Analyse complexe et problèmes de Dirichlet dans le plan : équation de Weinstein et autres conductivités non-bornées
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- Φιλύρη Αλεξόπουλος
- 6 χρόνια πριν
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1 Analyse complexe et problèmes de Dirichlet dans le plan : équation de Weinstein et autres conductivités non-bornées Slah Chaabi To cite this version: Slah Chaabi. Analyse complexe et problèmes de Dirichlet dans le plan : équation de Weinstein et autres conductivités non-bornées. Équations aux dérivées partielles [math.ap]. Aix-Marseille Université, 03. Français. <tel > HAL Id: tel Submitted on 9 Dec 03 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.
2 ttr é r t èq r t r r t r rs rs té é té t é t q s ré ré tr t é t q s t r t q t à t s s r t r ér t é t q s t r t q rs rés té t s t q t r é r tr s t r è s r t s éq t st t tr s t tés r é s r t r t ès r t r t rt r t r t ès r r rès s s r rt rs t t P rt t r r t r t rt r t r r r r t r r r Pr ss r rs rs té r t r r r r Pr ss r rs té r é s Prés t t s Pr ss r rs té r t r r t r r r P t q rt r té t îtr ér s rs rs té r t ss Pr ss r r t r r s îtr ér s rs rs té t r
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4 és é éq t st à ts s st éq t ré ss t s P t t s à étr P q s é r t L m [u] = u + (m/x x u = 0 ù m C tt éq t t r t t t r é s t r s s r m = r st rsq m = é éq t r st é r sé éq t r tt t r t t s s t s éq t s st é ér s s rés t ts s r m R s m C s r ss s t s s t s t s ér t rs st t rs st t s s s s r tés s é tr r r r s P s r t H + r m < r t é rè é s t s P s s s r s q q s r m C t s é étr r rt èr s t t r r s r é s r s r t rs r m C q s t s s P t r s t s r ss é s r èr t s s è r èt r ét q s sé r té s r s t r s r r r tt t r r s P s s r sér t rsq m R tr q tt st ê s s3 s rt s à r r r tr q s è rt r ét q st à s s s r té r t s r s s P s r r r t H + s s ù r ètr m st t r r t s r rés t t s s t t s r rés t r è rt s r s r s r à ts s r té t m s r s t t r r ç t s s é s r t t s P tr ss q tt ét s q à t s s s s t H + à r ré r s r èr rt ét ss t s q s P s t s t s s r s s s t s éq t w = αw α L r r < rés t t q s êtr t t r r s r été t st t s ré r té r s r té é s t s t s r s s r e s F s s rt s t ès s ré r tés t ù F st t r t ré r q r q t à r étr t q tt ss t s s s r t q r = P s t s t tr s t s s r s t s s t s éq t tr é rés r è r t à é s L p éré s s r s s ss s r s éq t s t t té à t t rt t à s W,
5 str t st q t t ts s t q t r s tr t t s P r tt s L m [u] = u+(m/x x u = 0 r m C s q t s s rt r r t s s t r r t t r m = r t s m = t r 3 r st q t s s t t s t s t st q t s r r 3 r s ts r m R t m C t r ss s t s t s r st r t rs t r st t s r s r t s t r r s r r P t r t H + r m < st s s t t r r t P r s r m C rt r s r r t s r r s r m C t t s t s r P q t t r s ss t r t s rst s s t t t r sts q s s r t r s s r r s s r m R s t t t s s s3 s s s tr r r s t s rt s t t s t t r r r s r P r r t r t H + m s t r s r r s t t s r t s rt r t r s r t t s ts r t t r t m s r s t r t r t t t t P q st t st rt st ss t s s t P t s r t s s t s t q t w = αw t α L r r < t t rs s r t r s t s r t s t r e s F r s r r t ss t s t r F rs t s r t t r t r r t s r = s t t t s r t s s t s t t t tr q t s s ss r t r s t s t t L p r t r s tr t t q t s s ts r t t s W,
6 s t èr s és é str t s t èr s tr t t s t s s ér t rs st à r ètr é s t s t t s à s étr Pré r s s t s t s t s t st t s s s r t é r s str t s t t s t é t s t s t s s s r té r r m t r r t t s t s r m C é rè é s t s P és t ts ré r s é rè é s t s P r m C r P ss r H + q Re m < t s r s r é s r s s xoy t s P r é s r s s t s r ss é s r èr t è s è s s s3 P r m ré é t t r r été s s3 s s t s t r r r s és t r è r t r ét s tr t éq t r éq t s r sq té Pr è r t r s P tr t t r rés t ts P r s t r s ér t s r é s r tér s t r r è rt
7 P r étr t q s t s s r s à s t r t q tr t t t s t é t s Pr rés t ts Ps r t s t t r t t r
8 tr t ét s ér t rs st L m := +(m/x x r tr st rt t s q r s rs é è s s t é sés r s P t t s à étr P q s t s t éq t st P r t r r èr s P r ètr m = rr s t s t s éq t r st é r sé éq t q st t sé r rés r s éq t s st s P r ètr m = s t r és t s 3 sé r s t r é r s r r t r ï P s ré sé t à rt r tr t s é s ét q s s r r t èr r r tér s r t èr s r ï u s t éq t (/x u = 0 rès t ès s étr r s r st q ï st r q s t r st t t rs s s s st s st s t éq t r r q st éq t é r s q é r t s t s s t s s s s rt s t s r q s s t r s s t s q s s t s s t s P m = t é étr q s st s r s t t s r ts és tr é r st r s t r ss s s é t s r è r t ét t é s s s r s u, n u s r r tér r t u st t s r r tér r r st t ét t t t é ér té r str r P u s t t r y u,? n u u, n u O x L (u = 0 t r è t êtr tt q é èr t s t ét s st q s r s tr t rés t r è r t r éq t ér é s rt s
9 té r r t à r t t à rés t rt r ér t r é r rt r è rt ét r r t t r t t r s s r s t r è r t sé ét s s t à r è rt st r étr r s t s s t t r s s P q t tr s t r r tr tt t ès ù ét s ss s r s r sq té éq t ré ss t s t s s t q s w = αw α L r r < tt t s st s é ér t s P s é s ss s r rs s q s r q t é r é st té t s tr tr q s t s tt t s tr s s r r té T t tr rt q r è r t r éq t w = αw ù s rt ré w s L p (T s t q s G p α s rés t ts é ér s t t s r L r (D (r > ré t r t ss t r t rt t s s tr t st r èr s r t q s r = st s éré P r éq t t té σ expw, st à r q σ s é r t σ = e h h s s s t s L t s ér é s r r s s s str t s s t r s t s L rr s w f Ref ù w ér éq t w = αw α = logσ / f ér éq t tr é f = ν f ν = σ t ù u = f ér éq t (σ u = 0 +σ s s t s r è r t à ts r és rt r éq t t té s éré st s str t t t q r σ = e h h W, st s ré r st t str t t s t P r rs t s t t té str t st ss r tré s tr sq s P ét és s t s t s s H + = {z C, Rez > 0} (x m u = 0 t q t z = x + iy H + x m st ré r é t ré r é t r st t str t t s t
10 tr t s t s s ér t rs st à r ètr é s t s t t s à s étr s tr s r s ét r ss ér t rs ér t s L m = + m x x m C é s s r r t H + = {(x,y R, x > 0} = {z C, Rez > 0} tt ss ér t rs és ér t rs ré ss t s P t t s à étr r ér t rs st été ét é èr ss 3 t s s s s m N m R s s s tér ss r s s tr s t s ù m C r s q q s ss s ù s rs m s r t r str t s à N Z R s s t s éq t s H + q s r s P r q t ré ss t s P t t s à étr L m u = 0 (EPSA s r t é s P t t s à étr ré é P s tér ss r ss s t s éq t L m u = δ (x,y ùδ (x,y és ss r (x,y H + t ss s t s éq t L m u = g ù g és r t ré èr é s rt H + s tr s r str r t r t s s s r rés t ts s ét t r t t s s s ér r st à r ér t rs n k= x i + m x n xn
