^snTkmzEni%^kqnK¢hjimzKnion·im}Yps\)n?o[\qpsrutwvTo[^-vTo[t/^]ilk5szK\q$kut©¢^-¦¥E^]5¢w^]i
(24) \5pskmY[zjj^Qi+p]¢p]\5^]nTkui%´¡n[twi ¢°ilkun[n¡E^c[jr}^][iyi%km»^Qiu}Y[n¡n[i t°vEruo ^]i*n[ ^s \qKn^snT¢wk hjilj^ ^%E\ilh?\qtw¢w¢wjKkuKz^QkmÄ*t°vT^so[k ^cn[¦^¥Eim^]zK ojo[ºWnru^ ^ «[zK^%ru\¥K^soru¢wruzKkmo[twzKtwn¢¢°jE^ ^ak|n?h?o[I\q^ap]jrmzEt°\vTo[^Ktw ®n[^ /´¡^+iukm}t |Y[®¤p]\qf?o[5tw¥¦\qKnT p k ¢w^]iyrup]imo[¢kukuiPvTo[^ n[zEoiyjp]\qzKnTkmruzKni¤ilo[r/^ciu}Y[ps\§n[zKo¡i¤©IzKo[kmt°imimzKn i +o[n[^ Kp]n[psr}¢wt°imkmtwzKnqjo\qz?[Zs¢w^ j^ ¶ zK¢w¢°n¼^sk =on·sK¢wso[¢*KojkmzE\$kmt°vTo[^-j^ ¢ twnjokun¡^-¢wtwn[pst°vTo[^K®³_ zkurm^nK¢hjim^-\qZsn^K § Nn[zKkmru^ zEn[nKtwiuimKn^E!§
(25) ¢°5[ru^s\qtZ]rm^*{|oilkmt´¡]$kmtwzKn kuY[pszErmt°vTo[^¨[^+^¨k|h?¡^+j^%\qzjjZ]¢^E® " Ê ] È $ # ³ ¯ tw¢U§ I^kmtkzE©i|k}Ks¢^E§ jtºIr}KkmtwzKn0§ ps¢wps\q^snTk}i¨´n[t°is§ \qz?[Zs¢w^7j^7¶ zE¢¢°n0§A\qpkuY[zjj^7j^¢° «zKn¡¬kmtwzKn)imtn[Eo[¢wtZ]rm^E§jp]vTo$kutzEn [^+¶ ^s¢w\qY[zK¢kmÁ.
(26)
(27) !"$#% ! &(' )! . #+
(28)
(29) ,.- / ] Ê Ë 0 Ê / 2435 [ / Ë 0 6#7078 Æ Ë 6 Ê # Æ ¸ j®w xyru^s¢wt\qtwnruh7ru^]imo[¢kuiz PKimh?\qjkmzKkmt°¨Kn¢whjilt°i ®%®¨®%®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® j® f?tw\q[¢t´^Q)[rmzE©[¢w^s\ ®®%®%®¨®%®®%®®%®®%®%®®%®¨®%®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® Ë 6 Ê # Æ :<; 0»Q Æ 07=Ej 0 Ê / ?Ê > @ Æ 0 6#7078 Æ A * ®wf?zE¢ojkutzEn)z AkuY[^+imt\q[¢wt¹´¡^] [ruzK©[¢w^s\"o¡iltwn[i|k}nr}jt°iuru^kmtwÁ]kmtwzKn ®%®¨®%®®%®®%®®%®%® * ®wK®w ¯zKru\o[¢°$kutzEn¸®®%®%®¨®%®®%®®%®®%®%®®%®¨®%®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® * ®wK® fTk}©[tw¢wt¹k|h i|kuojh ®%®¨®%®®%®®%®®%®%®®%®¨®%®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® * ® _ o\5^]rmt°sK¢0^jI^srut\q^]nEk}i* «zKrkuY[^+iltw\q[¢wt¹´^Q)im}Y[^]\q^®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® $C EË D # G FH0 Q @ 1 ®w xyruÆ ^]/ im^sÆnTkukmtwzKn)Æ n/ ´r}ilk KÆ n¢whjÆ ilt°i*z kmY[^\q^kuY[zj ®%®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® ® *< L¨¢wzK©K¢0zKn¡ilt°i|ku^snsh ®®%®%®¨®%®®%®®%®®%®%®®%®¨®%®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® M ®® N_ ruorm\5zEr ^]rm^]t°ilskmKtw\¢0^$jkmI^%^s¦ru£t\q¦^]hqnE «k}rmizE\9zKnkmkmYY[^%^+zKK©o[ilKku\qE^]¢wnE^"ku^]®%-® «zK®%ru\®¨o[®%¢°®$ku®%tzE®%n ®®%®%®®®%®%®®®%®%®%®%®¨®¨®%®%®®®%®%®®®%®%®®®%®%®%®%®® Æ > 0 Ê OMP Ë Ç,QÊ > @ Æ Ë $Ë 0« 0 Ê / #* @ J 6 R j®wS L¨¢wzK©K¢µ^]ilkmtw\qkm^²®®%®®%®%®¨®%®®%/ ®®%®/ ®%®%®®%®¨®%®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® j® * 0zj]¢µ^]ilkmtw\$ku^®®%®®%®%®¨®%®®%®®%®®%®%®®%®¨®%®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® j® ¢w¢oilkmr}$kutzEn ®®%®®%®®%®%®¨®%®®%®®%®®%®%®®%®¨®%®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® j® ¿a^]zE\q¡zTiltkmtwzKn)z kmY[^+E[jtkmtwzKnK¢0ilY¡I^a «on¬kutzEn ®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® j® ®w _co[\q^srutw]¢0¿atr}K¯[on¬kutzEn&®%®®%®%®®%®¨®%®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® j® ® ^]Ko[¢°rutwÁs^]-x1zkm^]nTkmt°¢ ®%®®%®®%®%®®%®¨®%®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® 9<TU@ W Æ V=Ê # #» Ë Ê Ë Æ # [®w xyru^]im^snT/ kukmtwzKn z kmY[^+¶czK¢w¢wKn b=zjj^s¢ª®®%®%®®%®¨®%®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® [[®® * XY[^]¢w^]zKkmrutwt¹zKkun)tw]©¡¢ ^sk|N £/j^][^srun)^]iuilo[twzKEn\qz^s nTkmkm^QY[X ^ LAtKn¢^s^]t°rm±?twn njiu}oY[^sku\qn¡^%^ n¡®®%¶c®%zK¢w®¢wK®%n®®%iu}®Y[^]®%\5®%^9®¨®%®%®®®%®%®®®%®%®®®%®%®%®%®® [® xyruz?z 1z n?o[\q^]rmt°sK¢µ¢zj}±?tn,®%®®%®®%®®%®%®®%®¨®%®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® Y\ 6$6 Æ / ZË 0 5 ®w_ tkuiu}Y[^+Kn f[}Y$kuÁ jkuY[^szErm^]\ ®%®®%®®%®%®®%®¨®%®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® \ ® X ^]}Y[n[t°sK¢µrm^Qilo¢¹k zK K^]zK\q^kurmh¼®®%®®%®®%®%®®%®¨®%®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%® \ ® * X ^]}Y[n[t°sK¢µ^]ilkmtw\$km^ ®®%®%®¨®%®®%®®%®®%®%®®%®¨®%®®%®%®®%®®%®®%®%®¨®%®®%®®%®®%®%®. *. +. ]^]`_"aBb ced!f. 1 9. . Q. ,B2. **. * ¦ . ,JI K. Q [[. 2 1. E K K * * *. :?: *(* * *. ?. 1[:. ** E. .
(30) . G . #" ?#.
