Modeling of topological derivatives for contact problems
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Modeling of topological derivatives for contact problems J. Sokolowski — A. Zochowski. N° 5599 16 Juin 2005. N 0249-6399. ISRN INRIA/RR--5599--FR+ENG. Thème NUM. apport de recherche.
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(27). Ωρ. B(ρ). ρ} ⊂ Ω O u = u(Ωρ ). . B(ρ) = {x : |x − O| <. ,. ρ. .. u(Ωρ ). Γ , ρ. Ω. . ∇u · ∇(v − u) ≥ 0 ∀v ∈ K. K(Ω) = {v ∈ H 1 (Ω)|v = g on Γ0 , v ≥ 0 on Γs } .. @. A3. Z. @. @. O. B(ρ). u(Ωρ ) ∈ H 1 (Ωρ ). Ωρ ρ ≥ 0 ρ. 1 E(Ωρ ) = 2. Z. ρ → 0+. k∇u(Ωρ )k2. kqs8KM`®YuXYaHJLkqXrVWKMj8Y[fhYa`ctL £8jlkp\lXr`nfp\ y Jj= pypyJ l· DD FE. RU. Ωρ. S L#E. ρ2 π |∇u(Ω; O)|2 + O(ρ3−δ ) 2. `n\fq_Y[HJXafpLKMLEL\JL*_[ pV]J\ltY[`fq\lkhGm(Hlt*L kp\l_u4XZYLfq_aLE_[jJsJ_[L*LE_Xat L*fq\d_aYa_[L;LEs¡t Ya`\`nfp\Y[HJYaLLE_aL;K bd]J`#kpmLE\qYPfp_[K fpBkm`n\JL`\dYaLE p_ukhm=f#qL*_Pk t*`_ut mnL `®Y[H Y[HJLt LE\dYa_[L khY E(Ωρ ) = E(Ω) +. T.. 0<δ <1. ΓR = {x : |x − O| = R}. ρ2 b(u(Ω), u(Ω)) = −ρ2 π|∇u(Ω; O)|2. @. 1 b(u, u) = − 2πR6. O. "Z. ux1 ds. 2. +. Z. ux2 ds. 2 #. 4jGkhHl_uL*kh_[KML L fpY[_[L*Lp_ L,t*kp\ sJL l\JLY[HJL\JL LE\JL*_[ pV ]J\4tYa`nfp\4khmls8L l\JL;sfq\ kh\4s sJL*jxL*\ls8`n\J fq\Y[HJL,XrKkpm GXa`nfpHl\lLX%]Jfh\lFt LEYamn`n`fpj8\lYakp`ctm fpjxL*_uk#Yafq_[XEXr ]J4HlXZ`nYutukhH \dYa`\l`ckht*mnm]4V¡s8s8LE`X%xYaLEHl_[LXPXa_[fhfp¢ït*K kpmYanLEHJs:LXaL*L*\Jj4LEkh_a_u qk#VYaL;s8s:L lkp\JXaVWL;KMs¡j8fpYa_fpYaXr`cLEt*m®X åkqs¨gZfp6 `n\d·YPL 6£W 8YakhLE\4\8s ¢ @. .. W@. ρ. E0 (ρ ; w) =. E0 (ρ ; w). Î ÉÉ ].
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(61). Ξ = {x ∈ Γs |u0 (x) = 0}. Z. [Ymn`YHJdL`n`n\JX MfpXamnHJfpf f \`\J\ `\ t fq\lls8{ `YaY[`nfpHl\:k#Y`cX`\XakhYaYaHJ`cXL lPL;s`n_[`nfqtuHJ_mnkpL mYXrjlkqt Lq khHJ\l`ctus H¦khkpmnxjJjxjJfqmn`XaL;`XYa`nYapfLPYaKMHJL;LMkpTWXa]Jfp_axL;fpX mnL*fhXa jllkp\lt*L`®Y[LLE\JL*_[ pV {¨ Ls8LE\JfhY[L kp\ls¡s8L l\lL YaHJL jxfqXa`®Y[`qLK L;kpXa]J_[L fh l\J`YaLL*\lL*_[ pV 3. @ @. D E. O(µ) = µ⊥ = {v ∈ HΓ10 (Ω)|hµ, vi = µ[v] =. .. Ξ. ,. u0 = u(Ω). @. u0 ∈ K. µ. hµ, vi =. Î ÉÉÂÂ ]. µ. TK (u0 ) ∩ O(µ) = CK (u0 ) ∩ O(µ) .. .. Z. Ω. v dµ = 0}. ∇u0 · ∇vdx .. 4.. .. HΓ10 (Ω) .