11 P P P P P P s s H + n = {x R n, x n > 0} r r à r tr t tt ss ér t rs été st s ù ét s m N t t rt r r s r s P t t s à étr q s r t èr t à H + s t π u(0, 0 sin m θdθ = π π π u(re iθ sin m θdθ (PM t r ss s t t t r s t s ss t t r t t r té r s t q s r t s r é s ét t r rs tr s P r m N t s t s r q s R m+ q s r r s s r s t s st t 3 st ét ss t r rr s q s r r s tr s P rr s ts à m t r r s ts à m r s t s é s t r ss s t t t s r é s r s rs ré s m t t tr P t s éq t s r t rs s t s t s s r t t r t P ss é ér s t P s tér ss ss r ts s P r st s tér ss ss r r étés é t s s r tés s P s t r r rés t t str t s P s t s s r q st é ér sé r r t s P r rs é r r s s t s t s s éq t s t q s à ts t q s à s t s tr t s s ssé r r s r r q ré q é ér sé r t q râ ér t rs s z t z s t r s r st q s ê èr q t r q t r q st rt ré t r r s t r t t t r r t q s s t s éq t s t q s t rt r s P s é r t s t s q é à t r q q t tr q é à t t r ss q q s t s s é r t t t t r t s q s t t à s t s r é étr q s r à ér t s r t r s s r tr t très tér ss t tr P s ré t à s tr s s s r sér s t t ér t r ˆLu = u+a x u+b y u+cu t ét r tt sér ss t à éq t t3 s rt s èr s P è s r m 0 r rés t t s s r té r s s t s s éq t s t rt r s t t rsq s r é q s r q r s t r è r q s st à r str r P s H + à rt r s ss s s rs t s s ér é s r r rt r st s t q s r s r rt tt t ès st ét é s s ù P st é s r s x s st ss ét é s s ù
12 P P P P P P st H + \[0,a] {0} a > 0 s tr t r r tr s q s t r q s s s rés t t rés t ts q ê s s s t s t t r m ré st rés t t t t t s t s t q s ér t s s t sé s s s tr ré é t tés s rés t ts r s rs s m s t r tr à tr ss s rés t ts st ê s rés t ts s r r s t rés t t r st t é rè é s t q r q t t P s r H + st à r r Ω \ K ù Ω st rt H + t K st t Ω êtr s P s Ω tr s H + \ K t t t rs r H + s t s ss t é rè t q r q s P s r H + t rs à t s r s r é s rs st t q t rq s q rés t t st s é t r P st éq t t q é é éré sq st t t té t rs s s r é s s rés t ts r r ét t rt t s r s rs rt èr s m m = ± tr ss r r s t Pré r s s t s t s t s t st t s s s r t é r s str t s t s s t s ér q s s r t à rs s s t t s t t r H + = {(x,y R, x > 0} r t R Ω st rt R n n N D(Ω és r s s t s C à s rt t s r Ω s rt t f é s r Ω s r té suppf := {x Ω, f(x 0} P r K t s s Ω t r D K (Ω s s ϕ D(Ω t s q s ϕ K rsq u st t é t ér t s rt Ω R n s ér é s rt s u s r t t r à t r té s u x i xi u r u xi i,n rsq α = (α,...,α n N n st t t r α = α + +α n α := α x αn x n = α x α x αn n r q str t T s r Ω st r é r s r D(Ω t s s s s t r t t t K Ω st N N t st C > 0 t s q ϕ D K (Ω, T(ϕ C sup α N n α N sup α ϕ. K t r r s T(ϕ =: T,ϕ t r D (Ω s t r s str t s s r Ω P P
13 P P P P P P a Ω é t str t δ a r a r ϕ D(Ω, δ a,ϕ = ϕ(a. f : Ω C t t té r té r s r t t t Ω r r rt à s r s dm n é t str t T f ss é à f r ϕ D(Ω, T f,ϕ = fϕdm n. r q s α N n st t t s T st str t s r Ω ér é α T st r é t str t s r Ω é r Ω ϕ D(Ω, α T,ϕ := ( α T, α ϕ. s té r t s r rt s tr t q s T f st str t ss é à t ss C N t s α N n st t q α N rs α T f = T α f. f C (Ω t T D (Ω é t str t ft r ϕ D(Ω, ft,ϕ := T,fϕ. t L ér t r ér t s r Ω r L = a α α α N ù N N s t ré é t st t é s r s t s α r α s t t q N t s t s a α s t s t s s C (Ω P r é t s T st str t LT s r str t LT = α N a α α T ér t r t L L s s s str t s ér t r q à str t T ss str t L T = ( α α (a α T. α N r rq q s f,g s t t s D(Ω rs t t T f f t T g g Lf,g := LT f,g = T f,l g = f,l g. t t t a Ω t L ér t r ér t s r Ω s t t L s r Ω a Ω t t str t é ér q T a t q LT a = δ a ù é té ré é t st é té s s s str t s s r Ω tt é té s réé r t ss ϕ D(Ω, ϕ(a = LT a,ϕ = T a,l ϕ. P P
14 P P P P P P s s st s s r t q s tt é t s s t s t s rt Ω rô r t s r L = s R st q s a = (a,a H + T a = 4π ln( (x a +(y a st s t t s D(R a str t U a é r U a = T a T ( a,a = ( 4π ln (x a +(y a (x+a +(y a s r rs s t t a s D(H + s s s D(R r s ϕ D(R t s q s ϕ D(H + U a,ϕ = ϕ(a ϕ( a,a U a,ϕ = ϕ(a ϕ( a,a = ϕ(a. rt r s r a Ω T a st s t t L a s Ω t s g D(Ω st t q g = L(ϕ ϕ D(Ω rs a Ω, ϕ(a = T a,g. t s s a Ω, ϕ(a = δ a,ϕ = L T a,ϕ = T a,lϕ = T a,g. s s t s t s r tt t rés r éq t Lϕ = g s D(Ω s g D(Ω t t s t é t s m N s R m s r té m s s t q m = P r m C L m st ér t r é s t r t t t u C (H + t r t t (x,y H + L m u(x,y = u(x,y+ m x u x (x,y. t s r r s t t s t s f(x,y = (f (x,y,f (x,y st t t r ss C s rt R t à rs s C rs div(f := f x + f y. ê s f : R C st t s r ss C s rt R t à rs s C rs f := ( f x, f. y P P
15 P P P P P P s t t s ér t r L m st ér t r q à t u C (H + ss t é s r H + r L m u(x,y = x m div(x m u(x,y. rés t rè r3 q s u st t é s rt s t H + t q div(σ u = 0 ù σ : H + R + st ss C rs st t v s t sstè éq t s é ér sé v x = σ u y v y = σ u x t v ér éq t é div( v = 0 tt r rq st t q σ s r s L m r m C ér t r é L m t L m ér t r é r r t t t u C (H + t r t t (x,y H + L mu(x,y = u(x,y x t sq ( mu(x,y = u(x,y m x x u x (x,y+ m x u(x,y L m u(x,yv(x,y dxdy = u(x,yl mv(x,y dxdy u, v D(H +, H + H + L m st ér t r t L m s s s str t s s r H + tt é t é s H + s tr s s sé t s rt Ω H + s s ù s t s q é s é t tr s r s q x t y é r r L m,x,y L m q s q s ér é s rt s s t é s r s x t y t q s tr s r s s t s éré s ét t é s s s é r s ér t rs R m S m t D ç s t P r u D(H + é t S m u t D(H + r (S m u(x,y = x m u(x,y. P r u D(H + é t Du t D(H + r (Du(x,y = u x (x,y. s ér t rs s é s ér t r s t s t Pr s t S m L m t L m D L m t L m q t r q S m L m = L m S m, L md = DL m. P P
16 P P P P P P Pr t r èr r t s (S m u x = mx m u+x m u x, (S m u xx = m(m+x m u mx m u x +x m u xx, (S m u yy = x m u yy. t t rs L m S m u = (S m u xx +(S m u yy + m x (S mu x = x m ( u m x u x + m x u. L m S m u = S m L mu è r t s t s ér t rs t L mdu = u x +m x x L mdu = x ( u x x t t t t ( u+ m x r é s r t s s rq s u = DL m u, x m C S m t L m S m s t s ér t rs t ts S m = S m t L m S m = (L m S m s s rés t t s é ér s r s s ér t rs L m t L m s ér s r σ : Ω C t ss C q s s ér t r s r C (Ω é r r u C (Ω P σ u(x,y = ù Ω st rt R rs σ(x,y div(σ(x,y u(x,y, ( ( Pσ = div σ. σ t s u,v D(Ω s s t s t ér t s s s str t s P σ u,v = Ω σ(x,y div(σ(x,y u(x,yv(x,ydxdy ( v = σ u dxdy Ω σ ( ( v = udiv σ σ Ω P P
17 P P P P P P = u,p σv é t S σ ér t r q à u C (Ω ss (S σ u(x,y = σ(x,y u(x,y. rs S σ P σ t Pσ ù P σ = div ( σ ( σ sq èr é t s s S σ P σ = P σ S σ s s é tés ré é t s s t r s r t èr s r ss t s r s t s u v s q r ér é s s t à s rt t s rt Ω s s s s r s t s s t s q tr s s tr st Pr s t Pr st Ω st rt r t t t H + t s u : Ω C st ss C rs r t t m C L m u = x m L m [x m u]. Pr t α C t s s u(x,y = x α v(x,y t u x = αx α v +x α v x, u xx = α(α x α v +αx α v x +x α v xx u yy = x α v yy. ( L m u = x α v + m+α v x + α(m+α v x x t s ss t α := m t t é té s s ù t t m st t r s t tr t ér t r T m q à t u é s rt Ω H + ss t v é s r s {x R m+, ( x +x m+,x m+ Ω} r v(x,...,x m+ = u( x + +x m+,x m+. ré é t é t s rés t t s t Pr s t P r u ss C s rt r t t t H + t m N m+ (T m u = T m (L m u Pr P s s v = T m u P r t t i,m+ v x i = x i u x +...+x x, +m v x m+ = u y t v x i ( x i = x +...+x ( +m x +...+x 3/ +m u x + x i x +...+x +m u x. P P