(31) A1# y #+ XY[^
(32) K^sn^sr}¢AszKnTkm^sTk¨z 1kmYtwi¨£*zKru±-t°i kuY[^
(33) imt\
(34) o[¢°$kmtwzKn=zK y£*¦¥E^%[ruzKKE$kutzEn)tnN\q^Qjtw-tn¡¢wojtwn[-im\qK¢¢ zK©¡i|k}K¢w^]i]® nzEruj^]r¤kuz
(35) I^srm «zKru\9ilo¡}Y5zK\q[o[ku$kutzEn0§j+´¡ruilkK[[ruzEE}Y-zEnilt°ilkui/tn)[¢h?twn[
(36) ilkuKn[r} km^Q}Y[n[t°vTo[^]iÂôn[tkm^^s¢w^s\q^snTk}i «zKr
(37) tnilkuKn^¦Ä^s¥E^sruh?£ Y[^sru^qtn²kuY[^jzE\twn³zK zK\q[o[ku$kutzEn0® *^]sKoim^z kmY^stwriltwÁs^E§ im\¢w¢yzK©¡i|k}K¢w^]iru^]vTo[twru^5\q^QilY²ru^´n[^]\q^snTk+tn¼zEru[^srkmz=©I^qkuK±K^]n³twnTkmzNEszEo[nTk %kut\q^7Kn \q©E^s\qszKzKru\qh[oojim^]k}$-kmtw «zKzKn¡r*km¢µY^ ^]7im^sst^]zKn\q[hE®ojk}$kutzEniiltwKn[t´¡]nTkm¢whqtnsrm^QKim^a©I^]]oim^¨z AkuY[^]im^%j^k}tw¢wi]§?£ Y[t°}Ytw\q[¢tw^]i nÀkmY[t°i£/zErm±=£*^5[rmzE¡zTil^q £*¦h)kuz)©Th?Eimiimo}YÀ ru^´n^s\q^snTk]®5½³^7}Y[z?zEim^
(38) )iltw\q[¢wt¹´^Q²zKnTku^?k kmz$n¡£7r}zEn ^ il7t°j^]twr/^s
(39) nTkc \qH¿ ^kuKY[^]zjzK[\qi ^Kku©[rmh5¢^+£ kmY[zq^]kurmK^¨±K¶ ^%^stw¢wnT\qkmz7Y[zKK¢]kmÁazE^]o[vTnTok ¡kmtw^]zKrln7 «^QY¬kmE¢wiPh)kuzzK©In¡^¨joim¬zKku¢wt¥Kn^Qqµ® kmY[Xtwn=Ytwi/£ £/truzE^]i rm±5tnNt°i/im+t\
(40) ´o[ru¢°il$k*kmiltwzKkm^]n¡ i z 1^s¢w^]kmruzK\En[^kutwa£¦¥K^crmzET$kmtwzKnA® imKs~a^5o[£ r¨t\qkmY¸^kmKYn¼z?²K[sjzKtnkmtwimzKtwiln¡kui]¢y§µzKimYnkmIY[^q^q «o[©nEilt°kmi¨twzKzn¼ zEnÀnEk}Eiltwh?n[\qtwn[j=kuz=kmt°il5twn[KnoK¢w¢Khjrmimttwk|i]hN§IkmKz=[Kjo[kmE^Q\qÀ^skmnTz)kakmkuYY[^7^il\[¢wrm¢yz¦jzEtw©\qi|k}KkmstwzK¢^En ® twXn7Y[^st°i¢w^]\qkmru^zKkmY\z?ÀEn[t°ia^ku¥Etw]^siPruhNtn¢wzKzEr}imj^+^srykuz)kmzimtwzEn[©jKkuo[K¢°tnrzKzEn¡\qilt°[i|¢ku^]^s\qns^sh
(41) nTk%£ \5Y^s^sn-kmY[szjzK[\qi[»ojzKk}r%$kmimtwtzKn[n¡Eo[¢¡¢°[razK «\o[ntw¬nkui¤tzEsnÀzKnT\qkuK^skmtn-Y[zjrm[^]^si}nTÄakmor}il^QnT k zErmn^sr}is®1XY[^sru^^jt°i|k}i
(42) ¥$Ei|k ¢wt¹ku^sr}$kuo[rm^%K©¡zEojkkmY[t°i imo[©j{|^]k]§im^s^¨ «zEr ^[\q[¢w^ C§ H® I^sn)ilo©il^s®=k Xz Y[^ D(0,jzK\R)Ktn²kuYzK$n k jz?^]0i^sn[k zD(0, k
(43) tnTkm^]R)ruim^]©Ik
(44) ^kmimYzK\q^[^7twimn[±7^]tzE 1YT©Ir}KzKjruY[twoz?i zjR³z£ t¹kukmY=Y[^^szKnTrukut^sEr tnA0®½³n ^-Ωn[zKkm^ ©IΓ^+n =zK∂D(0, £ Y[t°}Y5kmY[t°i¤ilkmo[h+kuK±K^QiP[¢°K^t°iPkuY[^ ¡zE¢h?EzKnK¢jzKI^sn7im^k Ω [[ruzEE}Y[twn[ D(0, R)R)\ Ω ® ½³^ imo[[IzEim^ kmY¡$k Ω twi im\¢w¢µzK©ilkuE¢w^a£ tkmY)©IzKo[nruh Γ j^]iurutw©¡^Q©?hkuY[^%IzK¢°r^]vTokmtwzKn . . .
(45) . obs. R. i ε. R. ε. obs. (Γε ) : r = εγ(θ).. ΓR. Γε. Ωobs. ¯ twKo[ru^q L¨^szE\q^kmruh7zK AkmY^%[rmzE©[¢w^s\ XY[^ zEojkm¢wtwn[^czK IkuY[t°i¤£*zKru±t°i¤EiP «zK¢w¢wz$£ is® ½À^¨zEnilt°j^]rP´r}i|k/+[rmzE©[¢w^s\ s¢w¢w^] l^[Kk kmYk¤kuK±K^QiPkmY^ «zKru\"zK 1qiuK[j¢w^¡zEtnTk [ruzK©¢^]\e a § b § V Kn Q^ £ tw¢w¢µ©¡^+j^s´n[^Q¢°$ku^sr¬Ä¬® ilo¡}Y-kuY$k ¯ twn (u , p ) ∈ V × Q Â|E®¦Ä a(u , v) + b (v, p ) = hf, vi ∀v ∈ V, . . . ε. ε. ε. . ε. ε. ex. ε. V,V 0. bε (uε , q) = 0 ∀q ∈ Qex .. $]. .
(46) .
(47) !"$#% ! &(' )! . n¼kuY[twi[ruzK©[¢w^s\)§1£/^7ku±E^7twnEkuzK]zEo[nTk%kmY[^[rm^Qil^]n^zK cNim\qK¢¢/zK©ilkuE¢w^£ t¹kuY·jt°\q^ku^sr ε §Pimz=kmY^ imzK¢wojkmtwzKn n$kuo[r}¢w¢h=j^]¡^]n[i¨zKn ®nNzKr}j^]rckuzj^Qimsrmtw©¡^kmY[t°iilzE¢o[kmtwzKn0§Wtk¨t°icIzEiuimt©[¢w^+kuz-k}±E^ ¢°K$[km¥¦^]Kr]nT®yku~aEn(u^kuY[zK, y^cpkm©Y[)E^5ilt° »i/Ez¬k¨ µkmkmYY^]imk ^cεru^]t°imi¨o[¥K¢¹k}^]isrm§Th=£*im^ \q[εKru¢zK¢1I©?zEh=im^ okmimz
(48) tnrm ^][K¢wnÀEK^ imh?kuY[\q^cjtwkmn[zKtkmkmt°t°
(49) ¢µKimnK¢wshj^ ilt°zi ku Y$Kk%r}£*n^