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(63) 2[ XF [. E @. hktET8khY`\l\t*L]4XrYaLMHJLkh\t fq`\¡kh\WlpY[XrL HJYa£L_ukpXaXaL t jlYYPkp_at*L;L Xr]lm®Y p`npLE\`n`cXX(W qjxV`fpqmnL*GVd\¡HlHJLEL*Wfqs8V:_a_[LE`nY[tpKHJ #LYa:HlKML jJL _[Y[f#_akp`cptsJLEkps¡jJK_[`nfh\khgZ_uLEst qYas8`n fp` x\¡ =LEG_afpLE\dHJ\dY[LEYaffp`ckh_[Y[L*mWHJK fhLxt*lK fpn{\JL*4YaL _[ 3`ntTWL;jJt_[Y[fh`fqgZL;\ t Y[l`fq· \ L fq`_BYaH Y[HJLXrLE\lXa`®Y[kh`W\4`s YZVMkp\lkhmnV8Xr`cX%fhXafpmn]8Ya`nfpG\4X&HJLY[fK#kpkhjJ_[`cjJk#`nYa\J`nfp \4khmx``\ \JL;bd{E]l} khPmn`®`cY[X`L;t*X*fp@\JG`ct*HJkpLEmnfpV_[L*sJK'` 3L*_[`cL*X\dYa]l`cXakhLElsmLq fq_`Ya} H Y[HJLs8` xLE_aLE\dYa`ckhm q`qL*\dV:YaHJL]J\l`nbd]JLXafpmn]8Ya`nfp\ Yaf Y[HJL#kh_[`nkhYa`nfp\lkpmx`n\JL;bq]4khmn`®YZV SK (u0 ) = TK (u0 ) ∩ O(µ) = {v ∈ HΓ10 (Ω)|v ≥ 0 on Ξ(u) ,. u0 ∈ K. @. I. K ⊂ H ( Ω) H 1 (Ω). 2t1 = ρ. 2. at (·, ·) = a(ρ; ·, ·) Π0 h. a(u0 , v) = 0}. 21. Π. DD E. , S S SK (u0 )YS E. . 01. . &@. kpkq\lXaXas ]JKMj8`YaY[H `fq\lXfh&GHJLEfp_[L*K Mkp_aL fpp_LE_akh` mnlL;skp\lsWV { J]J_aYaHJ`YaLEH _aKM{Efqz _a*Lq 8 fq%_LYaHlHJk¨Lq¨L kp_a`ck#Y[`fq\lkh m3`n,\JLELEbd\l]lt Lkpm` khYZmnV } 8lYafpHJ_LYaL*HJ£8LMjlt khfq\l\dXaq`nfpL*\ \J`nL*\4%t Lfqfpmn&f ,YaHlX*L@_[GLEHJkq`cs8XL*t _ fq%KMLjJ_amL*L;Yat*L;khXmnYaHJHJLL*_[jJL_[Y[fWHJfhL Z kplXrYa_ukptY_aL;Xr]lm®Y HJ`ntuH`nXk pL*\lL*_ukhm¢ `nEkhYa `nfpL \Y fhYaHJL`nKM4jJLm`ckt `t*YafpL\Wp]JL*\l£tY[kh`\lfqs¡\t YaHJmnfqLEXafpLE_[s¡L*K Xa]JlfqXa_L Y¨kpfh_a@`ck#k Y[`,fq`n\lm 4khm3LE_r`n\JYLEXabdjl]lkqkpt mL ` Ya`nLEXE l kp\lsmL*Y s8LE\JfhY[L YaHJL sJ]lkhmn`®YZLVXrjlHlkhkp`nm_a=`n\Jt* Mfp\lxXaL `cs8Y L*_%L*L*Ya\HJLfqmkhnf\ls `\l åkpHlK L*`n_[m LV:fhF#s8kpLE_a\J`ck#fhY[Y[`LEfqX\lYakhHlmxL`\ls8LE]lbdkh]lmkhfpmn` Ya`nLEX s8LEj4LE\ls8`n\J Mfp\ kjlkp_[kpK¢ L*YaLE_ {; x`nL\ kM{EXrz fqm]JYa`nHlfp`n\tuHYaf `nXYa{EHJ L tEkp XaL*LPY,]lfqX_ \J} fp*YaLp JYaHlkhYfp_ kp\ls QL fp_[fpL*8f#YupkhLE`n_E\ dmnL Y W Π0 h ∈ SK (u0 ) :. @. .
(64). hA0 w, vi = b(w, v). U .