18 P P P P P P m+ v = u+ m u x +...+x x, +m ù rés t t s r s t s t s r ttr r s s t s t s r L m t L m r m N t t r s r m Z r s t s st t s s r ss s r ttr t tr r q s r ss s t s r ss t t s s t s t s L m t L m r m C t s t s s s r té r r m t r r t r q s (x,y R δ (x,y st str t é r ϕ D(R, δ (x,y,ϕ = ϕ(x,y. t m t r s t Pr s t rt t s t m N P r (x,y H + t (ξ,η H + π E m (x,y,ξ,η = ξm π θ=0 [ (x ξ +4xξsin ( θ sin m θdθ ] +(y η m/ st s t t s H + r ér t r L m,ξ,η H + q s s s s str t s s H + t é (x,y L m,ξ,ηe m (x,y,ξ,η = δ (x,y (ξ,η. s s (ξ,η H + st é rs s s s str t s s H + L m,x,y E m (x,y,ξ,η = δ (ξ,η (x,y, q s q E m st s t t s H + ér t r L m,x,y t (ξ,η H + é Pr R m+ st t m N r q s t t s E(x = mω m+ x m, x Rm+, s s s str t s m+ E = δ 0 ω m+ ét t s r s èr té R m+ r t t t v D(R m+ v(t,...,t m+ = dτ dτ...dτ m+ m+ v(τ mω m+ τ R m+ ((τ t + +(τ m+ t m+ m/ ù τ = (τ,...,τ m+ P P
19 P P P P P P q t tt r t à v = T m u ù u D(H + s t s râ à r s t r t t (x,y H + u(x,y = (L m u( ξ + +xi m+,ξ m+ dξ dξ m+ ( mω m+ R m+ (ξ x +ξ + +ξm+ +(ξ m+ y m/ P r s r tt r ss té r s èr s r é s r s é r q s s t s ξ = ξcosθ ξ = ξsinθ cosθ = ξ m = ξsinθ sinθ m cosθ m ξ m = ξsinθ sinθ m cosθ m ξ m+ = ξsinθ sinθ m ù ξ = ξ + +ξ m+ θ m ] π,π[ t θ,...,θ m ]0,π[ r s ét r t tr é r stè r é s st t t rs r t t (x,y H + u(x,y = E m (x,y,ξ,η = ξm mω m+ r s t q I := π ξ m sinθ m sin θ m sin m θ η= π θ m= π ξ=0 π L m (u(ξ,ηe m (x,y,ξ,ηdξdη θ,...,θ m =0 θ m= π st s r s èr té s R m sq π π ω m = S m dσ = θ m = π t E m s é r t E m (x,y,ξ,η = ω mξ m sinθ m sin θ m sin m θ dθ...dθ m (ξ xξcosθ +x +(y η m/ π θ,...,θ m =0 sinθ m sin θ m sin m θ dθ...dθ m dθ m θ,...,θ m =0 mω m+ π θ=0 r t s t t q ω m = πm Γ( m π E m (x,y,ξ,η = ξm π θ=0 t t t r ss sinθ m sin θ m 3 sin m θ dθ...dθ m. sin m θdθ (ξ xξcosθ+x +(y η m/ ( (x ξ +4xξsin ( θ sin m θdθ +(y η m/ P P
20 P P P P P P s r t t (x,y H + t r t t (ξ,η H + ( m x E m (x,y,ξ,η = E m (ξ,η,x,y ξ t r r s t S m L m t L m rs s s s str t s ( (x m ( m x L m,x,y E m (x,y,ξ,η = L m,x,y E m (ξ,η,x,y = L ξ ξ m,x,ye m (ξ,η,x,y, t t r r L m,x,y E m (x,y,ξ,η = ( m x δ (ξ,η(x,y = δ (ξ,η, ξ P r m t r é t r s t ré é t é r st s r t t r s t s t Pr s t rt t s t m Z \ N P r (x,y H + t (ξ,η H + ( m ξ E m (x,y,ξ,η = E m(x,y,ξ,η x π = ξx m sin m θdθ [ π θ=0 (x ξ +4xξsin ( ] θ +(y η m st s t t s H + r ér t r L m,ξ,η t é (x,y H + t st s t t s H + ér t r L m,x,y t (ξ,η H + é Pr r t t m N u D(H + t (x,y H + u(x,y = (L m ue m (x,y,ξ,ηdξdη, (ξ,η H + t r r st r s t u(x,y = ξ m L m (ξ m ue m (x,y,ξ,ηdξdη. H + t t v(x,y = x m u(x,y t t x m v(x,y = ξ m (L m ve m (x,y,ξ,ηdξdη, H + s r t t m Z\N v D(H + t (x,y H + s t m = m ( m ξ v(x,y = (L m v E m (x,y,ξ,ηdξdη. H x + r s t st P P
21 P P P P P P t s t s r m C s r s s r r tr r q s r ss s ré é t s é ss t t s s t s t s s ér t rs L m r m s t r r t s r m C P s ré sé t s m rs E m = ξm π π θ=0 t t s q s m < rs t E m = ξx m π π θ=0 sin m θdθ [(x ξ +4xξsin ( θ +(y η ] m/ sin m θdθ [ (x ξ +4xξsin ( ] θ +(y η m s t t q s t E m és r t rs r rr s t s t q m m < Pr s t P r m C t (ξ,η H + és t r (x,y H + é (x,y H + \{(ξ,η} L m,x,y E m (x,y,ξ,η = 0. (ξ,η H + \{(x,y} L m,ξ,ηe m (x,y,ξ,η = 0. Pr P r té r s s t s f m (x,y,ξ,η,θ = [(x ξ +4xξsin ( θ +(y η ] m. P r tr r r èr é té r s t s t tr r q π θ=0 s s ér é s t f m L m,x,y f m (x,y,ξ,η,θsin m θdθ = 0. x f m = m (x ξ+4ξsin θ [(x ξ +4xξsin ( (= m(x ξcosθf θ +(y η ] m + m+ t xx f m = m [(x ξ +4xξsin ( θ +(y η ] m ++ s yy f m = + m ( m + ((x ξ+4ξsin θ [(x ξ +4xξsin ( θ +(y η ] m + m [(x ξ +4xξsin ( θ +(y η ] m ++ P P
22 P P P P P P t t f m = + m ( m + ((y η [(x ξ +4xξsin ( θ +(y η ] m + m [(x ξ +4xξsin ( θ +(y η ] m ++ + m ( m ((x ξ+4ξsin θ + +((y η [(x ξ +4xξsin ( θ +(y η ] m +. r ( (x ξ+4ξsin θ [ ( ] θ +((y η = 4 (x ξ +4xξsin +(y η 4ξ sin θ f m = r rq t q f m+ θ m [(x ξ +4xξsin ( θ +(y η ] m + m(m+ξ sin θ [(x ξ +4xξsin ( θ +(y η ]. m + xξsinθ = (m+ [(x ξ +4xξsin ( θ +(y η ], m + f m = m f m+ +m ξ x sinθ f m+ θ t r té r t r rt s t t π θ=0 f m sin m θdθ = m π = m x π θ=0 = m x θ=0f m+ sin m θdθ+m ξ x π θ=0 m(x ξcosθf m+ sin m θdθ π θ=0 x f m sin m θdθ, f m+ θ sin m θdθ ù rés t t s m r st t t t s r s m < è é té r s t é é t t t q S m L m t L m r s t r s t s t rt t s t s rès r s r té t s r t r tr r q s t t t s s t s t s r t t s s rs m C t s s t s rs t èr s rt r s tr s q rt t s s t s t s st r rt t s s t s t s t st r s ér t rs t q s s s s st s s r t q s r tt r s t s st t s q s s s té r s t q s P P
23 P P P P P P s t s st t s t t t é é t r s t s t t é rè r é t ér t s s st t s ss q s q é t st t s t s r é étr q s t f t g t s é s s r rt Ω R t à rs s t s t (x,y t ér t à Ω t q f st éq t à g (x,y t é r t f (x,y g f(ξ,η (ξ,η (x,y g(ξ,η s ér f g st é t g (x,y st à r r t t ε > 0 st α > 0 t q r t t (η,ξ Ω s (ξ x +(η y α rs f(ξ,η g(ξ,η ε g(ξ,η Pr s t t m C P r (x,y H + é E m (x,y,ξ,η (ξ,η (x,y π ln (x ξ +(y η Pr r tr t r s ù m st ré s ér r é à s s t s r ss s t r s t t = π E m (x,y,ξ,η = ξm π ( m ξ π d θ=0 π θ=0 sin m θdθ (+ksin θ m/ rq s q q d 0 k + s s r s t s t Pr s t rsq k + t m C Pr π θ=0 (+ksin θ m/ sin m θdθ [ (x ξ +4xξsin ( ] θ +(y η m/ sin m θdθ P s t u = sin θ tt té r t r 0 m 0 u m ( u m du (+ku m/ u m ( u m du ( k +u m/ 0 t r r t k + 0 u m d = (x ξ +(y η t k = 4xξ d. k + = m k m/ 0 u m du ( k +u = m/ 0 (u m/( ( u m m lnk. km/ du = t r u = k s t s 0 u m ( u m du ( k +u m/. u m du r s k ( k +u = th m tdt m/ 0 u m ( k +u m/ ( ( u m du 0 ( u m P P u du
24 P P P P P P th m t t rs q t + t q 0 dt r é t q q k + r s k 0 r s t rés t râ à r s t s s s m < st s r r s k th m dt dt = r s k k + 0 k + lnk. E m (x,y,ξ,η ( x m d 0+ π d m lnk km/ d 0+ π lnd. é rè t m C P r (x,y H + t (ξ,η H + π E m (x,y,ξ,η = ξm π θ=0 [ (x ξ +4xξsin ( θ sin m θdθ ] +(y η m/ s m ( m ξ t E m (x,y,ξ,η = E m(x,y,ξ,η x = ξx m π π θ=0 sin m θdθ [ (x ξ +4xξsin ( ] θ +(y η m s m < st s t t s H + r ér t r L m,ξ,η t é (x,y H + q s s s s str t s s H + L m,ξ,ηe m (x,y,ξ,η = δ (x,y (ξ,η. s s (ξ,η H + st é rs s s s str t s s H + L m,x,y E m (x,y,ξ,η = δ (ξ,η (x,y, q s q E m st s t t s H + ér t r L m,x,y t (ξ,η H + é Pr t m C t u D(H + t (x,y H + t ε > 0 t q D((x,y,ε H + ù D((x,y,ε st sq tr (x,y t r ε P s s I ε := L m (u(ξ,ηe m (x,y,ξ,ηdξdη = = H + \D((x,y,ε H + \D((x,y,ε (L m (u(ξ,ηe m (x,y,ξ,η u(ξ,ηl m(e m (x,y,ξ,ηdξdη r L m(e m = 0 s H + \D((x,y,ε é é t r s L m (ue m ul m(e = ξ (( ξ ue m u( ξ E m + m ξ ue m + η (( η ue m u( η E m. P P
25 P P P P P P s r r s r q s êtr t t Ω rt R t r st ss C r r t t n t r t r r s rt t à Ω t ds é é t r s r Ω r té ss t tér r Ω s r s X = (X,X : Ω C st t rs C rs Ω divx(x,ydxdy = Ω X(x,y n(x,yds râ à r q é à rt Ω = U\D((x,y,ε ùu st rt ré r H + t t s rt u s t s I ε = t [0,π] (ξ,η=(x,y+ε(cos t,sin t r s t tr q t [0,π] (ξ,η=(x,y+ε(cos t,sin t (( ( ξ ue m u( ξ E m + m ξ ue m [ [( ξ u+ m ξ u]cost+( ηusint cost+ +(( η ue m u( η E m sint εdt ] E m εdt ε 0+ 0 r lim ε 0 εlnε = 0 s t r r q lim ε 0 I ε st t tr r st lim ε 0 t [0,π] (ξ,η=(x,y+ε(cos t,sin t t tt t s r é à t I ε s s à rt r t t q m u(( ξ E m cost+( η E m sintεdt, t s J ε té r r t s é té ré é t s J ε = m u ξm sin m θdθ π t [0,π] ε m ( 0 (ξ,η=(x,y+ε(cos t,sin t +ksin θ m/ εcostdt+ }{{} π + m u ξm sin m θdθ π t [0,π] ε m+ ( 0 (ξ,η=(x,y+ε(cos t,sin t +ksin θ m/+ ε dt+ }{{} J ε, π J ε, π + m u ξm xsin θ sinm θdθ π t [0,π] ε m+ ( 0 (ξ,η=(x,y+ε(cos t,sin t +ksin θ m/+ εcostdt }{{} t ù r q k = 4xξ ε s s s r s t s s t s J ε,3 P P