(50) K£ ^tw\¢¢Po[j¢km^Qtwim[s¢wrmttw^]©Iru^i ©ThÀ)rm^Qjo^Q²ilE^ Q ®XY^7ilzE¢ojkutzEn (u , p ) z *kmY[^-szKrurm^QilIzKn¡jtnn[^]£ [rmzE©[¢w^s\£ tw¢¢¤kmY[^]n Q iu$kmt°il «h imo}YkmYk ¯ tn¡ (u , p ) ∈ V × Q Â|E® EÄ ∀v ∈ V, a(u , v) + b (v, p ) = hf, vi ε. ε. ex. ε. ap. ε. ε. ε. . ε. ap. ε. ε. V,V 0. n-zEru[^sr/kmz5zK©jk}twn-n?o[\q^]rmt°sK¢IimzK¢wojkmtwzKnA§T£*^aszKo[¢°-[twiuru^kutÁ]^ kmYtwi[ruzK©¢^]\ o¡iltwn[
(51) kmY[^[rmz¦jtw\qkmtwzKn imKs^]ij^snzkm^Q V n Q zK AKKr}n[E^%´n[tkm^^]¢^]\q^snTkui¨zKn² rm^]Ko[¢°r¨\q^]imY0®qXY[^5imzK¢wojkutzEnÀzK ¤kmY^ zErmru^]imIzKnjtwn[qjt°imsrm^skm^[ruzK©[¢w^s\ (u , p ) kmY[^]n)iu$kutwil´^Qi ilo}YkmY¡$k ¯ twn (u , p ) ∈ V × Q Â|E® * Ä ∀v ∈ V , a(u , v) + b (v, p ) = hf, vi h. bε (uε , q) = 0 ∀q ∈ Qap .. h ap. ε,h. ε,h. ε,h. . ε,h. ε,h. ε. h. h ap. ε,h. V h ,V h0. h. bε (uε,h , q) = 0 ∀q ∈ Qhap .. XY[t°i[ruzK©[¢w^s\j^]¡^]n[izEnNk|£*z im\¢w¢Pr}\q^ku^sr}is§ Kn ®Hk%t°i¨kmY?oi¨IzEiuiltw©[¢w^
(52) kuz szKnimtw[^sr%il^]¥K^]ruK¢ ru^sKtw\q^]i h Ez?^]ikmz 0 »Kilkm^sr
(53) kmY¡n ε §PzEr ε Kz?^Qi+kuz ε0 »Ei|kuh^sr
(54) kmYKn h ®³½³^ £ t¢w¢©¡^ tnTku^sru^]ilkm^]¸tn·kmY[^ imilku^]zEn¡n[r} rm^]KnTtwo\5\5^E^]§¤rmt°£ sKY[¢0^]n·[kuY[rm^=z¦j\qtw\q^]imY kmtwzKt°in)ilzTo[ºIr}^]ilru^)i «ruzEzK\q\ n?ruo[^]\q¸^skmruz³tw]kmY[¢0^N¢wzjim}\±Ttwn[¢w¡¢ §[zKtn)©ilkukuY[E^¢w^Kim^s®,nim^n,kmimY¡o$}k Y ²iltt°kmicoqkmtw©zKEn0 § [rmz¦jtw\$kmtwzKn¼zK u t¹ ε t°ikmz?zÀim\¢w¢*szK\qru^]¼kmz h ®²½À^ £ tw¢¢*[ruz$¥K^kuY$kQ§yo[nj^]rkuuY[^)zKn¡jt¹kutzEn «zEr¤Ktw¥K^]n
(55) ´?^Q ^q0zE§KnkmjY[t^]kmrmtwzK^ nN^sjkutwYil$kuiyk Kn[Kzz?kuY[^Qi ^srykuz´[j^] Kicκ »>Kilk0 Eimoi }Y5kukmY[Y^sn³k kui|k}n−[uKruk[[>ruz¦κ/jt\ln1/ε ® εXY[ <t°i¨ κ\5 h ^QniakuY$ko[nj^]r¨kmκY[> m k w t K z n zEn?¥K^sruK^QiA¥E^sruhim¢z$£ ¢whK® nqzKr}j^sr kmz+so[ru^kmY[t°iP[εruzK©[¢w^s\)§E£/0^twnEkurmzjjo¡^ hnE[jtkmtwzKnK¢[ilYK¡^ «on¬kutzEn ψ kmY¡$kPYKiAkmzru^s[ruzjjos^ykmY^*imtnKo[¢°r1Kimh?\5[kmzkutw/©I^sY¦¥?twzKr z [kuY[^^[Kk1imzK¢wojkmtwzKnA®Anj^]^]µ§kmY[t°i1©¡^]Y¦¥?tzEr t°ikmY[^%ru^]EilzEn-zK kuY[^%[ru^s¥?tzEoi¢°K}±7zK zEnimtwilkm^]nhE®yXYTo¡is§[£*^%zEnilt°j^]rkmY[^%n[^]£ K[[ruz¦jt\$kutzEnimKs^ imKn {ψ } Kn-kmY[^+imzK¢wojkmtwzKn (u , p ) zK AkuY[^+zErmru^]im¡zEnjtwn[qjtwiuru^ku^[rmzE©[¢w^s\ V =V ⊕ ilo}YkmY¡$k ¯ twn (u , p ) ∈ V × Q Â|E® ?Ä a(u , v) + b (v, p ) = hf, vi ∀v ∈ V ,. ε,h. ε. ε. ε,h. V. ε. h e. h. ε. ε,h. ε,h. ε,h. . ε,h. ε. ε,h. h e. h ap. ε,h. V h ,V h0. h e. bε (uε,h , q) = 0 ∀q ∈ Qhap .. `cimtwn[-kuY[t°ian^s£ jt°iuru^kmtwÁ]kmtwzKn0§IkuY[^5K[[ruz¦jt\$kutzEn=©¡^QzE\5^QizKn¡ilt°i|ku^snTk]®n[^s^]0§µ£/^5£ t¢w¢PilYz$£ªkmYk kmY^sru^¨^sjtwilkui κ > 0 tn[^sI^snj^]nTkcz ε n h imo}Y kmYk kuε − uε,h kH 1 (ΩR ) 6 κ. p ln 1/ε (ε1/2 + h).. n kmY[^%¢°Kilk rmk z kmY[t°i jzjo\5^]nTk]§[£*^£ t¢w¢0K^sk ©K}±kuz5kmY[^+¶czK¢w¢wKn)im}Y[^]\q^K§jz Ãku^sn oim^] tn)[r}Kkmt°^E® Hk kuK±K^Qi¤kuY[^ «zKru\ ¯ twn ilo}YkmY¡$k (u , p ) ∈ V × Q Â|E® EÄ a(u , v) + b (v, p ) = hf, vi ∀v ∈ V , ε,h h. . ε,h h. ε,h h ε. h. ε,h h. h ap. V h ,V h0. ε,h ε,h h bε (uε,h h , q) − Lh c(ph , q) = 0 ∀q ∈ Qap .. h. n5kmY[t°iP¢°Kilkyim}Y[^]\q^K§ L t°iPr}\q^ku^sr kmY¡$kyt°i1n[zk¤¨[rutzErmtj±?n[z$£ n5n
(56) kuY$kyYEi kmz%©I^ }Y[zEim^sn
(57) zKn5kmY^ ©Eilt°i z 1^s\q[twrutw]¢AszKnimtw[^sr}$kmtwzKn¡is®y½³^%£ tw¢¢ ilY[z$£ kmY¡$kafj}Y[^]\q^-Â|E® EÄ twi ru^s¢°$ku^]kmz7¶czK¢w¢wKn icfj}Y[^]\q^ ε,h h. ]^]`_"aBb ced!f.