(65). K ⊂V. ,. a(Π0 h, v − Π0 h) ≥ hh, v − Π0 hi ∀v ∈ SK (u0 ) ,. @. 2. w, v ∈ V WL @. @. V. 0. . @. ft = 0. =@. V. @. ?@. QI. C@. TI. V. V V. 0. h·, ·i. t ∈ [0, δ) δ > 0. yt ∈ K :. yt = Pt (ft ) y = Π0 (−A0 y0 ). at (yt , ϕ − yt ) ≥ hft , ϕ − yt i. @. . @. 0. . ∀ϕ ∈ K .. ft = 0. . yt = Pt (0). . 6 67(N . Y' "& ( T8. . F Y"& "< "66 H67" () : . . . at (·, ·) : V × V → IR ' "& 8B
(66) "& 2 [ t ∈ [0, δ) At ∈ L(V ; V 0 ) 6 H. C6F
(67) N ?R >$ 0 at (φ, ϕ) = hAt φ, ϕi ∀φ, ϕ ∈ V A ∈ L(V ; V 0 ). . At = A0 + tA0 + o(t). t > 0, t. ". . K ⊂V. S. (NX"&6
(68) R "V Y60 : ft = f0 + tf 0 + o(t). ? . {. L(V ; V 0 ) .. ". {;y. . V 0,. ft , f0 , f 0 ∈ V 0. ! "> " 5 2"&; ? 67< "A8N ?! < " X "& 6 <: Πf = P0 (f ) ∈ K :. a0 (Πf, ϕ − Πf ) ≥ hf, ϕ − Πf i. {;}. ∀ϕ ∈ K. Íé =Í. .
(69) {q{.
(70)
(71) ! "#$ %& $' () J 2 " ; "< %8
(72) ' <: 67 2 ∀h ∈ V 0 :. Π(f0 + εh) = Πf0 + εΠ0 h + o(ε). ". {. U . V. (N7"&6
(73) ? ?(T$& "V 0 9 "< "66 R" J? < : 0 ε > 0, ε. ?(N $"&6 )"& P67" ( < Π#: V → 0VR" (
(74) R6' 0 o(ε) h∈V V " ?! 67< "!#5 ? < " " 60 : N 0 V ; " $' < # [ [ (NX"&6
(75) . {;. t. t=0. . t > 0, t. yt = y0 + ty 0 + o(t). . 8 . ". {. FL . V,. y 0 = Π0 (f 0 − A0 y0 ) .. . }qy. {Ez. X6 9$' X ?4 ? J[ $ C < " = "
(76) :J67" < " "&! ? A )' " J ( ;
(77) " ':5 67 2"9> " . . a(ρ; v, v) = . ? "& :5$ " :. . Z. Ω. ev (O) = |∇v(O)|2.
(78) . k∇vk2 dx − πρ2 ev (O). 68/R ? 0
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(80) . . . u = u(Ω) = (u1 , u2 ). Ω. σ = (σ)ij i, j = 1, 2. . − σ=f Cσ − (u) = 0. u=0. uν ≥ 0,. I. σν ≤ 0,. σν uν = 0. στ = 0. . Ω, Ω,. . . Γ0 ,. Γc .. 2 2 σν = σij νj νi , στ = σν − σν = στi i=1 , σν = {σij νj }i=1 , 1 ij ( ) = (ui,j + uj,i ), i, j = 1, 2, ( ) = (ij )2i,j=1 , 2 {Cσ}ij = cijk` σk` , cijk` = cjik` = ck`ij , cijk` ∈ L∞ (Ω).. Î ÉÉÂÂ ]. [@. Ω. @.
(81). Γc. @. .
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(84) 2[ XF [. AI. /@ , @. Ω ρ
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(86).
(87) G. .. C. @. Ω B(ρ)
(88) Ωρ u = u(Ωρ ) = (u1 , u2 ). @. σ = (σ)ij i, j = 1, 2 . − σ=f Cσ − (u) = 0. uν ≥ 0,. ,. σν ≤ 0,. σν uν = 0. στ = 0 . @. @. . . . . @. . . Z. Ω. . . 0.. ).. . .. @. .. . .. @. . Ωρ. S. @. Ω. . a(ρ; v, v) = a(u, u) + ρ2 b(v, v) .. @. a(ρ; v, v). b(v, v) = −2πev (0) −. U U S @ I ?@. U . . . @. . 2.. a(ρ; ·, ·). b(v, v). . Γc .. 2 (λ + µ)(11 + 22 )2 + µ(11 − 22 )2 + µγ12 ,. .
(89) . . $U L . Γ0 , Γρ ,. 5@. a(u, u) =. . Ωρ , Ωρ ,. u=0 σν = 0. ,. S. cijk` ξji ξk` ≥ c0 |ξ|2 , ∀ξji = ξij , c0 > 0,. @. ρ2. ρ = 0+. . 2πµ σII δ1 − σ12 δ2 , λ + 3µ. a(ρ; v, w). ). @. . v. .. . v, w. 2a(ρ; v, w) = a(ρ; v + w, v + w) − a(ρ; w, w) .. Íé =Í.
(90) {;.