26 P P P P P P Pr s t rsq k + t m C π sin θ sinm θdθ m θ=0 (+ksin θ m/+ k + k m + lnk. Pr P s t u = sin θ tt té r t r 0 m 0 u m+ ( u m du (+ku m/+ u m+ ( u m du ( k +u m/+ 0 k + 0 u m+ = m k m/+ 0 u m+ du ( k +u = m/+ 0 (u m/+( ( u m du = t r u = k s t s u m+ ( u m du ( k +u m/+. u m+ ( k +u m/+ ( ( u m du 0 ( u m u du. 0 u m+ du r s k ( k +u = th m+ tdt m/+ 0 th m+ t t rs q t + t q 0 dt r é t q q k + r s k 0 r s t rés t r s k th m+ dt dt = r s k k + 0 k + lnk. Pr s t rsq k + t m C π sin m θdθ m θ=0 (+ksin θ m/+ k + mk m Pr P s t ré é t u = sin θ tt té r t m 0 u m ( u m du (+ku m/+ = m k m/+ 0 u m ( u m du ( k +u m/+. r 0 u m ( u m du ( k +u m/+ 0 u m du ( k +u = m/+ 0 u m ( k +u m/+ ( ( u m du s s st r r r r t tt é té P P
27 P P P P P P 0 u m ( ( +u m/+ ( ( u m u m m du ( 0 +u u du m/+ k k = 0 k + 0 ( u m ( +u m/+ k u m (u m/+ = 0 ( m u ( u m du m u ( u m du s r r s t m u ( u m du. ( u 3 m 0 u m+ ( k +u m +du m lnk. k + 4 ( t t rs râ à t 0 u m ( +u m/+ ( ( u m m du lnk. k + 4 k t r u = k s t s 0 u m du r s k ( k +u = k th m t m/+ 0 dt = k t m th ( r m s k. é t q q k + 0 u m du ( k +u m/+ k + k m. t t t t r s t rés t s à r t é rè r s t tr q J ε, ε 0+ m π 0 u m ( u m du k ( k +u m/+ k + m. π sin m θdθ m θ=0 (+ksin θ m/+ k + mk m t [0,π] (ξ,η=(x,y+ε(cos t,sin t u xm ε m m k m/(lnkεcostdt ( + m ε 0+ πx εlnε u(x+εcost,y +εsintcostdt t [0,π] (ξ,η=(x,y+ε(cos t,sin t P P
28 P P P P P P q t rs r s t tr q m J ε,3 ε 0+ π t [0,π] (ξ,η=(x,y+ε(cos t,sin t u xm m εm+(x k m/+(lnkεcostdt ( m ε 0+ 4πx εlnε u(x+εcost,y +εsintcostdt t [0,π] (ξ,η=(x,y+ε(cos t,sin t q t ss rs r s t tr q ε 0+ π m J ε, ε 0+ π t [0,π] (ξ,η=(x,y+ε(cos t,sin t t [0,π] (ξ,η=(x,y+ε(cos t,sin t u xm ε m+ m mk m/ε dt u(x+εcost,y +εsintdt ε 0+ u(x,y. r é q r t t m C t rt ré st str t t s t lim L m (u(ξ,ηe m (x,y,ξ,ηdξdη = ε 0+ H + \D((x,y,ε = L m (u(ξ,ηe m (x,y,ξ,ηdξdη = u(x,y H + q E m st t t s t t L m q q s t m C t rt ré st str t t s t r r m C t rt ré st str t t ér r à st t t t ss s ss rt s s r s s t s t s L m q q s t m C r s t s t st s s séq t é rè ré é t Pr s t t m C t Ω rt r t t t s H + t r st ss C r r rs r (x,y Ω t u ss C s Ω t t n t r t r r s rt t à Ω t ds é é t r s r Ω r té ss t tér r Ω s r s s u(x,y = Ω L m (ue m dξdη [ ( ξ ue m u( ξ E m + m ] Ω ξ ue m, ( η ue m u( η E m nds t ù té s s té r s u := u(ξ,η t E m := E m (x,y,ξ,η P P
29 P P P P P P Pr t s u st s C (Ω s s r (x,y Ω t ε > 0 t q D((x,y,ε Ω L m (ue m dξdη = (L m (ue m L m(e m udξdη. Ω\D((x,y,ε Ω\D((x,y,ε rès r r ré é t té tt r èr té r st é à Ω t [0,π] (ξ,η=(x,y+ε(cos t,sin t [ ( ξ ue m u( ξ E m + m ] ξ ue m, ( η ue m u( η E m (( ( ξ ue m u( ξ E m + m ξ ue m cost+ nds +(( η ue m u( η E m sint εdt, t rès q s s s r ré é t tt r èr r ss t q ε 0 rs [ ( ξ ue m u( ξ E m + m ] ξ ue m, ( η ue m u( η E m nds+u(x,y. Ω r tèr té r E m s (x,y tr q L m (ue m dξdη = L m (ue m dξdη, lim ε 0 Ω\D((x,y,ε t r s t st r é Ω é rè é s t s P s r s s r r tr r q t t P s r H + st à r r Ω\K ù Ω st rt H + t K st t Ω êtr s P s Ω tr s H + \ K t t t rs r H + r ss t s t s t s s êtr t r t r t t é rè é s t s s r q s s r ss s ér t s s s t s t s s t s rs m P r s t s r ss s r s rt ts ér ts s t r m s s s r r r sé ré t s s m < t m P s ré sé t s s r s s t s t s q s t s s r r H + st à r s s r s r é s t s à P r m < r ss E m (x,y,ξ,η = ξx m π π θ=0 sin m θdθ [ (x ξ +4xξsin ( ] θ +(y η m tr q E m ér tt r r été E m (x,y,, t rs q x 0+ t (x,y + P P
30 P P P P P P P r m π E m (x,y,ξ,η = ξm π θ=0 ér s tt r r été P r tr [(x ξ +4xξsin ( θ E m (x,y,ξ,η E m ( x,y,ξ,η sin m θdθ +(y η ] m/ st t rs s t t s H + t q ér tt r r été s s r s P r m < P r m F m (x,y,ξ,η = E m (x,y,ξ,η F m (x,y,ξ,η = E m (x,y,ξ,η E m ( x,y,ξ,η. és t ts ré r s u D(H + t s r (x,y H + é t U(x,y = u(ξ,ηf m (x,y,ξ,ηdξdη, H + rs lim U = 0 t r t t y R lim U = 0 (x,y + (0,y s U st ss C s r H + \ s u t r t t (x,y s u L m,x,y U(x,y = 0 Pr r rq q q (ξ, η st é F m (x,y,ξ,η = ξx m π π θ=0 sin m θdθ [ (x ξ +4xξsin ( ] θ +(y η m r m < rs F m (x,y,ξ,η 0 t r r rés t t (x,y + é ê s m [ π F m (x,y,ξ,η = ξm sin m θ [ π θ=0 (x ξ +4xξsin ( θ +(y η ]m [ (x+ξ 4xξsin ( θ +(y η ]m F m (x,y,ξ,η 0 t r r rés t t é (x,y + P P ] dθ
31 P P P P P P P r s t s t r rq r q r m < F m (x,y,ξ,η (x,y (0,y q q rés t t és ré ξx m π[ξ +(y η ] m/ π 0 sin m θdθ s s t t m (ξ,η s s rt u q st t s H + rt r st M > 0 t α > 0 é t q u t s q (ξ,η M t ξ α t y s R P s s r x [ α,α] f m (x = [ (x ξ +4xξsin ( θ +(y η ]m. é té s r ss ts s tr q r x > 0 ss 3 r t rt r f m (x f m (0 xsup f m [0,α] f m ( x f m (0 x sup f m. [ α,0] f m (x f m ( x x sup f 3M +α m x m [ α,α] α. Rem+ sup F m (x,y = O(x (ξ,η supp u y R q x 0+ è t é rs r r t rés t t q s s s (x,y (ξ,η s t t s s s H + rs L m,x,y F m (x,y,ξ,η = 0. rq rq s q s U st s D(H + rs L m,x,y U = u s q tt t té st s é ss r t r s U D(H + rt r t s r s q L m U = u s r s é t s é t t r s t q s t é t t u : H + R t s r H + é r t s t s t s lim H +u = 0 ε > 0, N N, n N, (x,y H +, x n (x,y n = u(x,y ε. P P
32 P P P P P P tr s t r s r t à s ér r q r H + H + st st t é s ts s r é s t s ts à t à r q t r rs s r r H + st rt s s t r r rs t s s tr r q tt t r r st s s rt q t r t rs t t t H + P s ré sé t s s r s t s t Pr s t t u : H + C s t s t s lim H +u = 0 lim u(x,y = 0 t y R, lim u = 0. (x,y + (0,y Pr s s r t st é t s s t t q t t tr s q lim H + u = 0 lim u(x,y = 0 t y R, lim u = 0 (x,y + (0,y t ε > 0 st A > 0 t q r t t (ξ,η H + ξ +η A u(ξ,η ε. ê r t t y R st α y ]0,[ t q r t t (ξ,η H + ξ +(η y < α y u(ξ,η ε. s t [ A, A] st t râ r r t s st α > 0 t q r t t y [ A,A] B(y,α s t s s s s B(y,α y y [ A,A] rt r s (ξ,η H + st t q 0 < ξ < α rs rs u(ξ,η ε q η s t s [ A,A] t r r r s t s ç s r tr r q s s t u L m (u = 0 s s r H + rs st t q t t r r rq s q rés t t st é é t r s m st t r t r str t t s t t s tr t t v é s r (R m+ R r v(x,...,x m+ = u( x + +x m+,x m+, rs r s t tr q v st r q s (R m+ R lim H + u = 0 lim {(0,...,0} R v = 0 r s t tr q v s r t r q s r R m+ t t t r t q lim H + u = 0 s tr s q lim x + v(x = 0 rés t q v st t q t t q u ss s é ér s s é è s ù m r s rs s q q s P P