(58) . G . #" ?#. Â|E® KĬ®cnrmkmt°o[¢°rQ§[kuY[t°ic£ tw¢¢ ¢w^]KNoi kuzqkuY[^szErm^skmt°s¢ [rmzTK}Y)z ¤¶czK¢w¢wKn=im}Y^s\q^]ic[rmz$¥?t°jtwn[kmY^ ukmr Y^]vT^so[zKrutwrm^^Qku¼tw]¥$¢0j¢wo[^]^ ¥K^] «¢zKzEr
(59) [\qkmY[^s^)nTkcKru©IKzK\5ojk ^skmku^]Y[r ^L¶czK¢w¢w® KnX Y[\5twi5zjt°jis§¤^]¢ kmzÀ/imkmzqY[^) »©IrQ§j^]kuilk5Y[^%zK a[ruzEzKo[©[r
(60) ¢w^s±?\ n[z$z£ ¢w´^]n[jK^Ktwn[§ykmY[kuY[^^+´Kr}z?i|kzjil^]¥¦rmKtwzK¢oo¡^i «zKrkuY[^+¶ zE¢¢°n¡-KruK\q^km^]rtn=n)ru©[tkmr}ruh\q^]imYYE rm^]\qKtn^] n zKI^sn[^Q vTo[^Qi|kutzEn0® ) ). #+2 y 6As ½³^5Krm^
(61) [rm^Qil^]nTkm¢whtwnTkm^sru^]ilkm^Q=twnÀn³KszKoilkmt°+jtºIr}KkmtwzKnNrmzE©[¢^]\)®bNzKru^+[ru^]t°im^s¢whK§W£/^
(62) £*KnTkckmz ilzE¢¥E^ twn Ω \ Ω kmY[^5¶ ^s¢w\qY[zK¢kmÁ5^]vTokmtwzKn ∆u + k u = f £ t¹kuYzEojkmEzKtwn[7r}Kjt°$kutzEnzEnjtkmtwzKn0®b=zKru^sz$¥E^srQ§ £*K^]^an[tw^s\qzEo¡i*zTil^%zKn¡5j¿ct¹kutwtrmzEt°n7}Y[zK¢wn-^k kmY[zE^nimj\qtkmKtwzK¢¢WnzEzK©ni|k}kmKY[s^¢^E©I§EzK©[o[o[nk*tk ruilh ^]^sΓ\iyzKku AYku$Y[k ^%5il\iltw\q¢wt¢W¢°zKr/©ilkmkuruE^]¢wkm^K\q®1^s½³nTk*^]szKnnim©¡t°j^^sKr jzEnjkmY[^QzK\q «zEz r kmn[YzE^!n C «YzKzK¢w¢\qz$£ zKtwEn[^sn[® ^]zKoi ©IzKon[ruhNzEnjtkmtwzKni]®%XY[^qi|k}n[KruN¥¦Krmt°$kutzEn¢A «zKru\o[¢°$kutzEnNz ¤kmY[t°i¨[rmzE©[¢w^s\twi ¯ twn i]® k u n¡ ∀v ∈ H (Ω \ Ω ) is® k v u ∈ H (Ω \ Ω ) =0 =0 ε,h h. i ε. R. . 2. ε. 1. Z. i ε. R. ∇u · ∇v − k 2. Z. 1. |∂Ωobs ∪Γε. uv +. Z. vTR u =. Z. i ε. R. |∂Ωobs ∪Γε. f v.. ¶ ^]rm^ ru^ «^]rui/kuz
(63) kuY[^+¿atrutw}Y¢^sk Ckmz _ ^]o[\n[n zKI^sr}$kuzKr* «zKrkuY[^ ¿ ¶ ^]¢\qY[zE¢¹kuÁ^]vTo$kutzEn0® T K[\5tkui n)n¡T¢whEkutw]¢µ^s?rm^QimimtzEn7kuY$kc]n ©¡^+j^QzE\5IzEim^]oimtnq¯zKo[rutw^sr^jKniltwzKnA§ Z X. £ [ Y s ^ u r ^ 1 H (kR) e T u= u = u(R, θ)e dθ, u k 2π H (kR) H_ k
(64) zKt°kmi^ckmY¢°k/Kiu£*ilt°^ sK£*¢PKrmnT^Qkyilo[kmz5¢k+kuzKYn¡$ilkt°jT^sr/t°rui©[t¹NkurusKzKrmnTh
(65) km\qtwn?^]o[imzKY[o^]i+i]§E¢twnn[z^]k¤Krn[^QzE¡^Q^]imiururukmtzK¢wrh
(66) «Eru[zK\ [kmH^]kmz%(ΓkuY[^a) imzE\nEku¢wz ¢¡zEH©i|k}Ks¢(Γ^ ) ®® XY[t°iPt°i1kmY[^crm^QKimzKn£ Y?h
(67) £/^ [ruzKIzEim^*kmz%oim^ ¨´¡kmtkmtwzKoiy[zK\twn5 «zKru\o[¢°$kutzEn ÂUil^]^ ÃÄ «zErPkmY[t°iyrmzE©[¢^]Ω\)§ ^^??kuku^s^snnjjtw^Qn[)%ilkuzEY[¢oj^akuimtzKzE¢wnojkut°i tzEimn5zKo[©?EhqYE%k twtwnn ilt°kmjY[^ ^+ΩimK»s£ ^ Y[t°}Y-i|kut¢w¢¡K¢¢wz$£ i H rm^]Ko[¢°rut¹k|h+ «zKrykuY[^aimzK¢wojkutzEn¡Ä¬®1XY?ois§EkmY^ i]® k v = 0 }. V = {v ∈ H (Ω ) R R \XY[^±E^ oimVoq¢jn¶ zKtwru¢w©¡\ ^]rlkvk kaimE=^K®PkvkXY[^twn[tkmt°+k∇vk ¢ ^[Kk armzE©[¢^]n\)5§jius^]vT¢°o[twr1¥¦K[¢ru^]zjnTjk okmz5k km(u,Y[^%v)rm=^]¥TtwzKo¡i/uvzEn[+^K§twi ∇u·∇v § imo}YkmYk ¯ tn¡ (u , p ) ∈ H (Ω ) × H (Γ ) ΩR \Ωiε. ΩR \Ωiε. ΓR. ΩR \Ωiε. R. R. R. (1)0 n (1) n. n. n∈Z. 2π. inθ. −inθ. n. 0. 1/2. R. −1/2. R. R. . i ε. 1. 1. 2 V. ε. Z . ΩR Z . 2 L2 (ΩR ). ε. 1 0. ∇uε · ∇v − k 2. Z. R. |∂Ωobs. 2 L 2 (ΩR ). . −1/2. R. uε v +. ΩR. Z. ΩR. ΩR. ε. vTR uε +. ΓR. Z. pε v =. Γε. Z. fv. ΩR. ∀v ∈ H01 (ΩR ),. quε = 0 ∀q ∈ H −1/2 (Γε ).. n kmY[^=im^]zEn ^]vTo$kutzEn0§y£*^)t\qIzEim^ Kn Y[zE\qzKK^]n[^szEoi5¿ctwrmt°}Y[¢w^k©IzKo[nruh¸zKn¡jt¹kutzEn zEn A KKr}n[E^c\
(68) o[¢¹kut¢tw^sr}is®P½³^%ru^s£ rut¹ku^ckuY[twi [rmzE©[¢w^s\"twn kmY[^%\qzKru^%zE\5¡K¬k £¦h ilo}YkmY¡$kcvTo[^ ¯ twn (u , p ) ∈ V × Q Γε. ε. . i ε. ε. ex. a(uε , v) + bε (v, pε ) = hf, viV,V 0. ∀v ∈ V. Γε. oimtn ÂU[®¦Ä. bε (uε , q) = 0 ∀q ∈ Qex. $]. .
(69) .
(70) !"$#% ! &(' )! . kuK±?tn[ Qex = H −1/2 (Γε ), Z Z Z uv + a(u, v) = vTR u, ∇u · ∇v − k 2 ΩR ΩR ΓR Z bε (v, q) = ∀(v, q) ∈ V × Qex , qv, Γε Z hf, viV,V 0 = f v.. ∀ u, v ∈ V,. sK^]n\5©Itwn^+NimY[kmz$Y£ kn-EkusY$zEk ru[tn[7t°i*kmkmz7Y^kuY[{|o[^5\qrm^Q=ilo[z¢ AkuiakuY[z^% n[ zEHrm§W\kmYtw¢0i¨[[^sruruzKt¥$©[$¢wku^st\¥E^at°zKi¨ £/^]¢§j¢ £ ¡zTY[ilt°}^QY)µ®%rm^Q`cKim[tinXL¨ru^s^snN «zKru\o[¢°[§Wtk ΩR. . pε. uε. £ tkmY n kmY[^%zEojkmEzKtwn[
(71) nzKru\qK¢µ¥K^Q¬kmzEr*kmz Ω . ~an[^%zK kuY[^
(72) Ej¥$nTkuKK^Qi/zK 1kmY[t°i ´¡kmtkmtwzKoiajzK\Ktn) «zErm\
(73) o[¢wkmtwzKn)t°ikmz7rmz$¥?t°j^+jtwrm^Q¬km¢wh7kuY[^% «o[n¡¬kmtwzKn kmbNY¡$?ka£*^s¢w¦¢w¢
(74) hj i*i ^]kuvTY[o^
(75) iukmtwzK\qn^+i]®yruzKf?¢wtw^n «zK^¨rakmYkmY^¨^
(76) zE©¶ il^]ku¢K\qs¢Y[^ zE¢¹kuÁt°^Qi vEimo¡\q$kmKtw¢zK¢CnN§j£*E^¨i ^]kuY[n^
(77) zEso[o[nTrmkmru^]^sr/nTkckuY[^%kcvTkmo[Y^]^
(78) ilkmimtwo[zKrln »Ez A^+kuzKY[ 1^%km©IY[^s^
(79) Y£ ¦¥?trutwzK^+r* «pzEzr kmY^+zKo[¢^ (u , p ) Ei ε → 0 ®yXY[^n[^s?kcil^QΩ¬kmtwzKn)Kjrm^Qimim^]i/kuY[twicvEo^]ilkmtwzKn0® pε = [. ∂uε ∂uε ]|Γε = ∂n ∂n. i ε. |Γε. ε. ε. . i ε. ε.