(91)
(92) ! "#$ %& $' (). ïG\HlY[L HJt*LfpXa\WkppK L*£L XrL*Y,k¨V`nXYaHlsJLL lJ\J`L;mn` s\JLE`\kp_YaHlfp`n_[X,K tEkpXaL dV `cXs8L Y[L*_[K `n\JL;s_[fpK'YaHJL fq_aK]Jmnkfp_ p fq ]JL _YPjJ]4]JX,_[t jxfqfq\lXaXrLE`cXs8LEkh_jJjlYaHJ_af¨L£8`nfpKMmnkhf Ya`nfp`n\J\ #kh_[fp`nkhYaY[`nHJfpL\lkpXafpmmn`]8\lYaLE`nbdfp]l\ khmn`YZV HJY[`cf:tuHt fqjJ\q_[Yuf#kpdt `cYs8L;jlX_afqk:JXrmLE] K t*`qLEy \dYa¢ mnV pzjJ*_[ LEt*`nXaLfp_ `nG\lHls8LLEj4_[LELE\lXa]Js8mLEY\qfpY,8fpYu kh`n\J LEs`nXY[HJL fpmnf `n\J 4 qfq_Xr`nKMjJm`ct `YZV LkpX[Xr]JKMLYaHlkhYYaHJL mn`\lLEkh_fq_aK `cX W qy 3. A@. b(v, w). .. b(v, v). K = {v ∈ H 1 (Ω)2 |vν ≥ 0 on Γc ,. @. v=g. 5@. uρ ∈ K :. @. ?M. . u(Ωρ ). uρ. a(ρ; u, v − u) ≥ L(ρ; v − u). @. . on Γ0 } .. .
(93) S. ∀v ∈ K .. L(ρ; ·). ρ. 6 Q "< :P(N J ?= ?X "V > ?"? " ; 9067 " . uρ < J8T ?9 ρ(N# . ρ. 0+. uρ = u(Ω) + ρ2 q + o(ρ2 ). . in H 1 (Ω)2 ,. . A C#?2 %$ < = = 67< " #) ?P "#$ &
(94) '2( 67" 6 P 67< " P ?J 2 "V q "&4 "&6u(Ω) <: q ∈ SK (u) = {v ∈ (HΓ10 (Ω))2 |vν ≤ 0. on Ξ(u) ,.
(95) "Y' : ?. a(0; u, v) = 0}. a(0; q, v − q) + b(u, v − q) ≥ 0. ∀v ∈ SK (u) .. p GHlLjl_afWfhfh@GHJL*fq_aLEK" `nXXa`nK `nmckh_Yaf Y[HJLjJ_[fhfp(GHlL*fp_[L*KS{h lkh\4s:Y[HJL*_[L fq_aL `cXfpKM`®YaYaL;s. 8 . H "&
(96) "& / 1 2 R"&. . U . J 9
(97) " /":!67" < " J . . Ξ(u) = {x ∈ Γs |u(x).ν(x) = 0} ?A 67 " P < " X "& 6 <: u∈K. u ∈ H (Ω). q }. ρ=0. "B ?N0 "& ; = " 'N " ? "B T" ( ) − u k = o(ρ2 )
(98) 8 J 0 8O<?$ / ?/ $
(99) 58
(100) ku(Ω = ρ5 ?/ρ 2# H (T8 " ? 2
(101) J 0 7# <
(102) "O $"& J/ "" C> ? ; ? 5 25# " ? u u(Ωρ ) F " <:/ ! ? J 0 7#
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(104). b(v, v). .. . ?@. N@. 2πev (0) =. ,. π(λ + µ) π 2 R6. Z. (v1 x1 + v2 x2 ) ds ΓR. 2. +. 2 12k (1 − 9k)(v1 x1 − v2 x2 ) + 2 (v1 x31 − v2 x32 ) ds + R ΓR Z 2 12k µ 3 3 (1 + 9k)(v1 x2 + v2 x1 ) − 2 (v1 x2 + v2 x1 ) ds + 2 6 , π R R ΓR. µ + 2 6 π R. Î ÉÉÂÂ ]. Z. . .
(105) L .
(106) {. WS.