33 P P P P P P Pr s t t u C (H + t q L m u = 0 t lim H +u = 0 rs u 0 s r H + Pr P r (ξ,η H + s r N N φ N (ξ,η = θ (Nξθ ( ξ N ( η θ N ù θ t θ s t s t s C s r R à rs s [0,] t t s q θ (t = r t θ (t = 0 r t θ (t = r t [, ] t θ (t = 0 r t R\],[ s s s q t t s s ér é s s t s θ t θ ts {,,,} s t s θ θ u st t C s r r t H + s t L m u = 0 rs uφ N st t ss C t à s rt t s r H + t (x,y H + q s r é s t t s t r P r N ss 3 r râ à r s t s E m st r é r F m u(x,y = u(x,yφ N (x,y = L m (uφ N F m dξdη H + r t L m (uφ N st t q t s s r té F m u(x,y = [L m (uφ N +ul m (φ N + u φ N ]F m dξdη H + = u[l m (φ N F m div (F m φ N ]dξdη H + = u[l m (φ N F m div (F m φ N ]dξdη D D 8 = u[l m (φ N F m + F m φ N ]dξdη D D 8 ù D,...,D 8 s t s s é ts s ts [ D = N, ] [ N N, N ] [, D = N, N ] [ ] N,N, P P
34 P P P P P P [ ] N D 3 =,N [ D 5 = N, ] N [ ] N D 7 =,N N [ N, N ] [, D 4 = [ N, N η [ ] N,N, D 6 = ] et D 8 = N, N ] [ ] [ N N,N,N [ N, ] N [ N, N, ], ], [ N, N ]. N/ D 5 D D 6 ND φ N D 3 ξ N N/ N D 8 D 4 D 7 N/ N r s D i lim H + u t 0 rs u N := sup (ξ,η D D 8 u(ξ,η N + 0. s s r r s té r s rt t s r D,...,D 8 P r s s r s s s s ts q t s r s st t s s t r s t N r D s s sup φ N ξ = O(N t sup φ N η = 0. r D D 4 s s ( sup φ N ξ = 0 t sup φ N η = O N r D 3 s s sup φ N ξ ( = O N t sup φ N η. = 0. P P
35 P P P P P P r D 5 D 8 s s ( sup φ N ξ = O(N t sup φ N η = O N. r D 6 D 7 s s ( sup φ N ξ = O N r D D 5 D 8 s s t sup φ N η ( = O N. r D D 3 D 4 D 6 D 7 s s sup L m (φ N = O(N. sup L m (φ N = O (. N Pr P r (ξ,η D φ N (ξ,η = θ (Nξ t φ N ξ (ξ,η = Nθ (Nξ φ N η (ξ,η = 0, q s sup D φ N ξ L m φ N (ξ,η = N θ (Nξ mn θ ξ (Nξ, = O(N, sup D φ N η = 0, sup D L m (φ N = O(N sq s ér é s θ s t r é s t q r (ξ,η D ξ N P r (ξ,η D φ N (ξ,η = θ ( η N t q s sup D φ N ξ = 0, φ N ξ (ξ,η = 0 φ N η (ξ,η = η N ( θ, N sup D st ê s r D 4 L m φ N (ξ,η = ( η N θ, N φ N η ( = O N (, sup L m (φ N = O D N P r (ξ,η D 3 φ N (ξ,η = θ ( ξ N t φ N ξ (ξ,η = ( ξ N θ N φ N η (ξ,η = 0, P P
36 P P P P P P q s sup D 3 φ N ξ L m φ N (ξ,η = ( ξ N θ ( m ξ N N ξ θ, N ( = O N, sup D 3 φ N η = 0, P r (ξ,η D 5 φ N (ξ,η = θ (Nξθ ( η N t φ ( N η ξ (ξ,η = Nθ (Nξθ N L m φ N (ξ,η = N θ (Nξθ ( η N q s sup D 5 φ N ξ = O(N, st ê s r D 8 sup D 5 φ N η ( sup L m (φ N = O D 3 N φ N η (ξ,η = η N θ (Nξθ (, N + ( η N θ (Nξθ N m ( η ξ Nθ (Nξθ N ( = O N, sup D 5 L m (φ N = O(N. ( P r (ξ,η D 6 φ N (ξ,η = θ ξ ( η N θ N t φ N ξ (ξ,η = ( ξ ( η φ N N θ θ N N η (ξ,η = ( ξ N θ N L m φ N (ξ,η = ( ξ ( η N θ θ + ( ξ ( η N N N θ θ N N m Nξ θ q s sup D 6 φ N ξ ( = O N, sup D 6 φ N η ( = O N st ê s r D 7 ù ( η θ N, ( ξ ( η θ N N (, sup L m (φ N = O. D 6 N s s t t st r s q t tés s t s r i {,..., 8} F m dξdη, ξ F m dξdη et η F m dξdη. D i D i D i P r Re m < s s r i = : ( F m dξdη = O, D i N r i =,4 : D i F m dξdη = O ( N, D i D i ( F m ξ dξdη = O N F m η dξdη = O(N. P P.
37 P P P P P P r i = 3 : D i F m dξdη = O ( N, D i D i F m ξ dξdη = O(N. r i = 5,8 : ( ( F m dξdη = O, F m D i N D i ξ dξdη = O N ( F m η dξdη = O. N r i = 6,7 : F m dξdη = O ( N, F m D i D i ξ dξdη = O(N, F m η dξdη = O(N. Pr P r m < s s D i, F m (ξ,η = ξx m π π θ=0 sin m θdθ [ (x ξ +4xξsin ( ] θ +(y η m. st st t C t q r t t (ξ,η H + t F m (ξ,η C ξ [(x ξ +(η y ] Rem. ê s s F m ξ = F π m ξ ξx m π (m θ=0 [(ξ x+xsin θ ]sin m θdθ [ (x ξ +4xξsin ( ] θ +(y η m, t r t θ [0,π], [(ξ x+xsin θ ]sin m θ ] +(y η m [ (x ξ +4xξsin ( θ (ξ x+xsin θ ( (x ξ +(η y Rem = ξ xcosθ ( (x ξ +(η y Rem ξ +x ( (x ξ +(η y, Rem st st t C t q r t t N ss 3 r t r t t (ξ,η H + t F m ξ C [ (x ξ +(η y ] Rem + ξ(x+ξ (. (x ξ +(η y Rem P P
38 P P P P P P F m η = ( m(η yξx m π π θ=0 sin m θ [ (x ξ +4xξsin ( ] θ +(y η m, st st t C 3 t q r t t N ss 3 r t r t t (ξ,η H + t F m η C 3 ξ η y 3 Rem. râ à s é tés s s st r s té r s s ér t s t s s r s s D i r D é té D F m dξdη = O( ξ= N ξ= N η= N η= N ξdξdη [(x ξ +(η y ] Rem η= N = O(/N η= N P s st t D F m ξ= ξ dξdη = O( N ξ= N dη [ (x N +(η y ] = O(/N. Rem η= N η= N [ (x ξ +(η y ] Rem ξ(x+ξ + ( dξdη (x ξ +(η y Rem η= N = O(/N η= N dη [ (x N +(η y ] +O(/N = O(/N. Rem r D é té D F m dξdη = O( N ξ= N N η= N ξdξdη [(x ξ +(η y ] Rem N N = O( ξ= η= N N [ = O(N P s é té D ξ N η y dξdη = Rem O(N η= N ] (N y Rem ( N y Rem N F m η dξdη = O( ξ= N N η= N dη η y Rem = O(N Rem+. ξ η y 3 Remdξdη P P
39 P P P P P P N = O(N η= N dη η y 3 Rem = O(NRem r D 3 é té D 3 F m dξdη = O( N ξ= N N η= N ξdξdη [(x ξ +(η y ] Rem N = O( ξ= N N η= N N = O(N η= N N = O(N η= N P s é té D 3 F m N ξ dξdη = O( ξ= N N η= N ξdξdη [ (x N +(η y ] Rem dη [ (x N +(η y ] Rem dη [+(η y ] Rem = O(N. [ (x ξ +(η y ] Rem ξ(x+ξ + ( dξdη (x ξ +(η y Rem N = O(N η= N dη [ (x N +(η y ] Rem = O(N+O(N 3 N N +O(N 3 η= N η= N dη (x N 4 Rem dη [ (x N +(η y ] Rem = O(N+O(N Rem = O(N. r D 4 s st s D r D 5 é té D 5 F m dξdη = O( N ξ= N N η= N ξdξdη [(x ξ +(η y ] Rem = O(/N [ N = O(/N η= N dη (η y Rem (N y Rem ( N y Rem ] = O(/N 3 Rem. P P
40 P P P P P P P s é té D 5 F m ξ= ξ dξdη = O( N ξ= N ξ= N η=n = O( ξ= η= N N é té D 5 η=n η= N [ (x N F m ξ= η dξdη = O( [ (x ξ +(η y ] Rem ξ(x+ξ + ( dξdη (x ξ +(η y Rem Rem +(η y ] + ( = O N N ξ= N ( (x N η=n η= N ξ(x+ξ Rem +(η y ( ξdξdη η y = O 3 Rem N dξdη r D 6 é té D 6 F m dξdη = O( N ξ= N N η= N ξdξdη [(x ξ +(η y ] Rem = O(N [ P s é té D 6 F m ξ=n ξ dξdη = O( N = O(N η= N dη (η y Rem (N y Rem ( N y Rem ξ= N η=n η= N ξ=n η=n [ = O( ξ= N η= N ] = O(N +Rem. [ (x ξ +(η y ] Rem ξ(x+ξ + ( dξdη (x ξ +(η y Rem ξ(x+ξ + (η y Rem (η y 4 Rem ] dξdη P P
41 P P P P P P N = O(N+O(N 3 η= N é té D 6 F m ξ=n η dξdη = O( ξ= N dη (η y 4 Rem = O(N+O(NRem = O(N. η=n η= N ξdξdη η=n η y = 3 Rem O(N η= N = O(N Rem. dη η y 3 Rem r D 7 s st s D 6 r D 8 s st s D 5 P r m t t s s st t s t s s s t s Pr P r m s s ( π F m (x,y,ξ,η = ξm sin m θ π θ=0 [(x ξ +4xξsin θ +(y η ] m/ [(x+ξ 4xξsin θ +(y dθ. η ] m/ r t t (ξ,η H + [ (x+ξ 4xξsin θ ] m/ +(y η = [ x +ξ +xξcosθ+(y η ] m/, rs r t t (ξ,η H + [ (x+ξ 4xξsin θ ] m/ +(y η ( (x ξ +(y η Rem t st st t C t q r t t (ξ,η H+ t F m C ξ Rem ((x ξ +(y η Rem. tt st t st r s t s s t r r r s t r s rt ts s r D s s r é té s t réé r t F m F m (x,y,ξ,η = ξm π π θ=0 sin m θk m (x,y,ξ,η,θdθ ù K m (x,y,ξ,η,θ = [(x ξ +4xξsin θ +(y η ] m/ P P
42 P P P P P P [(x+ξ 4xξsin θ +(y η ] m/. P r (x,y H + é θ [0,π] é t η R é é t t g m s r [ /N,/N] /N < x r g m (ξ = [(x ξ +4xξsin θ +(y η ] m/. tt t st é sq rt (x ξ +4xξsin θ +(y η = x +ξ xξcosθ+(y η (x ξ +(y η t q r r t r st ss ré r (x /N > 0 s s K m (x,y,ξ,η,θ = g m (ξ g m ( ξ K m (x,y,ξ,η,θ ξ sup g ξ x +x m m ξ [ ξ,ξ] [(x ξ +(y η ] Rem, + q q q st st t c t q ξ Rem+ (ξ,η D, F m c [ (x ξ +(η y ] + Rem. ê s s F m ξ = mf m + mξm ξ π t r t θ [0,π], = t râ à θ [0,π], = π θ=0 ( sin m (ξ x+xsin θ θ [(x ξ +4xξsin θ +(y η ] m + [(ξ x+xsin θ ]sinm θ [(x ξ +4xξsin θ +(y η ] m + (ξ +x xsin θ [(x+ξ 4xξsin θ +(y η ] m + ξ xcosθ [(x ξ +(y η ] ξ +x Rem + [(x ξ +(y η ] Rem [(ξ +x xsin θ ]sinm θ [(x+ξ 4xξsin θ +(y η ] m + ξ +xcosθ [(x ξ +(y η ] ξ +x Rem + [(x ξ +(y η ] Rem (ξ x+xsin θ [(x ξ +(y η ] Rem + + (ξ +x xsin θ [(x ξ +(y η ] Rem + +. P P dθ.
