Singular behavior of the solution of the Helmholtz equation in weighted L p -Sobolev spaces

Transcription

1 Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces C. De Coster and S. Nicaise Colette.DeCoster,snicaise@univ-valenciennes.fr Laboratoire LAMAV, Université de Valenciennes et du Hainaut Cambrésis, France 6th Singular Days Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 1/2 WIAS Berlin, Aril 29, 2010

2 Outline of the talk The Problem Some Embeddings and consequences The main result Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 2/2

3 The roblem Let be a olygonal domain of R 2 with a Lischitz boundary. On this domain, we consider the heat equation u t u = f in ( π,π), u(x, t) = 0 on ( π,π), u(, π) = u(,π), in, where f L (( π,π),l µ()) (described below) with 2. Goal: Find regularity results for a large range of values of µ. Tools: Uniform estimates for Helmholtz eq. and theory of sum of oerators. Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 3/2

4 Some references [Kozlov, 88]: = 2, full asymtotic exansion. Tool: Fourier analysis [Grisvard, 95]: > 1,µ = 0, decomosition into regular and singular arts. Tool: Theory of sum of oerators. [Nazarov, 01, 03], [Solonnikov, 01], [Pruss-Simonett, 07], [Amann, 09]: > 1, µ large enough to avoid the singularities. Tools: Estimates of the Green function/theory of sum of oerators/blowing u. Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 4/2

5 Reduction In order to give existence and regularity results for such a roblem, we first localize the roblem. Reduce to the truncated sector = {(r cosθ,r sin θ) 0 < r < 1, 0 < θ < ψ}, ψ (0, 2π]. Use the theory of the sum of oerators, hence we need first to study the Helmholtz equation u + zu = g in, u = 0 on, (1) where g L µ() with 2 and z π + S A where Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 5/2

6 π + = {z C R(z) 0}, S A = {z C z R and arg z θ A }, for R > 0 and θ A ] π 2,π[ fixed. Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 6/2

7 Some definitions For > 1,µ R: weighted s. with homogeneous norms: L µ() = {f L loc () rµ f L ()}. and Vµ k, () = {u L loc () u Vµ k, () < }, u := D γ u(x) r (µ+ γ k) (x)dx. Vµ k, () γ k In H0() 1 we will denote its semi-norm by u 2 H = u Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 7/2

8 Embeddings and consequences Le 1. Let 2 and µ satisfies, µ < 2 2, if > 2, µ 1, if = 2. (2) 1. L µ() L 2 1(), 2. L 2 1 () (L µ ()) = L q µ(), 1 q + 1 = 1, 3. H 1 0() L q µ(). Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 8/2

9 Proof 1. follows from Hölder s inequality and the fact that r 1 µ L s () if 1 µ > 2. s 2. consequence of the first one by using duality. 3. a. = 2, it is well known (see Thm in [CDN book]) that H0() 1 L 2 1(). We then conclude observing that, for µ 1: r 2( µ+1) L (). b. > 2, we use the embedding H0() 1 L 2 1() and the second assertion. Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 9/2

10 Coro 1. Let 2 and µ satisfies (2). Then for all g L µ() and all z π + S A, the roblem w H 1 0(), u w + z uw = gw, (3) has a unique solution u H 1 0(). Rk. u H 1 0() is a weak solution of (1): u + zu = g in, u = 0 on. Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 10/2

11 Some Inequalities Le 2. Let 2, µ satisfies (2), z π + S A, and u H 1 0() be the solution of (3). Then u H 1 0 () g L µ (), (4) (1 + z 1/2 ) u L 2 () g L µ (). (5) Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 11/2

12 Proof For Rz 0: Alying (3) with w = u we have u 2 H + z u 2 = 0 1 gū. (6) By Lemma 1, taking the real art of (6), we obtain u 2 H + Rz u 2 g 0 1 L µ u H 1 0. (7) The result follows as Rz 0 and using Poincaré inequality. Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 12/2

13 Coro 2. Let g L 2 (), z C with Rz 0, and u H 1 0() be the solution of (3). Then (1 + z ) u L 2 () g L 2 (). Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 13/2