(80)
(81) !"#$ % #
(82) &'## (
(83) . n©IzE^]rus[zK^s\qra^]kmiz ¥EzK^s©jruhk}iltwn\-K¢¢C\q® zE0rm^^
(84) k%^oji[¢wi|twsk}t¹k%rmkj^]£ iutkmruY²twjimkmzKtwzK\qnÀ^qzt ynTkmkuoY[t¹^5kutim¥EzK^¢wojtwkujt^QzEKn i]®qu ½Ì§µ£*Y[^s^5n £ t¢w¢P©I^]nzEK\q¢h?^]Ási^
(85) ¥Kt¹k}^]i¨rmhN©¡^]imY\¦¥?¢wt¢UzE§Ar¨kmY[E^ i ε z´K©¡^]¢wi| k}K¢w^ Ω jt°imK[¡^Qr}iilz
(86) t¹k im^s^s\i*ru^]KimzKn¡©[¢w^ kmzimo[[IzEim^ckuY$k u K^skuiε¢wzEim^sr/kmz
(87) kmY^¨¢wtw\5tk |twnstwj^]nTk § imzK¢wojkmtwzKn,kuz¼kmY[^³jtºIr}KkmtwzKn [ruzK©[¢w^s\ £ t¹kuY[zKo[kn?h,zK©ilkuE¢w^¸Â«£*^N£ t¢w¢a[^´n[^Àt¹k[rm^Qt°il^]¢h ©I^s¢wz$£cĬu§?tU® ^K® u ' u ® \ ¬kuo¢w¢htktwi*IzEiuiltw©[¢w^ kmz5[ruz$¥K^ kuY$k*kmY^sru^a^s?t°ilkui κ > 0 tn[^sI^snj^]nTkzK ε ilo¡}Y kmY¡$k ε. i ε. . 0. . ε. ε. 0. kuε − u0 kH 1 (ΩR ) 6 p. κ. XY?oi]§AkuY[^-twnt°j^]nEk´^s¢°¼[ruz$¥?twj^Qi%)´r}i|k5[[ruz¦jt\kmtwzKn³z §1©[o[k
(88) n[zKk¥K^]rmhÀ[rm^Qt°il^E®n·zKr}j^srkuz Kz ©[kutwn\qzErm^
(89) [rm^Qt°iltwzKnA§µ£/^qsKnoim^kuY[^qKimhT\qjkuzkutw5nK¢hjimtwicukmY¡$k%£*EicrmzE¡zTil^Q=tn * C®Hk%szKnTkuKtn¡i kmY^ «zK¢w¢z$£ twn[5rm^Qilo[¢k]® TU@ Æ?Ê Æ 2*)7, .( ! ? (0/ ? # 1 ! ' 21 # !"$ ?# +-,. κ>0. ε. , ,. ln 1/ε ε. ε. . ,. kuε − u eε kH 1 (ΩR ) 6 κ ε1/2 .. 3 ,. U [® EÄ ½³^atw\q\5^Qjt°$km^]¢h-rm^]\5twn7kmY[^j^´¡n[t¹kutzEni*n[^]s^]iuimKrmh
(90) «zKr
(91) Kz?zj7zE\q[rm^]Y[^sn¡iltwzKn7zK 0kmY^¨K©¡z$¥E^ «zKru\o¢w® TU@ Æ ÈuÊ54 > / K 0 Ê / χ • nÀkuY[^ru§^]ilkzK /kuY[t°i+ «jzKzjr o[\q^]nEk nχ : Rt°ic→[^]Rru^]Eruil^tw «n[^s¡r}i¨®¤½³kmz=^ £ Ctw¢¢ £ «rmo[tkmn^ kmtwzKn¼ilo¡}YÀkuY$k χ(r)§W=n=1K «¢wzKimz r r 6 1/2 χ(r) = 0 χ (x) = χ(|x|/ε ) tn¡i|ku^]K zK χ ® r > 1 χ χ TU@ Æ 0 / 0»Ë Æ / 8 Æ #»Ë u • − π u0 (0) −2π u0 (0) ε u eε (x) = (1 − χε1/2 (x)) u0 (x) + 2 v0 (x) + χε1/2 (x) V (x)1ΩR \Ω¯ iε (x) ln ε − λ0 ln ε − λ0 0. +. 0. ε 0. 0. ]^]`_"aBb ced!f. ∞. ε δ. δ.
(92) G K u0. ru^ «^sr}i*kmz5kuY[^%tn¡t°j^snTk ´^]¢w)j^s´n[^])Ei¤kuY[^%o[n[t°vTo[^ «o[nkmtwzKn=iu$kmt°il «hTtwn[ u0 ∈ H 1 (ΩR ) ∆u0 + k 2 u0 = f. twn. zKn. •. #" ?#. ΩR \ {0}. ∂ΩR \ ΓR u0 = 0 ∂ u0 ΓR ∂ n + T R u0 = 0. zEn. Æ > [Y8 Æ #UË v ½À^+j^s´n[^¨kmY^ »r´^]¢w)Ki*kuY[^%o[n[t°vTo[^+imzK¢wojkmtwzKn z kmY[^%[ruzK©¢^]\"©¡^]¢z$£%® ¤Æ ¸ 2*)7, +-, .( ! 7 U J v ∈ H (Ω \ {0}) 1 TU@. 0. . 0. . 1 loc. . R. Ω \ {0} ∆v0 + k 2 v0 = 0 R #" ∂Ω \ Γ v0 = 0 R R . ∂ v0 ∂n. + T R v0 = 0. #. ΓR. v0 − Y0 (kr) ∈ H 1 (ΩR ).. /hil^]r}$kutzEn-z ¥$rutwK©[¢^Qis§jtk t°i¡zTimimt©¢^¨kmzrmz$¥E^ kmY¡$k v K[\5tkui kmY[^%o[ntwvTo[^+j^QzE\5IzEimtkmtwzKn . 0. v0 = Y0 (k|x|) + L10 (v0 )J0 (k|x|) +. X. L1p (v0 )J|p| (k|x|)eipθ +. X. L0p (v0 )Y|p| (k|x|)eipθ. nzKr}j^]rckuz±E^s^sÀn[zk}$kutzEnizKn¡ilt°i|ku^snTka£ tkmY * H§µ½À^qn[zKkm^ J = 1 n Y = γ + £ Y[^sru^ γ = 0, 57721 . . . t°ikmY^ N o[¢w^sr zEnilkunTkQ® -Æ ¸jGFP nkuY[^+sEil^¨£ Y[^]rm^ Ω twi*kuY[^+jt°il±7zK 1s^snTkm^]r 0 nruEjtwoi R §j£*^%Y¦¥E^ |p|>1. . |p|>1. 0,0. 0,0. 2 π. 2 π. ln( k2 ). §. R. (1). v0 (x) = −iH0 (k|x|) = −iJ0 (k|x|) + Y0 (k|x|). •. L10 (v0 ) = 1/i. Kn. L0p (v0 ) = L1p (v0 ) = 0 |p| > 0. Æ / Æ jY8 Æ #»Ë V ½À^¸j^s´n[^³kmYtwi) «on¬kutzEn ¥?t° ¢^]\5\,z q^jt°i|ku^sns^¼n o[n[t°vTo[^sn[^Qimi]® ^]\5twn kmY¡$k ® {x(r, θ) ∈ R | r = εγ(θ) } ¤Æ [¸ 2*) 2 W .( ! 7 J TU@. ε 0. Γε =. . 2. . J . +-,. . 1 ΩN = {x(r, ! θ)' , ∈ , R2 | r > γ(θ) } V0 ∈ Hloc (ΩN ) 1 l0 (V0 ) ∈ C Ω ∆V0 = 0 N V = 0 # ∂Ω 0 N
(93)
(94) 2
(95)
(96) 1 J ln r) − l (V ) lim sup (r; θ) − (Y +
(97) = 0.
(98) V 0,0 0 0 0,0 r→+∞ 0 π θ∈[0,2π]. [ . $]. .
(99) .