43 P P P P P P s st t s é s r t é té tr t q st st t C H + t ( F m ξ ξ Rem C [(x ξ +(y η ] Rem t q r t t N ss 3 r t r t t (ξ,η + ξ Rem (ξ +x [(x ξ +(y η ] Rem +. s s é r r tt é té s r D t s t é té t s t s q st s st t s C t C é t s N t s q r t t (ξ,η D ( F m ξ ξ Rem C F m η = m(η yξm π [(x ξ +(y η ] +Rem π θ=0 C ξ Rem + [(x ξ +(y η ] +Rem ξ Rem (ξ +x [(x ξ +(y η ] Rem + ( sin m θ [(x ξ +4xξsin θ +(y η ] m + [(x+ξ 4xξsin θ +(y dθ. η ] m + ê st st t C 3 t q r t t N ss 3 r t r t t (ξ,η H + t F m η η y ξ Rem C 3 ((x ξ +(y η. Rem + râ à s é tés s s st r s té r s s ér t s t s s r s s D i r D é té D F m dξdη = O( ξ= N ξ= N η= N η= N ξ Rem+ dξdη [ (x ξ +(η y ] + Rem [ ξ= ( Rem+ ( ] Rem+ N = O(N ξ Rem+ dξ = O(N ξ= N N N P s st t D F m ξ= ξ dξdη = O( = O(/N Rem+. N ξ= N η= N η= N ξ Rem dξdη [ (x ξ +(η y ] + Rem. P P
44 P P P P P P ξ= N = O(N ξ Rem dξ = O(/N Rem. ξ= N r D é té D F m dξdη = O( N ξ= N N η= N ξ Rem dξdη ((x ξ +(y η Rem N N = O( ξ= η= N N ξ Rem dξdη y N Rem = O(N, r q t té té ré st ré t ré é t N s t s r st O(N P s é té D N F m η dξdη = O( ξ= N N η= N η y ξ Rem dξdη ((x ξ +(y η Rem + N N = O( ξ= η= N N ξ Rem N dξdη N = O( y η Rem+ ξ= η= N N N Rem dξdη = O(N. y Rem+ N r D 3 é té D 3 F m dξdη = O( N ξ= N N η= N ξ Rem dξdη ((x ξ +(y η Rem N = O( ξ= N N = O(N Rem+ η= N P s é té D 3 F m N ξ dξdη = O( ξ= N N N η= N η= N ξ Rem dξdη ( (x N +(y η Rem dη ( = O(N. (x N +(y η Rem ( ξ Rem [(x ξ +(y η ] Rem + ξ Rem (ξ +x [(x ξ +(y η ] Rem + dξdη N = O(N Rem η= N dη [(x N +(y η ] Rem +O(N Rem+ N η= N = O(N+O(N = O(N. dη [(x N +(y η ] Rem + P P
45 P P P P P P r D 4 s st s D r D 5 é té D 5 F m dξdη = O( N ξ= N N η= N ξ Rem dξdη ((x ξ +(y η Rem P s é té D 5 N = O(/N Rem+ η= N F m ξ= ξ dξdη = O( N ξ= N η=n η= N ( dη y N Rem = O(/NRem. ξ Rem [(x ξ +(y η ] Rem + ξ Rem (ξ +x [(x ξ +(y η ] Rem + ξ= N η=n ( ξ Rem = O( ξ= η= N y η + ξrem (ξ +x Rem y η Rem+ N = O(/N Rem. dξdη dξdη é té D 5 F m ξ= η dξdη = O( N ξ= N η=n η= N η y ξ Rem dξdη ((x ξ +(y η Rem + ξ= N η=n = O( ξ= η= N N ξ Rem dξdη y η Rem+ = O(/NRem+ r D 6 é té D 6 F m dξdη = O( N ξ= N N η= N ξ Rem dξdη ((x ξ +(y η Rem P s é té D 6 N = O(N Rem+ η= N F m ξ=n ξ dξdη = O( ξ= N η=n η= N + dη ( N yrem = O(N. ( ξ Rem [(x ξ +(y η ] Rem ξ Rem (ξ +x [(x ξ +(y η ] Rem + ξ=n η=n ( ξ Rem = O( ξ= N η= N y η + ξrem (ξ +x Rem y η Rem+ dξdη dξdη P P
46 P P P P P P η=n = O(N+O(N Rem+ η= N dη = O(N. y η Rem+ é té D 6 F m ξ=n η dξdη = O( ξ= N η=n η= N η y ξ Rem dξdη ((x ξ +(y η Rem + η=n = O(N Rem+ η= N dη = O(N. y η Rem+ r D 7 s st s D 6 r D 8 s st s D 5 s t s t ré t s rés t ts t s s s s ré é ts i sup Di L m φ N D i F m dξdη ( ξ φ N, η φ N D i ξ F m D i η F m O(N O(/N (O(N,0 O( N O(/N O(N (0,O( N O(N O(/N O(N (O(,0 N O(N O(/N O(N (0,O( N O(N O(N O(/N (O(N,O( N O( N O( N O(/N O(N (O(,O( N N O(N O(N O(/N O(N (O(,O( N N O(N O(N O(N O(/N (O(N,O( N O( N O( N t sé t ér r q r q i {,..., 8} s q t tés sup L m φ N F m, sup ξ φ N ξ F m et sup η φ N η F m D i D i D i D i D i D i r st t r é s rés t q q N + s u(x,y = o( u 0 t r r s t P P
47 P P P P P P é rè é s t s P r m C s r s t é rè é s t s P s s r t r é s é rè s t s s r t t ér t sq s tr t té st ré é à tr rs r Ω é rè é s t t Ω rt H + t K s s s t Ω t u C (Ω \ K st s t L m u = 0 s Ω\K rs u q é s t r u = v +w ù v C (Ω st s t L m v = 0 s Ω t w C (H + \K st s t L m w = 0 s H + \K lim H +w = 0 Pr P r t t s s s E C tρ > 0 é te ρ = {x C, d(x,e < ρ} E ρ st s E t r s s q Ω st rt r t t t H + s t ρ ss 3 t t t s rt q K ρ t ( Ω ρ s t s ts st t ϕ ρ D(H + à s rt s Ω\K t q ϕ ρ s s Ω\(K ρ ( Ω ρ y K ρ Ω ( Ω ρ K ρ Ω\{K ρ ( Ω ρ } ρ x r ϕ ρ s rt r s t r é r P r z = x+iy Ω\(K ρ ( Ω ρ t t F z (ζ := F m (x,y,ξ,η t L ζ := L m,ξ,η pour ζ = ξ +iη, râ à r s t u(z = uϕ ρ (z = F z (ζl ζ (uϕ ρ (ζdξdη Ω ρ P P
48 P P P P P P = F z (ζl ζ (uϕ ρ (ζdξdη + F z (ζl ζ (uϕ ρ (ζdξdη ( Ω ρ K ρ = v ρ (z+w ρ (z. r r t tr rs q v ρ st s t L m v ρ = 0 s r Ω\( Ω ρ t q w ρ st s t L m w ρ = 0 s r H + \K ρ ss lim H + w ρ = 0 s s t t q σ < ρ rs ré é t t t é s t u = v σ +w σ s r Ω\(K σ ( Ω σ s r s q v ρ = v σ s r Ω\( Ω ρ t w ρ = w σ s r H + \K ρ P r r r rq s q s z Ω\(K ρ ( Ω ρ rs v ρ (z+w ρ (z = v σ (z+w σ (z t r ( éq t L m u = 0 s w ρ w σ st s t ( s H + \K ρ q st é à v σ v ρ s Ω\(K ρ ( Ω ρ t v σ v ρ s r s t ( s r Ω\( Ω ρ s w ρ w σ s r s t ( s H + t lim H +w ρ w σ = 0 r s t s rs t r s t v ρ = v σ w ρ = w σ, P r z Ω t é r v(z = v ρ (z r t t ρ ss 3 t t t s rt q z Ω\( Ω ρ ê èr r z H + \K s w(z = w ρ (z r ρ t t s s s rr és à é s t s té u = v +w s s t t q Ω s t H + q q t q u st s t L m u = 0 s r Ω\K s t a H + t R ss 3 r r q K D(a,R t q D(a,R s t r t t t s H + t ω = Ω D(a,R r rq q K st s s s t rt ω r t t t s H + t q u st s t ( s ω \K q t rés t t é tré r s s s rts r t t ts s s u(z = ṽ(z+ w(z r z ω\k ù ṽ st s t ( s ω t w st s t ( s H + \K s t s s t lim H + w = 0 t s q ér V = u w st s t ( s Ω\K t q s r s t ( s K r st é à ṽ s ω s u = V + w r t é s t s té u ré é t s tr é s t u = v +w v C (Ω L m v = 0 t w C (H + \ K L m w = 0 t lim H + w = 0 rs V v = w w s Ω \ K t w w s r à H + st s t L m (w w = 0 s H + t ér lim H +(w w = 0 râ à r s t t t w = w s V = v q è r r t é rè é s t r P ss r H + q Re m < P r Re m < t r r s t à rt Ω = H + r t r r r rés t t té r t r ss u s t L m (u = 0 s H + ss t s rs u s r s r é s P P H + RE M <
49 P P P P P P s rés t ts t t t t été t s r s s s s é èr t ér ts H + \ (0,a] {0} a > 0 r s t s s s ù m = ± s t q s q s r r r s s tr P s ré sé t s s Pr s t t m C t q Re m < t u : R R t t t r é rs st q P U ss C s r H + t q lim (x,y + U(x,y = 0 t t q r t t y R lim (0,y U = u(y. s s s r t t (x,y H + ù C m = m π U(x,y = C m x m η= π θ=0 sin m θdθ = Γ ( m m π Γ( m Pr t s f(x,y = x m u(η dη (x +(y η m P r tr r r r t r (x +(y η m s t s t tr r q L m f = 0 r ér t s s s té r t x f = ( mx m (x +(y η m ( mx m (x +(y η m xx f = m( mx m (x +(y η m ( m(3 mx m (x +(y η m + ( m(4 mx3 m (x +(y η 3 m s yy f = ( mx m (x +(y η m f = m( mx m (x +(y η m t é t q L m f(x,y = 0 U(x,y = C m x m η= u(η dη (x +(y η m P r t r t = y η t t x U(x,y = C m + ( m(4 m(y η x m. (x +(y η 3 m t= m( mx m (x +(y η m = C m x u(y txdt (+t m η= P r t é rè r é s t tr r q u(η dη (+( y η x m C m t= dt (+t m = m π π θ=0 sin m θdθ t= dt (+t m =. P P H + RE M <