14 The domain Def 1. Let 2 and µ R. Then we define D(,µ ) = {u H 1 0() u L µ()}. Rk. If µ satisfies (2) and 2 2 µ kλ, k N. Then [Maz ya-plamenevskii, 78] D(,µ ) = V 2, µ () H 1 0() + San {η(r)r λ sin(λ θ) 0 < λ = kλ < 2 2 µ}, η cut-off fct s. t. η = 1 near r = 0 and η(1) = 0. Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 14/2

15 An existence result Le 3. Let 2, µ satisfies (2) and µ > λ, with λ = π ψ, z π + S A, u H0() 1 sol. of (3) with g L µ(). Then u D(,µ ). Proof. As Le 1 H0() 1 L µ () for all µ > 2, we distinguish different cases: 1. µ > 2 and therefore u = g zu L µ() < µ < 2. Take µ = µ + 2. Since µ > 2, u H0() 1 is solution of u = g zu L µ (). Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 15/2

16 This imlies that u V 2, µ (), because the set {λ = k π,k Z : 0 < ψ λ < 2 2 µ } = (the assumtion µ > λ λ > 2 µ). Accordingly r µ 2 u L () u L µ(), due to µ 2 = µ. This guarantees u = g zu L µ(). The general case follows by induction. Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 16/2

17 An a riori estimate Le 4. Let λ = π, 2, µ > λ satisfy (2) and ψ 4( 1)λ 2 + 2µ 2 µ2 > 0. (8) Let z C with Rz 0, u D(,µ ) sol. of (3) with g L µ(). Then Rz u L µ g L µ and Iz u L µ g L µ. Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 17/2

18 Proof Some integrations by arts v = r µ u satisfies v 2 v v 4 v 2 ( v) 2 µ µ r 1 v r v 2 v + z v = r µ g v 2 v, R( r 1 v r v 2 v) = r 2 v. r 2 v Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 18/2

19 Setting w = v /2, these two identities lead to 4( 1) w 2 + ( 2µ 2 D µ2 ) r 2 w 2 D +Rz w 2 R( g v 2 v). ( Poincaré in θ 4( 1)λ µ µ2 ) D D D w 2 λ 2 1, w H r 0() 1 2w2 r 2 w 2 +Rz w 2 R( g v 2 v) D D Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 19/2

20 The main result Thm 1. Let 2, and let µ > λ satisfies (2), (8) and, for all k Z 2 2 µ kλ. Then, for all z π+ S A, u D(,µ ) sol. of (3) with g L µ(), i.e. weak sol. of (1): u + zu = g in, u = 0 on, admits the decomosition u = u R + c λ (z)p λ (r z)e r z r λ sin(λ θ), (9) 0<λ =kλ<2 2 µ with u R V 2, µ (), c λ (z) C and P λ (s) = k λ > 2 2/ µ λ. k λ 1 i=0 s i i!, Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 20/2

21 The main result: uniform estimates u R V 2, µ () + z 1/2 u R V 1, µ () + z u R L µ () g L µ (); 0<λ <2 2 µ c λ (z) (1 + z 1 1 µ+λ 2 ) g L µ (). Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 21/2

22 Sketch of the roof Lemma 4 g zu L µ () g L µ (), hence u can be seen as a solution of u = g zu in, u = 0 on, and by standard regularity results: u = u 1R + c λ (z)r λ 0<λ =kλ<2 2 µ sin(λ θ), with u 1R V 2, µ (), c λ (z) C. Factors P λ (r z)e r z : to have uniform estimates in z. Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 22/2

23 One alication Thm 2. Let 2, and let µ = 1 λ satisfies (8) and, for all k Z 2 2 µ kλ. Then f L ((0, );L µ()), a sol. of t u u = f in (0, ), u = 0 on {t = 0}, that admits the decomosition u = u R + (E(r, ) t q λ )r λ 0<λ =kλ<2 2 µ sin(λ θ), (10) with u R L ((0, );V 2, µ ()) W 1, ((0, );L µ()), q λ W 1 1 µ+λ 2, (0, ) and E(r,t) = rt e r 2 4t. Singular behavior of the solution of the Helmholtz equation in weighted L -Sobolev saces. 23/2