(100) !"$#% ! &(' )! . ½À^+]¢w¢µn[^]Kr/´¡^s¢°-kuY[^ «o[nkmtwzKn. ®PHk t°i [ruz$¥K^Q7twn ¦ µkmYk. V0ε (x) = 41 V0 ( xε ) Z ∂V0ε = 1. Γε ∂n. X Y[twi+¢wEi|k%[ruzKI^srmk|hNt°i%tnTku^sru^]ilkmtwn[=n³£*^Y¦¥K^}Y[zTil^]nkuY[^7\o¢¹kut[¢wt°s$kut¥E^5 »K¬kuzKr+©I^k|£*^s^]n V n ilzÀkmYkt¹kt°iimkmt°i|´^Qµ®·Hk£ tw¢¢ ©I^)t\qIzKrmkuKnEktwn·kuY[^ «zK¢w¢z$£ twn[kmz²±K^]^s·twn,\5twn kmY^ ^jVn¡iltwzKnz V »n¡zK V Ä*tn kuY[^%n[^stwKY?©IzKruY[z?zj7z 1twnj´n[tk|hK®PHk t°iKtw¥K^]n©?h X V £ Y^sn r → +∞. ÂU[® * Ä 2 e , V ∈C V (r, θ) = (Y + J ln r) + l (V ) + π r XY[^%ilo[\ zEnT¥E^sruK^Qi¤twn H n[zKru\"zKn)n?h-szK\qE¬k il^skcilo st^]nTkm¢wh »r ¦£¦hq «rmzE\9kmY^%zKrutEtn0® -Æ ¸jG FP nkuY[^+sEil^¨£ Y[^]rm^ Ω twi*kuY[^+jt°il±-z P^snTku^sr 0 nr}Kjtwoi ε §[£/^Y¦¥E^. ε 0. 0. ε 0. 0. 0. 1 0. 0,0. 0,0. 0. |p|>1. 0,p ipθ |p|. 0,p. 1. i ε. V0ε (x) =. •. |x| 1 ln 1ΩR \Ωiε . 2π ε. Æ Ê / ]] / λ ½À^%n[z$£ j^´n^kmY[^+szKnilkuKnEk twn?¥KzK¢w¥K^Q7twn kmY[^%\$k}}Y[^]^jKnimtzEn TU@. 0. . λ0 =. π1 l0 (V0 ) − L10 (v0 ) . 2. -Æ ¸jGFP nkuY[^+sEil^¨£ Y[^]rm^ Ω twi*kuY[^+jt°il±-z P^snTku^sr 0 nr}Kjtwoi ε §[£/^Y¦¥E^ i ε. λ0 = γ + i. π k − ln 2 2. ½³^twnT¥?tkm^kmY[^
(101) rm^QKj^]r kmzo[nj^]ruilkuKn «zKru\√o[¢°NÂCj® Ä £ tkmY=kuY[^+ «zE¢¢wz$£ twn[7IzKtwnEk¨z P¥?t^]£%®c¯Kr «ruzK\ kmY^ im\qK¢¢µzE©ilkuKs¢^E§j\qzKru^¨rm^Qt°il^]¢hq «zEr |x| > ε §jkmY[^+imzK¢wojkutzEnzK kuY[^%^[Kk [rmzE©[¢w^s\&iu$kutwil´^]i /¢wzEim^akuz5kmY[^+im\¢w¢µzK©ilkuE¢w^K§? «zKr. uε (x) ' u0 (x) + |x| <. √. ε/2. − π2 u0 (0) v0 (x). ln ε − λ0. §?kmY^+ilzE¢ojkutzEnzK AkuY[^%^[Kk [rmzE©[¢w^s\&iu$kutwil´^]i. uε (x) '. −2π u0 (0) ε V (x) ln ε − λ0 0. zn¼ kuY[K^)rlkuiltw\so[¢wK¢w¢*r]§ zK©£/il^kuEzE©j¢w^KkuK®tn¸½³K^ n²£ ^tj¢w¢/[¢wo¡twsil^ t¹k¢w^sjkm^][r+rm^QkmimYimtwtwi5zKn³twnjz « zKrukm\Y[^-$km©¡twzK^]n¼Y¦tw¥?n·tzEzEr%ru[z^s r+kmY[kuzÀ^}ilzEY[¢z?ojzTkuilt^-zEn²ÀzKiln²\kmKY¢¢w^7^srq©¡zEil¡o[Kn[^-Krmzh AKKr}n[E^c\
(102) o[¢¹kut¢tw^sr}i* «zKrkuY[^+¿ctwrutw}Y[¢w^kcszKn[t¹kutzEnzEn Γ ® £ Y[zE½³¢^+^ j´zEn\K¢¢wtwhqn rmΩ^QsK®y¢¢IHk n[t°zKi kmIY[zE^]iur/ilKtw©[im¢wh?^\qkmjzkmzEzK©jkmt°ku Kt^]n)ilkmtw\il$Y¡km^Eru®P¡X^]r Y[^]^cilrukm^]tw\qimo[¢kmk/^+z¦ A£X¦Yh5^szK «rmruzE^s\"\9kmjY®wc^%t°zKi*©ilnkuE^]il¢wkm^ tw\Ω$ku®¤^cXtwn7Y[t°kmi Ytw^i kmY^+ilo[©[{|^]¬k zK AkuY[^ «zK¢w¢z$£ twn[5kmY^szKru^s\)® TU@ Æ?Ê Æ 2*) 2 ' , , 1 # , , # +-, , .( ! 7 + ( # / . ε. i ε. R. / [ #". Ω0. % ΩJ0 ⊂[ Ω R/B ? ( Q #1. . ' , , ". ε. ε. kuε − u eε kH 1 (Ω0 ) 6 κ ε.. ]^]`_"aBb ced!f. Ω0 ∩ Ωiε = ∅. . κ.
(103) ]. G . . #" ?#.
(104) %
(105) % . Hk t°in[z$£ IzEiuiltw©[¢w^¨kmz\±E^qimt\q[¢w^%ru^s\ru±©Kim^]zKnkmY^%rm^Qilo[¢kuizK AkuY[^%[ru^s¥?tzEoi im^]¬kutzEn0§ pε = [. ∂e uε −2πu0 (0) ∂V0ε ∂uε ]|Γε ' [ ]|Γε = ∂n ∂n ln ε − λ0 ∂n. |Γε ,. U [® ?Ä ©ThÀkmY^. imtns^ χ = 1 zKn Γ )® XYTo¡i%£/^7[^]t°j^kuz=rm^][¢°K^qkuY[^-imKs^z AKKr}n[E^
(106) \
(107) o[¢¹kut¢tw^sr}i Q im\qK¢¢w^sr imE^ Q j^s´[n ^Q-©?h ilo¡}Y-kuY$k q˘ ∈ C } ⊂ Q = H (Γ ). ∂V Q = { q = q˘ ∂n XY[t°iPn[^]£,imKs^YKiPjtw\q^snimtwzKnqzKn[^E§Eimz¨kmYkyt¹kyt°iPIzEiuiltw©[¢w^kmz%t°j^snTkmt «h q = q˘ £ t¹kuY q˘®1f?z%£*^ j^snzkm^ kmY^n[zErm\ z yn=^]¢^]\q^snTkcz [ruzK©[§W¢w^s\n £*kuY[^5t°Ki rmn[^5zEtwrmnE\ ku^sru£ ^]til¢wkm¢A^Q©IN^
(108) twn[^t°´i¨n[km^QY[)^]nÀ©?h ru^|q| «zErm=\
(109) o[|˘q¢w| kmtC^Q® ^K=®tkmnYkm^Y[\q^5zj «zEj¢o[¢wz$¢wo£ i tnzK |q|Eia zE\5¢^sNnTo\©I^srQ®
(110) XY[^5Q q˘ \n[n^sr imo}YkmYk ¯ tn¡ (u , p ) ∈ V × Q ÂU[® EÄ ∀v ∈ V, a(u , v) + b (v, p ) = hf, vi ε 1/2. ε. ex. ap. ε 0. ap. −1/2. ex. ε. ∂V0ε ∂n. ap. ε. ε. ε. . ap. ε. ε. V,V 0. bε (uε , q) = 0 ∀q ∈ Qap .. XY[t°icrmzE©[¢^]\ ´[kuiatnkmY[^5il^sklkutn[ z PkuY[^
(111) Kn¢whjilt°i zK ¤¯*® *ru^sÁsÁ]t KnNb¼® ¯[zErlkutntn H®¨½À^5£ t¢w¢1s¢w¢ t¹k% imt\q[¢wt¹´¡^] [ruzK©[¢w^s\)®1n=K[jtkmtwzKn £*^+zKn¡ilt°j^sr*kuY[^ «zK¢w¢z$£ twn[5n[zk}$kutzEni ^sk ˘b (v) = b (v, ∂V ) ∀v ∈ V. Z = {v ∈ V | b (v, q) = 0 , ∀q ∈ Q } ∂n _ zKkm^+twnNKrlkutwso[¢°r kuY$k b (v, q) = ˘b (v) q˘® XY[^