(107) 2[ XF [. `YaH @. 12k 3 3 (v x − v x ) ds, 1 2 1 2 R2 ΓR Z 12k µ 3 3 (1 + 9k)(v1 x2 + v2 x1 ) − 2 (v1 x2 + v2 x1 ) ds, = πR3 ΓR R. σII =. kp\ls. σ12. µ πR3. . Z. (1 − 9k)(v1 x1 − v2 x2 ) +. Z 9k (v1 x1 − v2 x2 ) − πR3 ΓR Z 9k (v1 x2 + v2 x1 ) − δ2 = πR3 ΓR. 4 3 3 (v x − v x ) ds, 1 1 2 2 3R2 4 3 3 (v x + v x ) ds. 1 2 2 1 3R2. δ1 =. !#(+- ! (
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(109)
(110) . L Y]lX,t*fp\lXa`cs8L*_YaHlLJ]Jm L*\lL*_[ pV]J\4tYa`nfp\4khmfhYaHlL fp_[K . G. E(u) =. 1 2. Z. qz. k∇uk2 dx,. HlL*_[L X[k#Y[`nX lL;X`\lXa`cs8LYaHJLs8fqKkh`n\ YaHJL khjJmckpt*L LEbd]lk#Y[`fq\ d Hlfp`YamnL:HfhXr]J`YaY[HlkpLJmn#Lkh4_[`nfqkp]JJ\lmnL:sJkp_[_akqV¡s8`nt ]lfqX \ls8`Yat `nfpfp\d\lYuXEkhP`n\JiLE]Js _P`n q\fqkpYaHJm=L`nXPs8Yafpf KXZkpY[]l`\=s8 VfqYaHJ_LXr`n`KM\ 4jJ]Jmn` L*t*\l`®YZt*V L%fpLY[jJHJ]8LYMXaK`®YuXkhmnt LEt*\d`_uYat L*]J_ mckhk#Y_ %L:sJf\JfhY kh\dYYafXrYa]4s8V¦kh\4#skh_[_[`cL*khjJJmcmnkpL:t*s8L fpKkp`\lXEW V¡Xaf YaHJ%LMLLEbd`cXa]Jfp`nmc#k#khYamnL L*\dY[Y HJ`nL X£8jJHJ_[fpLEmnLX[Xr`n`n\lfp\Xr`cs8f#LMqL*Y[_ HJL:_[`\J Bi\ L kïq\XaXaY[]JHJKML L lHJ_[Xr fpYPKMXr`®fqYaY[L* pH j L*kh\l\ L*fpLkp]4sJK X%s8f8NP`Yas8L*`n`fp]JV\4Kkhkhmx\l4\¡fq]Jt*B\lfpTWsJ\l`nkpsJ\l_a`®t Y[V L `fq`\d\lY[X*L* p_ukhmlYaLE_aK f#qL*_ 4{ L*£8HljJL*_a_[L;L XaXa`nfp`c\ XBYaHJLfp]8Y kh_us \Jfp_[KkhmlqLEt Yafp_Bfp\Y[HJLPxfp]J\4sJkh_[Vfp LPKk¨V t*fp\lt*L*\dYa_uk#Y[Lfp\Y[HJL W @. u. .. Ω. Ω ⊂ IR2. S . ∆u = 0. @. 7-. @. ρ. @. C.. @. 3. Z. H@. k∇uk dx =. n. E(u). ΩR = @ Γρ = ∂B(ρ). ΓR = ∂B(R). E(u). 2. Ω. @. @. x=0 C(ρ, R) = { x | ρ < kxk < R } @ Ω \ B(R). . @. Z. 2. ΩR. k∇uk dx +. Z. 2. B(R). k∇uk dx =. Z. 2. ΩR. k∇uk dx +. B(R). ER (u) =. Z. u. ΓR. @. Z. u. ΓR. S . ∂u ds, ∂n. S . ∂u ds. ∂n. Íé =Í.
(111) ¨{ y Y[PHJL*L_[LXafpY[mnHJ]8L Ya`nfp¨\kpm]lfhLE Xfh fp\ kp_aL`\ p`npL*\¦dV¡Y[HJL P`n_[`ntuHJfpmn\L YYaf¡N,LE]JKkh\J\ KkhjJjl`\J 4 3\lkhKMqL*mnV WV:KMGL;HJkhL:\lX]Jfp\ltY[Y[`HJfqL\ _aLEmnkhKYa`nk¨fpV \ 4L t fp\4XZY[_a]lt YaL;s `YaH YaHJL:HlL*mnj fhe(fp`cXaXafp\ pLE_a\lL*m6 Y[HJL:jlkp`_ t*fp\lXrYa`Ya]JYaLEXjxfpmckh_,t*fWfp_us8`\4k#YaL;Xkh_[fp]J\ls WY[HJL*\
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(113) ! "#$ %& $' (). I. ∂u/∂n. ΓR. ∆v = 0. B(R),. v=u. S . ΓR ,. ∂u/∂n = ∂v/∂n. O@. v. G. H3. (r, φ). 0. fq_,kh\WV. 1 π. v(r, φ) =. r≤R. Z. ∞. π. 1 X r n u(R, ψ)[ + cos n(φ − ψ)] dψ 2 n=1 R −π. LXaHlkpm=kqXaXa]JKMLfp_,k KMfpKML*\dYYaHlkhY ,. 1 ∂v (r, φ) = ∂r π =. Z. 1 πr. π. u(R, ψ)[ −π. Z. ∞ X. n. n=1 ∞ X. π. u(R, ψ)[ −π. r<R. ` xLE_aLE\qY[`nkhYa`n\J p`npLEX 21. S