50 P P P P P P P r r r rq s rès q t= dt (+t m ù B st t êt r t q m π π θ=0 ( = B, m = Γ(/Γ( m Γ ( m sin m θdθ = m ( π m B m, m = m P s t s t r t r t Γ ( Γ(z = π / z Γ(zΓ z + π mγ ( m Γ( m. t r ré rr Γ(z + = zγ(z t t rés t t st à r q Γ(/Γ ( m Γ ( m m π mγ ( m Γ( m =. té é r s t r é r s t rq rr t s s r q st st t r r r s t s Re m t s m st t r t r t s u C (H + s t L m (u = 0 s H + rs t v é s r R m+ r v(x,...,x m+ = u(0,x m+ t v(x,...,x m+ = u ( x + +x m+,x m+ st t r q s r (R m+ R rt r s m r s t tr q v s r t r q s r R m+ q t rs à é t q v st t t q t q u 0 q r q r è ét r t s t L m (u = 0 s t q u t rs 0 à t q s rs u s t s s r s r é s st r è q s s s s s s s t r è r t st séq rt t té éq t L m u = 0 r H + s tr t r s s s Rem s r r tt r s t s r t s s t s q s t s t r t t sé s s q s à tr ss s s s tr é ts r r s r t é t q s s r r s t t s s é t s t r s t s s s r é s r s s xoy s tr s s sstè r é s t sé r s s s r t rt èr t té à ét L t L s s ù ét é st sq tré H + PP
51 P P P P P P s t s ér q s s r s r è s tré r és t s t èr èt s r é s r s t été ré sé s s ré t t ès s r t α > 0 s ér s r s t é A = ( α,0 t r é t é B = (α,0 r s s r s ét t t q s t t ré r sstè r s t M st à st t t t rès ln ( MA MB P r é t r èr r é r st s r é r st τ := ln MA MB. θ = ÂMB. y M(x, y A( α, 0 θ O B(α, 0 x r r é s r s s r é s r s s t rs r é s r é s rtés s (x, y xoy r t q (α x iy ( α x iy = e τ+iθ (x α+iy = e τ+iθ [(x+α+iy] q s rès t t s rt s ré s t r s r x α = e τ [(α+xcosθ ysinθ] y = e τ [(α+xsinθ+ycosθ] x[ e τ cosθ]+ysinθ = α[e τ cosθ+] xe τ sinθ+y[ e τ cosθ] = αe τ sinθ. rés t sstè éq t s s r t t r é t t x = α s τ τ cosθ, y = αsinθ τ cosθ. r R > 0 t s s a = R +α sq tr (a,0 t r R rr s à ( a a τ τ 0 = ln R + R = arg ch a R. PP
52 P P P P P P s r t st H + = {(τ,θ : τ ]0+ ], θ [0,π[}. s s τ = τ 0 s t s r s tr (αcothτ 0,0 t r α/s τ 0 q q r t t τ 0,τ t s q 0 < τ 0 < τ s {τ τ 0 } st sq r é t s 0 < τ < τ st é t r s H + sq r é {τ τ } y τ = /3 θ = π/6 τ = /3 τ = / θ = π/3 τ = / θ = 0 τ = θ = π τ = x θ = 0 θ = 5π/3 θ = π/6 τ = 0 r s α = t s P r é s r s s q s t s t r s s t s r é s rtés s s t s r é s r s t é rè s t st r m = r s s s s ét s rs m s q q s é rè u ér L m u = 0 s rt H + t s s s s v m (τ,θ = s m τ( τ cosθ m/ u(τ,θ ù r é t s m τ( τ cosθ m/ = exp ( m ln s τ m ln( τ cosθ rs ( v m τ + v m θ +cothτ v m τ + 4 (m 4 s v m = 0. τ Pr s s u τ = α [ ] τ cosθ u s τ sinθ u ( τ cosθ x ( τ cosθ y PP
53 P P P P P P t s t s s s [ ] u s τ sinθ θ = α u τ cosθ u +. ( τ cosθ x ( τ cosθ y u x = α u τ = α ( τ cosθ 4 α + ( τ cosθ 3 t ( ( τ cosθ u s τ sinθ u, τ θ [ ( τ cosθ u x + s τ sin θ u y ] ( τ cosθs τ sinθ u x y [ s τ(cos θ+ τ cosθ u x +sinθ( τ +cosθ τ ] u y [ u θ = α ( τ cosθ 4 s τ sin θ u x +( τ u cosθ y ] +( τ cosθs τ sinθ u x y [ α + s τ( cos θ cosθ τ u ( τ cosθ 3 x +sinθ( τ τ cosθ u ] y t rt r s s t t q ( ( τ cosθ u L m,x,y u = α P s s t t u τ + u θ = α ( τ cosθ [ ] u x + u. y τ + u m( τ cosθ + θ s τ( τ cosθ u(τ,θ = ( τ cosθm/ v m (τ,θ s m τ t s L m,x,y u t F(τ,θ t t u τ msinθ τ cosθ u. θ t r m θ = r m (τ,θ = r m θ = m ( τ cosθm/, s m τ sinθ τ cosθ r m m 4( τ cosθ ( cosθ τ +msin θ r m PP
54 P P P P P P s t r m τ = ( τ cosθs τ ( τ +(m τ cosθ m r m r m τ = 4( τ cosθ s τ [ 4 τ 3 τ cosθ+(m τ cos θ+ +(m τ +(4 m τ cosθ+(m cos θ+m(m ] r m. éq t L m,x,y u = 0 s réé r t ( v m r m τ + v m θ t + v m τ + v m θ ( r m τ + m τ cosθ s τ τ cosθ r m + ( r m θ msinθ τ cosθ r m ( r m +v m τ + r m m( τ cosθ + θ s τ( τ cosθ r m τ + r m m( τ cosθ + θ s τ( τ cosθ è r tr t é rè r m τ msinθ τ cosθ r m τ + m τ cosθ s τ τ cosθ r m = r m cothτ, r m θ msinθ τ cosθ r m = 0 r m = 0 θ r m τ msinθ ( r m τ cosθ θ = 4 (m 4s r m. τ t t r v m s s r v m (τ,θ = A m (τb m (θ s t à r s sé ré s rt r éq t s t s t r v m t t A m A m +cothτ A m A m + 4 (m 4 s τ = B m B m. t r r t ét t t θ t t τ é t q s t r s s t st ts t n C t q tt st t s t é à n rs ( A m +cothτa m + 4 (m 4s n A m = 0, τ B m +n B m = 0. t B m ét t t r t π ér q r θ st é étr q t st t n t é ss r t êtr r t r PP
55 P P P P P P P r ét r éq t s t s t r A m s t s t t t C m ér rs éq t s τc m( τ+ τ C m( τ+ q t s réé r r s s r A m (τ = C m ( τ. ( τc m( τ τ C m( τ+ (n ( 4 n (m 4s C m ( τ = 0 τ 4 ((m / C m ( τ = 0. τ (LAH tt éq t st é éq t r ss é r q rq s q s s s s z = τ t u(z = C m ( τ rs ù ] ( z u zu + [ν(ν + µ u = 0 z (LA ν = n t µ = m. tt éq t st é éq t r ss é s ré t r µ = 0 à éq t r ( z u zu +ν(ν +u = 0. (L s s é r ss s s t s é t s tt éq t q s t s t s r ss é s r èr t è s è s t s r ss é s r èr t è s è s s s r s r s r rés t t té r s t s r r èr t è s è r z = τ > P µ ν ( τ = ν s µ τ Γ( µ νγ(ν + ν > t (µ+ν < 0 0 ( τ + θ µ ν s ν+ θdθ P ν µ ( τ = µ s µ τ π ( πγ µ 0 ( τ + s τ cosθ µ+ν sin µ dθ θ µ < µ < P µ ν ( τ = π s µ τ Γ ( µ τ 0 [( ] ν + θ ( τ θ µ+/dθ PP
56 P P P P P P Q µ ν( τ = eiπµ π s µ τγ(ν +µ+ s µ θ µ Γ(ν µ+γ(µ+/ 0 ( τ + s τ θ ν+µ+dθ µ > (µ ν < 0 t µ+ν + / Z µ < Q µ ν( τ = Q µ ν( τ = e iπµ π s µ τ eiπµ Γ ( e (ν+ θ µ τ ( θ τ µ+/dθ t (µ+ν + > 0 ν Γ(ν +µ+ Γ(ν + s µ τ π 0 ( τ +cosθ µ ν sin ν+ θdθ ν > t µ+ν + Z r s t s s s s r t s s t s ér é s r s t s r r t r P µ ν = P µ ν. Q µ ν (z = πeiπµ cos(πνp µ ν +sin[π(ν +µ]q µ ν sin[π(ν µ] r ν µ Z rt r r ν = n n Z s s q q s t µ C P µ ν Γ(ν µ+ Γ(ν +µ+ Q µ ν = Q µ ν e iπµ Γ(ν +µ+q µ ν = e iπµ Γ(ν µ+q µ ν, [ = P ν µ π e iπµ sin(πµq µ ν s s s s r s r t s t s r ss é s r èr t è s è r ], π Q µ ν( τ = e iπµ Γ(µ+ν + P ν (cothτ, s τ µ P µ ν ( τ = ie iπν Γ( ν µ π Q ν s τ µ (cothτ. s s ss s r s ré rr r s t P ν µ+ ( τ = (ν µ τ Pµ ν ( τ (ν +µp ν ( τ µ s τ (ν µ+p µ ν+( τ = (ν + τ P µ ν ( τ (ν +µp µ ν ( τ. (z dpµ ν (z dz = (ν +µ(ν µ+(z / P ν µ (z µzp ν µ (z. (z dpµ ν (z dz = νzp µ ν (z (ν +µp µ ν (z. PP