(112) ¢twn[^Qr «zErm\ ˘b t°icimzK\q^+±?twn)z ¤¦¥E^sr}Etn[5zKI^sr}$kuzKrQ® nj^]^]À£ Y[^snÀkmY^5zE©ilkuKs¢^
(113) twi% jt°il±W§ Γ t°i str}¢w^5z /s^snTkm^]r)KnNr}Kjtwoi ε §0nÀ£*^
(114) Y¦¥E^ ˘b (v) = R ½³^ v(ε, £ t¢w¢θ)dθ [ruz$¥K® ^*kuY[^ ^jt°i|ku^sns^ nqon[twvTo[^]n[^]iuiPzK IkuY[^cimzK¢wojkmtwzKnqzK ¡kuY[^cimt\q[¢wt¹´¡^]q[ruzK©[¢w^s\ ÂU[® Ä®1Hk¤twi ´r}i|kn[^Q^]iuiuruh
(115) kuz5imY[z$£ÌkuY$k Kn-twnj Himo[)zEnjtkmtwzKn-t°iimkmt°i|´¡^]7©?h a ®Pb=zKru^sz$¥E^sr¤kmY[t°itn[ Himo[szKn[t¹kutzEn YEi*kmzq©¡^%on[t¹ «zErm\ £ t¹kuY)rm^QilI^]k*kuz ε «zKrkuY[^%[ruzK©[¢w^s\"kmz©I^+i|k}©[¢w^K® ¤Æ · .( 2 ! ) : [ /B ? ( Q #1 ' ". ε. . ε. ap. ε. . ε. ε 0. ε. ε. ε. ε. ε. 1 2π. 2π 0. +-,. α0 > 0. , ,. ε. inf sup. |a(u, v)| > α0 . kukV kvkV. Ê0Ê?> x¤rmzj^]^]³©?hÀszKnTkmr}Kjt°¬kutzEn0§Aimo[[IzEim^qkmY[^]rm^-^jtwilkui=il^QvEo^sns^ ε → 0 § n¡³Kn[zkuY[^sr (u ) imo}Y-kuY$k «zEr n?h n § ^k sup a(u , v) → 0 u ∈ Z , ku k = 1, kvk f?twn^ (u )nt° ic©Ii|kuzKrmo[zEn¡n[jK^]¢wh¼)twn zKn?H¥E^sru(ΩKtwn[=) §Itn£*^sKn=^?kmkmr}z$K£k r} «rmµzE§P\ im¦(uhE§ )®¼7X ilo±E©^ il^QvTo[^sns^+£*^]±?¢whszKn?¥K^]rmEtn[qtwn H (Ω ) i]® k w = 0 zEn 5n[^]tEYT©IzKruY[z?zj-LzK (Ω0 })®)XY[^snA§A «zEr n uim!o 7tw^snTku¢vh³∈¢wKrmHE^K§ (Ωv ∈ \Z {0})® ¯=rmzE\{wkmY∈^ H (Ω ) £*^]±)zEn?¥K^sruK^]n^E§j£/^
(116) zK©jk}twn v) = 0 ®af?twn^ v t°ic}Y[zTil^]nNru©[t¹kuruKrmtw¢wh-twn H (Ω* \ {0}) §o¡il^kmY[^ »KkqkmYk5kuY[t°i7il¡K^ t°ij^snim^)a(u, tn »imt\q[¢w^)rm^Qilo¢¹kqkmY¡$ktwi¦¥$Kt¢°©[¢w^ tn ÃÄkuz²im^s^kmY¡$kkmY^ [¶ ru^]^s¢¥?\qtwzKY[ozEi
(117) ¢¹kuÁ%^]vT^]ovTKo¢tk|kmh²twzKY[nzKkm¢°Y[ik «zKrqn?h ® \ vHi ∈q(ΩHzKn¡)(Ωil^QvEo)^s§¤n£ sY[^K§tw}Y t\q[¢wtw^]iqK]zKr}jtwn[®y=¯ kutwzNnkm¢wY¢hE^ § £*^s¢w¢*¡zTil^Qjn[^]iui
(118) z u=0 ku k → 0 u∈Z ε v∈Z ε. . n. εn. n. 1. 1. n. 1. n. n V. R. 2. R. v∈Z εn. V. n→+∞. n. 1 00. R. R. 1. 1 00. R 1. n. n→+∞. R. εn. R. R. n L2 (ΩR ). k∇un k2L2 (ΩR ) 6 k∇un k2L2 (ΩR ) + <e{. Z. ΓR. n→+∞. un TR un } = a(un , un ) + k 2 kun k2L2 (ΩR ) .. $]. .
(119) E.
(120) !"$#% ! &(' )! . ?f twn^ a(u , u ) → 0 §jkuY[twi ¢^QK[ikuz ku k ®*XY[t°i zEnEkuruEjt°¬kuizEo[r twn[t¹kutwK¢ Eimimo[\qjkmtwzKnA§ → 0 imz
(121) kuY[^%ru^]imo[¢¹k zK kuY[^%¢^]\q\q5t°i[ruz$¥K^]0® n¸Kjt¹kutzEn¼£/^7K¢wimz=Y¦¥E^qkmzN[ruz$¥K^5kuY$k
(122) Kn¼nj |f?o[·szKnjtkmtwzKn²t°iiu$kutwil´^]²©?h b ®=¯[zEri|k}©[tw¢wt¹k|h iu±K^E§TkmY[t°i zEnjtkmtwzKn YKic¢°ilz
(123) kmzq©I^%o[n[t «zKru\5¢whiu$kutwil´^Q-£ tkmY)ru^]im¡^Q¬kkmz ε ® ¤Æ+-, · .( 2 ! )) 1 [ /B ? ( Q #1 ! ' , , ". n. n. n H 1 (ΩR ). n→+∞. n→+∞. ε. β0 > 0. ε. sup. |bε (v, q)| > β0 |q| kvkV. Ê0Ê?> Hk t°i ilo st^]nTk kmzzEnilt°j^]r 5´[j^] «o[nkmtwzKn z kmY^¨zErmtwKtwn0®PXY?oi]§?kmY[^]rm^%Y[zE¢wi v∈V. . sup. ∀q ∈ Q.. u ∈ C ∞ (ΩR ). k}±?twn[5kmY[^+¥$K¢o[^ 1 zKn=5n[^]tEYT©IzKruY[z?zj. |q| |bε (v, q)| > kvkV kukV. f?twn^ u t°i*´[j^])Kn jz?^Qin[zkcj^]¡^]n zKn ε §?kmY[t°i [ruz$¥?tw[^]i/kmY[^+[^]imtru^]rm^Qilo¢¹kQ® v∈V. ¯ tn¡¢w¢hE§0£/^qn[^]^]kmz)[ruz$¥K^5zKnTkutn?o[tk|h=[ruzKI^srmkmtw^]i]§µo[n[t «zKru\£ t¹kuY²rm^QilI^]kakuz ε ®
(124) kmY[t°i£ tw¢¢yK¢¢wz$£ oi¨kuz [¢h¢°Kiuilt°sK¢Iru^]imo[¢kuizK ^s?t°ilkm^sn¡^%no[n[t°vTo[^sn^]iui «zKr iuK[[¢^ CIzKtwnTkrmzE©[¢^]\is® ¤Æ · .( 2 ! ) O ? (0/ ? # 1 ! ' . +-,. κ>0. , ,. ε. (i) (ii). |a(u, v)| 6 κkukV kvkV p |bε (v, q)| 6 κ ln 1/ε kvkV |q|. ∀u, v ∈ V, ∀(v, q) ∈ V × Qap .. Ê0Ê?> ^koi7szKns^snTkmr}$ku^zEn (ii)\ ® `ailtwn[¼o[kHzº ku^]}Y[n[t°vTo[^K§/t¹k7t°i imo!7n[tw^Q^svEnTo¡k/ku¢wz
(125) t¹k|hrm(i)z$¥E^ Kkm©¡Y[zEt°i*oj[k ruazKIt°^si5rmk|kmhrut «¥?zEt°r¢Cv® ∈AD(Ω ®y`ailtwn[ 0^s\q\ ®¨twn7kmY^¨K[¡^]njtµ§T£*^¨±Tnz$£ kmYk kmY^sru^¨^sjtwilkui κ > 0 tn[^sI^snj^]nTkcz ε ilo¡}YkuY$k ) . R.