(114) S. 1 r n−1 cos n(φ − ψ)] dψ R R. n. n=1. r n R. π. dy. cos n(φ − ψ)] dψ. S . ∞ 1 ∂ X r n =− sin n(φ − ψ)] dψ [ u(R, ψ) πr −π ∂ψ n=1 R Z π ∞ n X ∂u r 1 (R, ψ)[ sin n(φ − ψ)] dψ. = πr −π ∂ψ R n=1. Z. S
(115) S . fqmHlnf `mnL:`n\J`n\q Y[L*Y[_a q`c_[t khYa`n\Jfp _ WV¦jlkp_rYuX LHlk¨pL]lXaLEs HlL*_[LMYaHJLt*fp\dYa`n\W]J`YZVfp f#pL*_ ,jJjJmnVW`\l YaHJL ,. @. G. |t| < 1. . ∞ X. @. u. ∞ 1 X [(teiα )n − (te−iα )n ] 2i n=1 1 te−iα teiα = − 2i 1 − teiα 1 − te−iα t sin α = = H(t, α) 1 − 2t cos α + t2. Lkh_[_a`npL k#Y LXaHlkhmn\JL*L;s¡khmcXrfYaHJL#kpm]JL fp @. ∂v 1 (r, φ) = ∂r πr. HlL*_[L @. Z. ∇v(0). v(r, φ) =. 1 2π. π. Î ÉÉÂÂ ]. ∂u r (R, ψ)H( , φ − ψ) dψ. ∂ψ R. L Y,]4X_aL _a`YaL WV:KML;kh\lXfpe(fp`cXaXafp\ pLE_a\lL*m6 −π. Z. W@. π. u(R, ψ)K( −π. K(t, α) =. . tn sin nα =. n=1. ,. ΓR. S
(116) S. r , φ − ψ) dψ, R. 1 − t2 . 1 − 2t cos α + t2. d}. S . HG. S7U S$L .
(117) ;{ } NPf.
(118) 2[ XF [ @. qz. 1 ∂v ∂v ∂v (0) = lim (r, φ) cos φ − (r, φ) sin φ , r→0+ ∂r ∂x1 r ∂φ ∂v 1 ∂v ∂v (0) = lim (r, φ) sin φ + (r, φ) cos φ , r→0+ ∂r ∂x2 r ∂φ. kp\ls. S . ∂K r 2 ( , φ − ψ) = cos(φ − ψ), ∂r R R 2 ∂K r ( , φ − ψ) = − sin(φ − ψ). lim r→0+ ∂r R R lim. r→0+. PL*\lt*Lp lkhÏYaLE_,s8` 3L*_[L*\dY[`nkhYa`n\J kh\ls¡Xa]JlXrYa`Ya]8Y[`fq\¡`\dYa_[f qz L fp8Yukh`n\Y[HJL l\lkpm=_[LEXa]Jm®Y I. 01. . S7U . ∂u (0) = ∂x1 ∂u (0) = ∂x2. L Y,]lX\lf _aL*Ya]J_[\Yaf @. S . Lt fq\lXa`ns8LE_ ,. ER (u) Z Er (u) =. 1 π. Z. @\@. q`qL*\ fp_. kp\ls. Er (u) r<R Z π ∂u ∂v u ds = u(r, φ) (r, φ)r dφ. ∂r ∂r Γr −π. π. Z. π. NPf L fplXaL*_[pLYaHlkhY Kk¨V:xLL £8jJ_[LEX[XaLEskqX Er (u) =. A.. yh. Z Z π 1 1 ux1 ds, u(R, ψ) cos ψ dψ = πR −π πR3 ΓR Z Z π 1 1 ux2 ds. u(R, ψ) sin ψ dψ = πR −π πR3 ΓR. ,ÏYaLE_PXr]llXZY[`®Y[]8Ya`n\J q} . S ?@. −π. u(r, φ). −π. Γr = ∂B(r). WV. yJ{. HlL*_[L GHlL*_[L fp_[L yp %Ya_ukh\lXrfp_[KXBY[f @. . yp. 1 ∂ H0 (t, α), 2 ∂α. . H0 (t, α) = log(1 − 2t cos α + t2 ).. . π. y. π. ∂u ∂ r (R, φ)u(r, φ) H0 ( , φ − ψ) dφdψ, ∂ψ ∂φ R −π −π Z π Z π 1 ∂u r ∂u =− (R, φ) (r, φ)H0 ( , φ − ψ) dφdψ. 2π −π −π ∂ψ ∂φ R. 1 Er (u) = 2π. . yq. r ∂u (R, φ)H( , φ − ψ) dφdψ. ∂ψ R. H. H(t, α) =. . Z. Z. S. Íé =Í.