57 P P P P P P t s s r s r tt t r t t s rs P µ ν ( τ t Q µ ν( τ r t t τ > 0 t t s (µ,ν C r s t s t r ss à µ t τ és rt t s t s r ss é s r èr t è s è rsq ν st r ν = n n Z t q n + Pr s t t τ > 0 t µ C és t s s rs s ν st r ν = n n Z s s q ν +, P µ ν ( τ eτ/ π s τ ν µ / e τν q ν, P µ ν ( τ e τ/ π s τ ( ν µ / e τν q ν +, q ν, Q µ ν( τ e iπµ e τ/ π s τ νµ / e τν Q µ ν( τ e iπµ e τ/ π s τ ( νµ / e τν. s éq s s t s t r s r r rt à τ st à r r s s t t t r [τ 0,τ ] 0 < τ 0 < τ Pr P µ ν ( τ = s s r t t ν q st r n n N r Γ(ν + [ e (ν+ τ +e πi(µ (ν+ τ] [ +O Γ(ν µ+ π(ν +s τ t r t r tr q q ν + Γ(ν + Γ(ν µ+ ( πν ν+/ e ν ν+/ ν π(ν µ ν µ+/ e = (ν µ µ e µ ν+µ ν µ ( ( = (ν µ µ e µ exp ν + ( ln µ ν µ ν ( ]. ν P µ ν ( τ ν µ πνs τ e τ e τν = eτ/ πs τ ν µ / e τν, q s r èr st t s s t t r t t à rt r r t P µ ν = P µ ν tr s è st t é r t t r Q µ ν( τ π s τ νµ / e iπµ e τ(ν+/ t r èr é t q r ν = n n Z s s Q µ ν = Q µ ν. PP
58 P P P P P P r tèr r s éq ts é s r s s r ss s t s s P ν µ tqµ ν à s t s r é étr q s t s t s st t s s t s r é étr q s t r s r r rt à rs r ètr s s s s r r t t q t é ( τ cosθm/ s m τ { cos(nθ sin(nθ } P m ( τ n Q m ( τ n st s s s s t s à tér r sq τ τ t tr t é st s s s s t s s τ τ 0 q st é t r s H + sq t ù 0 < τ 0 < τ t st r m = st à r r µ = s ét s s ù m st q q n Z. é rè t m C t 0 < τ 0 t u s t ss L m u = 0 s sq τ τ 0 t s t v s t ss L m v = 0 s H + \ {τ > τ 0 } q st é t r s H + sq {τ > τ 0 } q s s r r H + lim H + v = 0 rs st s t s (a n n Z t (b n n Z s l (Z q s t ê à é r ss r t s q u = + n= a n Q m n ( τs m τ( τ cosθ m e inθ t v = + n= b n P m n ( τs m τ( τ cosθ m e inθ. s t (a n st q s r r èr sér st r s t t t [τ,τ ] τ 0 τ < τ sq τ > τ 0 t r s sér st r s t t t [τ 3,τ 4 ] 0 < τ 3 < τ 4 τ 0 é t r sq τ > τ 0 s H + s m st ré t m < rs s t (b n st q Pr t é s t θ u(τ 0,θ( τ 0 cosθ m/ s m τ 0 sér r r r r rt à θ é t r r r u(τ 0, u(τ 0,θ = s m τ 0 ( τ 0 cosθ m ù a n st s t l (Z q s t s t a n = π π 0 + n= a n e inθ, ( τ 0 cosθ m/ s m τ 0 u(τ 0,se ins ds. PP
59 P P P P P P tt t ét t ss r r rt à θ é t q s t (a n n st à é r ss r q n + t ũ(τ,θ = s m τ( τ cosθ m + n= Q m ( τ n a n Q m ( τ n 0 ï u s r r τ = τ 0 s râ à r s t s s q n + Q m ( τ n Q m ( τ n 0 s τ 0 s τ e n (τ 0 τ t t éq t st r s r t t t [τ,τ ] 0 < τ 0 τ < τ rés t q sér t s é ss t ũ r r t s r t s s ts [τ,τ ] sq τ τ 0 st ê s ér é s r r rt à τ t θ q s r t s ss à s t s r ss é s r é ré é t rt r t ũ st t é s r sq τ τ 0 t q ï u s r r τ = τ 0 râ t q s t éq t t q st ét r é èr q r s s rs r é r é t q ũ st q P s sq τ τ 0 q ï u s r r τ = τ 0 P r t v r st t t t t é s t θ v(τ 0,θ( τ 0 cosθ m/ s m τ 0 e inθ sér r r r r rt à θ é t r r r v(τ 0, v(τ 0,θ = s m τ 0 ( τ 0 cosθ m ù b n st s t l (Z q s t s t b n = π π 0 + n= b n e inθ, ( τ 0 cosθ m/ s m τ 0 v(τ 0,se ins ds. tt t ét t ss r r rt à θ é t q s t (b n n st à é r ss r q n + t ṽ(τ,θ = s m τ( τ cosθ m ï v s r r τ = τ 0 + n= P m ( τ n b n P m ( τ n 0 e inθ PP
60 P P P P P P s râ à r s t s s q n + P m ( τ n P m ( τ n 0 s τ 0 s τ e n (τ τ 0 t t éq t st r s r t t t [τ,τ ] 0 < τ < τ τ 0 rés t q sér t s é ss t ṽ r r t s r t s s ts [τ,τ ] é t r sq τ > τ 0 st ê s ér é s r r rt à τ t θ q s r t s ss à s t s r ss é s r é ré é t rt r t ṽ st t é s r é t r sq τ > τ 0 t q ï v s r r τ = τ 0 s s r r q lim τ 0+ṽ = 0. m < s s q n N t râ à P m ( τ = n m πγ ( m s m τ π 0 lim P m ( τ = 0 τ 0+ n t s r n > Rem s s P m n Rem s Rem Rem ( τ π Γ ( m τ π s Rem τ π Γ ( m 0 π 0 ( τ + s τ cosθ n+m sin m θdθ ( τ + s τ cosθ n+rem sin Rem θdθ ( τ + s τ n+rem sin Rem θdθ C m s Rem τe (n+rem τ n> Rem sup τ [0, τ 0 ] P m ( τ n b n P m ( τ n 0 râ t q r s t s P m ( τ n 0 n + e inθ < + n m π s τ0 e nτ 0. s s é r q lim τ 0+ ṽ = 0 r st à r r té é s t ré é t s s ù m R m < é r r r s t q ét r t q Q m ( τ n ( τ cosθ m/ A := Q m ( τ n 0 s m τ e inθ n Z := (a n n Z PP
61 P P P P P P B := r s s3 P m ( τ n P m ( τ n ( τ cosθ m/ s m τ e inθ n Z := (b n n Z r r s t r è r t L m u = 0 s D((a,0,R ù u = ϕ s r D((a,0,R st é r u(τ,θ = s m τ( τ cosθ m + n= Q m ( τ n c n Q m ( τ n 0 ù {τ = τ 0 } rr s r tr (a,0 t r R t ù ê c n = π π 0 e inθ ( τ 0 cosθ m/ s m τ 0 ϕ(a+rcoss,rsinse ins ds. v(τ,θ = s m τ( τ cosθ m + n= P m ( τ n c n P m ( τ n 0 st s t L m v = 0 s H + \D((a,0,R é à ϕ s r D((a,0,R ù c n = π π 0 e inθ ( τ 0 cosθ m/ s m τ 0 ϕ(a+rcoss,rsinse ins ds. s m st ré t ér m < rs v ér lim H + v = 0 t t v ré é t str t st s s t r è r t L m v = 0 s H + \D((a,0,R q s s r H + s s s3 P r m ré s s t ré é t q s s t s t r r r L m u = 0 r t èt s tt s t tr r q r m ré tt st ê s s3 rt s rt é t t r r été s r s é t é ér t r r été s s s s3 r é t t X s rt t (x n n N s t X P P M
62 P P P P P P t q st s t q s rt s t s3 s st st t s c, C > 0 t s q r t t s t (a n n Z t r t r s s c a n a n x n C a n. n n n s s ù (x n n Z st èt r s s3 tr s r t s r { x i,x j } i,j st é tr r ss é à {x i } i s t s r q s t {x i } i s t s s3 t é t êtr r é t tr r Pr r été {x i } i st s s3 r s rt s {x i } i st èt s t s rt t q s tr r é t ér t r rs t r é s r l (N s s3 s s t s t r r r ta tb s s s t s éq t L m [u] = 0 r s t t à tér r sq τ > τ 0 t à tér r tr sq τ > τ 0 < τ 0 < τ Q m ( τ n A := ( τ cosθ m/ e inθ := (a Q m ( τ n 0 s m n n Z τ B := P m ( τ n P m ( τ n ( τ cosθ m/ s m τ t C ré s s ré é t s e inθ n Z n Z := (b n n Z C := (c n n Z := (c n = a n t c n+ = b n n Z é t s r é s r s r {0 < τ 0 < τ < τ } s r té A t C r t s r s t r f,g L ( A f,g = π π 0 s m τ 0 f(τ 0,θg(τ 0,θ ( τ 0 cosθ θ m r s t s t + π π 0 s m τ f(τ,θg(τ,θ ( τ cosθ θ. m Pr s t C r s s3 r s rt L ( A P P M
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N m i i = 1,..., N m i Fi x N 1 F ij, j = 1, 2,... i 1, i + 1,..., N m i F i = j i F ij + F x i mi Fi j Fj i mj O P i = F i = j i F ij + F x i, i = 1,..., N P = i F i = N F ij + i j i N i F x i, i = 1,...,
Διαβάστε περισσότερα