(126) Z
(127)
(128). b=zErm^]z$¥K^srQ§
(129) Z
(130) |v(εγ(θ), θ)| =
(131). Y[^]n^ Z. Γε. εγ(θ) R. 2π 0. |v(εγ(θ), θ)|dθ. 6. Z 2π ∂V0ε
(132)
(133) v
(134) 6 κ |v(εγ(θ), θ)| dθ. ∂n θ=0
(135) Z R dr 1/2 Z R ∂v 1/2 ∂v
(136) ; (r, θ) dr
(137) 6 | |2 rdr ∂r εγ(θ) r εγ(θ) ∂r Z. 2π 0. Z. R εγ(θ). 1/2 1 drdθ r. p 6 κ ln 1/ε kvkH 1 (ΩR ) ,. Z. 2π 0. Z. R εγ(θ). |. 1/2 ∂v 2 | rdrdθ ∂r. £ Y[t°}Yh?tw^s¢°[i (ii) ® ½ÌtkmY 0^]\5\Ei[® * § j® Kn²j® ?§0£*^5]n²[[¢whNXY[^]zKru^s\¾K®wml¬®wqz y£ Ytw}Y³[ruz$¥Tt°j^Qic^s?t°ilkm^sn¡^K§ o[n[t°vTo[^sn^]iuin¡¼i|k}©[tw¢tk|h³z kmY[^ ilzE¢o[kmtwzKn¼z [ruzK©¢^]\ ÂCj® Ĭ®XYtwi
(138) sn¼©I^7£ rut¹kmkm^]n¸tn¸kmY[^ «zE¢¢wz$£ twn[ \n[n^srQ§ . ]^]`_"aBb ced!f.
(139) Q. G . ¯zKr KnTh. f ∈V0. §jkmY^sru^¨^sjtwilkui qo[ntwvTo[^%zEo[[¢w^. a(uε , v) + bε (v, pε ) = hf, viV,V 0. (uε , pε ) ∈ V × Qap. #" ?#. ilo}YkmY¡$k]§. ∀v ∈ V,. bε (uε , q) = 0 ∀q ∈ Qap .. b=zErm^]z$¥K^sr/kmY^ «zK¢w¢z$£ twn[5^]ilkmtw\qkm^Qi%«o[n[t «zKru\&£ tkmY)ru^]im¡^Q¬kkmz εÄ/Y[zE¢w ^k |p | 6 (1 + )kf k . ku k 6 kf k Hk-t°i-IzEiuimt©[¢w^)kmz·[rmz$¥?t°j^Nn Eilh?\qjkuzkmt°NKn¢whjilt°i5 «zKr u §cimtw\5tw¢°rkuz¼kmY[^zKn[^= «zEr u § tnÌzEru[^srkuz j^skm^sru\qtn^%t¹k}i©¡^]Y¦¥?tzEr*£ Y^sn ε E^kui¥E^sruh7il\K¢¢C® TU@ Æ?Ê Æ 2*)): .( ! ? (0/ ? # 1 ! ' 21 # !"$ ?# ε. 1 α0. V. ε. V0. kak α0. 1 β0. V0. ε. +-,. κ>0. , ,. ε. ε. ,. ε. ke uε − uε kH 1 (ΩR ) 6 κ ε1/2 .. XzEY[nt°ili-kmruruo^]¬imkuo[t¢zEk-nt°iz µimkmt\qY^atw¢w\Kr7$k}ku}z Y[^]X7Y[^^]jzKru^s\ n¡iltw[zK®nÀzK ņTtkmkuYi^a[n[ru^Qz?zr¤ ´ «zK^s¢w¢°¢z$£ YEi7iykmkuY[z^³©Iim^¨K\q}Y^NKru¢Etwn[¬ku^Q^sisru® tÁ]^]¶ qz$£*Kiy^s¥KkmY[^]r]^§/il «zEzK¢ro[kmkmtwzKYn ^ z *-imt\q[¢wt¹´¡^]NrmzE©[¢^]\)®+XY[^szErm^]\[® * hTtw^s¢°u[iakmY¡$k¨kmY[^
(140) ´r}i|k¨km^]rm\iaz ykuY[^
(141) \ku}Y[^QN^jKnimtzEnNz u Y[zEtwKtn¡Y[^]t°rcj^ku^s£ ru\t¹kuiuY-Ä® kuY[\ ^n[´^sr}£%i|k§µkmX^]Y[rm\^]zKi*ru^sz\ kmY[j^%® * ^sjrmz$¥?nt°jimtw^]zKian-7zK K¢wuzK©¡Â»©[¢Aoj^Qk i|kukmt\Y^]imkm^a^)k|£*«zKz5n ^ΩjÄ n¡©[ilo[twzKk¨ntik¨jt°iazq¡n[zTzKimk imt©zK¢tw^nskmtwz j^¨¢z? «zKzE± r «zKr imzK\q^¢wz?]¢µ^Qi|kut\$ku^%¦£*¦hq «ruzK\9kuY[^%zK©¡i|k}K¢w^K®P¶c^sru^¨t°i 5ru^]imo[¢¹k imt\qtw¢wKrkmzXY[^]zKru^s\ [® ?® TU@ Æ?Ê Æ 2*)1 ' , , 1 # , , # +-, , .( ! 7 + ( # / . ε. ε. ε. R. / [ #". Ω0. % ΩJ0 ⊂[ Ω R/B ? ( Q #1. . ' , , ". ε. Ω0 ∩ Ωiε = ∅. ε. . κ. . kuε − u eε kH 1 (Ω0 ) 6 κ ε.. A/ * /% #+ #+! As)) #+2 ½³£ tw^a¢¢I£ rm^stw «¢¢W^sr/kmz%[¢kuhY[^ilkui|Kkun^s[7zKr}q µkm´Y[n^at¹ku\5^a^Q^]il¢YA^]\5§?K^]nnTk*£ tw¢[¢¡rmkmz¦^]jntwq\ku$z
(142) kmtwzK[nq® /kuzzEnxyilrut°zKj©^]¢r ^]\ÂCj® Ĭ®P§?iln-zE\qkmY[^ ^akmrurmt°^Qi|n[k*Ezo[ 0¢wkuY[kmtwt°zKi¤nKzK¡ ^]r h® XY[t°i1kmrut°n[Eo[¢wkmtwzKnqsKn
(143) ©I^czE\q¡zTil^QzK ¡kurmt°n[E¢^Qi1zKryvToEjruKn[K¢w^]i](T® ¯zK)ryKnTh h §Kku±E^ (K, Σ , P ) Ω ³zK¢w¢w^]¬kutzEn·zK ´n[tkm^)^s¢w^s\q^]nEk}iqz czEru[^sr q Kiuimz?stwkm^Q¼£ t¹kuY T ® A^k V = {u ∈ C (Ω ) | u ∈ ®½³^5[ku^sY[n[^zKkmK^ iuilIzjt°km$Yku^5^]n?©o[E\
(144) il©¡t°i ^]zr zim *Y[I^sK^/ru «^so[^Qnic¬zKkut yzE «nrui^s^]Â [zK\)t°i § E(aimimzj)t°$ku^]¨kmkuz Y[^]im^Ä® j¯^szKErPrm^]^]n?iah z y «ru^s^QjzK§\)j^]§0n[Kznku ^ P } (ϕ ) §kmY[^\qjt\¢AruEjto¡iz ytwrus¢^Qi zEnEk}twn[^] twn K §ϕKn)j^snzkm^ h = jatwK\ (K) ®*½³K^j∈^´¡Tn[^ ρ(h) = ρ min ®T ρ n h = max h ® ½³^£ tw¢w¢ \±E^kmY[^ «zK¢w¢z$£ twn[-Eimimo[\qjkutzEn zEn=kmY^%kmrutwKn[Ko¢wkmtwzKn V , XY[^sru^^jt°i|k}i σ , σ > 0 imo}Y-kuY$k «zEr n?h h > 0, σ < ρ(h) < σ ® ½³^³iu¦h·kuY$kkmY[^Àkmrut°n[Eo[¢wkmtwzKn T twi ru^sEo[¢wKr]® XY[twi\q^]Kni7kuY$k t¹k=jz?^]in[hzKk zEnTkutwn KnTh,\q^]imY ru^´n[^]\q^snTk]§WtwnNrmkmt°o¢wKrcn[zKk¨twn=kmY^
Documents relatifs
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
usual route to school and provide us with some details about their travel routine (e.g., schedule, mode of transportation, people accompanying them, etc.); 3) recent interventions
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