(119) { j4GkpHlX[LXYa]lf \lYatHJY[`Lfq\mn` K `Y `YaH HlkqXBfq_ fp8Yukh`n\J`n\Jkh\ `\dY[L* p_ukhlmL Xr`n\J q]Jmnkp_a`YZVMfhYaHJLYZVWj4L lXaf L Kk¨V ypy fpNPXaHl f kpmfq%]l\ XaL LYakpHJ\dL qY%kq`qY[XrfL*VW\=KMY[ k j8p`®YY[Lfh`Ya\d`c`ntYam BfML tu£8kpHljltEkht kp\lfq\l p]JXrL`n\dfpYBYa\¡HJYaHls8LLLE#_a`nkp`n\ pml]JL;s:]JLLE`nfp\l\ t Lqfhy 6&Y[HJNLkhKMXrïK\ L*mnkpVpfqm Jx_[s8`HJ LEfp_mnL Yaf`nXL*Ya#HJkpL m]lPXakhfpX[YamnXaL]8]JYaYaKM`nHlfp\``n\JX fhtu HlYaHlkh\JL q#L khmn]JLL `\ fq\ fp\ yh} Y[HJL*\F lkpX[Xr]lK `n\J L Hlk¨pL y fq_ PL*_[L y kp\lsYaHJLE_aL*fp_[L yhz T8]JlXrYa`Ya]8Y[`\J MY[HJ`cX,t fq_a_[LEt Ya`nfp\`n\dYaf yJ{ p`npL;X
(120)
(121) ! "#$ %& $' (). H0 (1, α). @. u. α=0. r→R. ER (u) = −. @ @ =@.
(122) G. ΓR. 1 2π. Z. π. −π. Z. log |α|. π. −π. @. ρ < 21 R. B(R),. )@. 7-. vρ = u. . B(ρ). 3. )@. vρ. ΓR ,. ?@. ∂vρ =0 ∂n. ρ2 sρ = r. . ∂sρ ρ2 =− 2 ∂r r. ρ2 r2. π. $U . L . ∂v ∂v (0) cos φ + (0) sin φ , ∂x1 ∂x2. . Z. u(r, φ). pfNP f yh jlkpX[Xa`\J YafYaHJLm`nKM`®Y `YaH. . LPfq8Y[kp`\`\ d`nL }p GHl`nXkpmnf ,kh\4X s]lXYuk Y[qfL t fq\lXr`cs8LE_ YaHJLfqmnf `n\J khjJjJ_[f¨£8`KkhYa`nfp\= L Y X[k#Ya`cX lLEX q `n\lXa`nsJL }l{ GHlL*\ %LKk¨Vt*fp\lXa`nsJL*_ `\k J£8LEss8fpKkp`\ kp\lskpsJs 4fq]J\lsJkp_aV:YaL*_[KX }q @. @. . @. Ω \ B(ρ). r→R. ρ2 δER (u) = − 6 πR. @. G. kh\4sMY[k d`n\J `\dYaf kptEt fp]l\qY%YaHlkhY. "Z.
(123) G. ux1 ds. ΓR. E(ρ ; uρ ) =. H@. uρ. E(ρ ; uρ ) =. Î ÉÉÂÂ ]. ).. 1 2. Z. . ∂v ∂v (0) cos φ + (0) sin φ . ∂x1 ∂x2. ∂v ∂v (0) cos φ + (0) sin φ r dφ ∂x1 ∂x2 −π Z π Z π 2 ρ ∂v ∂v =− (0) (0) u(r, φ) cos φ dφ + u(r, φ) sin φ dφ . r ∂x1 ∂x2 −π −π. δEr (u) = −. . Γρ ,. vρ = v + sρ + o(ρ2 ), ∂vρ ∂v ∂sρ = + + o(ρ2 ), ∂r ∂r ∂r. I. . ∂u ∂u (R, ψ) (R, φ)H0 (1, φ − ψ) dφdψ. ∂ψ ∂φ ∂u/∂n D E. ∆vρ = 0. r > 21 R. U. 2. +. Z. ux2 ds. ΓR. uρ. 1 2. Z. Ωρ. u=v 2 #. .. 7@. @ . .. S . k∇uρ k2 dx.. Ωρ = . ΩR. 1 1 k∇uρ k2 dx + ER (uρ ) + δER (uρ ) + o(ρ2 ). 2 2 ΩR. .
(124) { G HlHl`n`ntuXH KMYaL;HJkhL\llX ]JY[m Hlk#L*Y\J`nLE\ _a qåVkpt `nY X,% p`nLpL*H4\k¨pWL V Yafs8f `YaHXafpKMLM]J\lt Ya`nfp\ X[k#Y[`nXrVW`\l d `n\ Ffp_ FL.
(125) 2[ XF [. @. @. 2G. @. S . wR. ΩR. Z 1 E(ρ ; wR ) = k∇wR k2 dx 2 ΩR Z π Z π ∂wR ∂wR 1 (R, ψ) (R, φ)H0 (1, φ − ψ) dφdψ − 4π −π −π ∂ψ ∂φ "Z 2 Z 2 # ρ2 wR x1 ds + − wR x2 ds . 2πR6 ΓR ΓR. }p . [YGHJHlLL G`nl\ HJ_ulLXZ]JY=_[LEYL*\ljJt _[fLLEY[XafpL*L*F_[\dKMYaY[HlkhXFLYa p`nfpXr`nKp\ L@khL*}pmn£8 kqHJ%tfqkhYFmmnLqL*f £8,jJ,X%_amnL;]l=XaX%fhXa`nYafpfYa\HJYaLE_ufpKkh_=\lY[XrkhHJ_[fpLBL_[`nK bd\d]lY[kqYakpHJts8YL_uk#s8Y[Y[fqkq`nKX tPMkhfp`n_[\_[fpKK X ¨Xafhkpfp\lmnsd`nYa\JHl L&YaYaHJHlL `_uxsfpkh]JjJ\ljlsl_akhf¨_[£8V `nKM#kpkhmYa]JL;LX jl_afq JmJLE]JK Y fp_`YaHkhL*jl\JmnLEkq_at qLBV L;bdjl]lkpk#_[Y[kp`KMfq\L Ya`n_[\`n*#LEkhs¡_[`nWkpVJmnLBs8fpïK\¦kpåkp`\ t YE Xr`nY[\lfPt*LY[HJY[LHJXaL kplK _uLXrYjJY[_[L*fp_aJKmnL*XK _aLE`n\jJ_[Y[LEHJXaL L*\dJYP£8LEL*s\lL*s8_[fq pKV khfp`n\ _ Y[sJHJ`nX[L t _[L Y[Hl`fp;k#mnLYa`nfpfh\ fh L K k¨GV HlLE`nXMpL*_[\ LEXak¨]JqmY[fpX `cs _[]lfpKXr`n\Jfp 4XrLE_a#k#khY[`\lfqs \ Y[Y[HJHlL¦k#Y_[L*fpKM]JY[L;XaXr`nHJs8`nL \J fp Y[HJ`n\ Lfq]J_[\ls8LEt Ya_`nfpYa\f pL Y qt*fp`q`n\lL*\t `cs8dL;V X `YaH p`npL*\WV YaHJLXafpmn]8Ya`nfp\fhPY[HJL¡4fq]J\lsJkp_aV#kpm]JLjJ_[fpJmnL*K `YaH J]Jm LE\JL*_[ pV } L*Gkq£Wt*Hlt*YaLL*fp_[]J\l\dkpLEY[km X,%Yaffqfp_ `_[\dK Yakh_[\lf8fpsMs8%]l]lY[t*XaHJ``n`n\J\JX jJ4Xr_afqfqfq]JJ]J_umn\lL*t sJKLkpY[_aL*`cVMX _[K$t LEfqkq\lt*Xr`ns8fpm\l`V¦Yat*`nfpfpL*8\d\lYaYuX&_ukhfpk#`n\J\Y[LELEs¡s fqW \VYaY[XHJk L W`JL,\J£8 ¡XrLELE#s¦Lpkh dt _[K `c`n_[k#f8t*Yas8m`nLfp` 4\tEfhk#Ya`nkhfp\l\Msfpjl3khl_u]Jkhm KM (kpL L*Y[sJ\l_as8L*`n*_[`n\J pL;sV WV G HJLmckpXrYfq_aK]JmnkKMk¨V xL L*£8jJ_aL;XrL;s khmcXaf¦`n\kh\lfhYaHlL*_fp_[K L YM]lXsJL*\JfpYaLdV YaHJL LE\JL*_[ pV:sJL*\lXa`®YZVk#YYaHlLj4fq`\dY YaHlL ]J\ltY[`fq\ `nXHlkp_aKMfq\J`nt`n\ JYaHJLE\ Y[HJLL £8jJ_[LEX[Xa`fq\lX%fp_,s8LE_a`n#k#Ya`npL;X .. @. 7-. ΩR. Y@. Ω. Ωρ. C3. .. .. ΩR. . Ω B(R). @. w. 1 E0 (ρ ; w) = 2.
(126) G ?@. wR. G. ρ. @. ΩR. )@. @. @. . C@. Ω. . Z. ρ2 k∇wk2 dx − 2πR6 Ω. G. "Z. wx1 ds. ΓR. ∂Ω. .
(127) G. 2. +. Z. @ ).. wx2 ds. ΓR. wR. G. 2 #. S. .. E0 (ρ ; w). 2.. 2G. ΓR. ρ. eu (x). x. eu (x0 ) = k∇u(x0 )k2 .. 3. u. 1 u/1 (x0 ) = πR3. B(R). Z. u · (x1 − x1,0 ) ds,. 1 u/2 (x0 ) = πR3. Z. u · (x2 − x2,0 ) ds.. kp_aLL £JkqtYE ï\W`L fpFY[HJ`nXE 8fq_aK]Jmnk } tEkh\4L_[L _[`®YaYaLE\ `n\YaHJLL;bq]l`#khmnL*\dYfp_[K 3. W@. ΓR (x0 ). S . 1 E0 (ρ ; w) = 2. @. Z. ΓR (x0 ). }dy. . 1 k∇wk2 dx − πρ2 ew (0). 2 Ω. Íé =Í.
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