Generalized Lax-Milgram theorem in Banach spaces and its application to the mathematical fluid mechanics.

Transcription

1 Second Italian-Japanese Workshop GEOMETRIC PROPERTIES FOR PARABOLIC AND ELLIPTIC PDE s Cortona, Italy, June 2-24, 211 Generalized Lax-Milgram theorem in Banach spaces and its application to the mathematical fluid mechanics. Hideo KOZONO Mathematical Institute, Tohoku University June 21, 211

2 jointly with Taku YANAGISAWA(Nara Women s Univ.) 1. Introduction Lax-Milgram Theorem. X: Hilbert space a(, ) : X X C: bilinear form (i) (bi-continuity) M > such that a(u, ϕ) M u ϕ, u X, ϕ X, (ii) (coercive estimate) δ > such that δ u 2 a(u, u), u X F X,!u X such that Remark. a(u, ϕ) = F (ϕ), ϕ X. u δ 1 F X.

3 c.f.application. Ω R n : bounded domain, Ω C (E) { Lu = f in Ω, Bu = on Ω, L: elliptic operator, B: boundary operator Example 1; Dirichlet Problem L = + c(x), c(x), Bu = u Dirichlet condition X = H 1 (Ω) a(u, ϕ) = ( u, ϕ) + (cu, ϕ), u, ϕ X Lax-Milgram Theorem F H 1 (Ω) H 1(Ω),!u H 1 (Ω) s. t. ( u, ϕ) + (cu, ϕ) = F, ϕ ϕ H 1 (Ω), i.e., u is a unique weak solution of (E).

4 Example 2; Neumann Problem L = + c(x), c(x) >, Bu = u ν X = H 1 (Ω) a(u, ϕ) = ( u, ϕ) + (cu, ϕ), Neumann condition u, ϕ X Lax-Milgram Theorem F H 1 (Ω),!u H 1 (Ω) s. t. ( u, ϕ) + (cu, ϕ) = F, ϕ ϕ H 1 (Ω), Question. L r -case? Problem. F W 1,r (Ω),!u W 1,r (Ω) s.t. ( u, ϕ) + (cu, ϕ) = F, ϕ ϕ W 1,r (Ω), 1/r + 1/r = 1.

5 Simader s approach (LNM 256, Springer 1972) a(u, ϕ) = ( u, ϕ) + (cu, ϕ), u W 1,r 1,r (Ω), ϕ W (Ω), (i) (bi-continuity) M > s.t. a(u, ϕ) M u W 1,r (Ω) ϕ W 1,r (Ω) (ii) (variational inequality) u W 1,r (Ω) C sup ϕ W 1,r (Ω) u W 1,r 1,r (Ω), ϕ W (Ω); a(u, ϕ) ϕ W 1,r (Ω) F W 1,r (Ω) W 1,r (Ω),!u W 1,r (Ω) s.t., u W 1,r (Ω), a(u, ϕ) = F, ϕ, ϕ W 1,r (Ω) The same for the 2m-order elliptic boundary value problem.

6 2. Generalized Lax-Milgram Theorem. (X, X ), (Y, Y ): Banach spaces a(, ) : X Y C: bilinear form Question. F Y, w X such that a(w, ϕ) = F (ϕ), ϕ Y with the estimate w X C F Y?

7 Assumption. (i) (bi-continuity) M > s.t. a(u, ϕ) M u X ϕ Y, u X, ϕ Y ; (ii) (decomposition by direct sum) N X {u X; a(u, ϕ) =, ϕ Y }, N Y {ϕ Y ; a(u, ϕ) =, u X} R X : closed subspace in X, R Y : closed subspace in Y s. t. X = N X R X (direct sum), Y = N Y R Y (direct sum). (iii) (variational inequality) C > s.t. u X C ϕ Y C ( sup ϕ Y ( sup u X a(u, ϕ) + P X u X ϕ Y a(u, ϕ) + P Y ϕ X u X ) ), u X,, ϕ Y, P X : X N X, P Y : Y N Y : projections.

8 Theorem. (generalization of the Lax-Milgram theorem) (X, X ): Banach space, (Y, Y ): reflexive Banach space a(, ) : X Y C: bilinear form satisfying the Assumption. F N Y, i.e., F Y with F (φ) =, φ N Y, w X such that with a(w, ϕ) = F (ϕ), w X C F Y, ϕ Y Remarks. (i) X need not be reflexive. (ii) Lax-Milgram theorem on the Hilbert space Theorem (iii) dim.n X <, dim.n Y < R X X, R Y Y : closed subspaces s. t. X = N X R X, Y = N Y R Y (direct sums). Theorem Fredholm alternative!

9 (iv) Fichera-Faedo s theorem W : normed space X, Y: Banach spaces S : W X, T : W Y: bounded linear operators k > s.t. T ϕ Y k Sϕ X, ϕ W F Y, g X s.t. g, Sϕ X,X = F, T ϕ Y,Y, ϕ W Lax-Milgarm theorem on the Hilbert space

10 3. Applications.; L r -de Rham-Hodge-Kodaira decomposition on Riemannian manifolds with the boundary. ( Ω, g): compact n-d Riemannian manifold with Ω C. ν T Ω τu = ν (ν u), νu = ν u for u l (T Ω), 1 l n, ν : l (T Ω) l 1 (T Ω): the interior product defined by (ν u)(x 1,, X l 1 ) = u(x 1,, X l 1, ν) for X 1,, X l 1 T Ω. u = τu + ν (νu) for all u l (T Ω). d : l (T Ω) l+1 (T Ω): the exterior derivative, l =, 1,, n 1, : l (T Ω) n l (T Ω): the Hodge star operator, l =, 1,, n, δ : l (T Ω) l 1 (T Ω): the co-differential operator, l = 1,, n, δ = ( 1) n+1 d χ n, χu = ( 1) l u for u l (T Ω) (u, v) Ω u v, for u, v l (T Ω).

11 E r d (Ω)l 1 {u L r (Ω) l 1 ; du L r (Ω) l }, E r δ (Ω)l {v L r (Ω) l ; δv L r (Ω) l 1 }, l = 1,, n, 1 < r < τ : u E r d (Ω)l 1 (Ω) τu W 1/r ( Ω) l 1 = (W 1 1/r,r ( Ω) l 1 ), ν : v E r δ (Ω)l νv W 1/r,r ( Ω) l 1 = (W 1 1/r,r ( Ω) l 1 ), s. t. the generalized Stokes integral formula holds: (du, v) (u, δv) = τu, νv Ω, l = 1,, n, {u, v} E r d (Ω)l 1 W 1,r (Ω) l or {u, v} W 1,r (Ω) l 1 E r δ (Ω)l. Xd r (Ω)l+1 {α W 1,r (Ω) l+1 ; dα = in Ω, να = on Ω}, Vδ r {β W 1,r (Ω) l 1 ; δβ = in Ω, τβ = on Ω}.

12 Theorem 3.1. ( Ω, g): compact n-dimensional Riemannian manifold, Ω C. 1 < r <, l = 1,, n 1. (i) ω L r (Ω) l, α X r d (Ω)l+1, β V r δ (Ω)l 1, h C (Ω) l L r (Ω) l with dh =, δh = s. t. with ω = δα + dβ + h α W 1,r + β W 1,r + h r C ω r. α X r d (Ω)l+1, β V r δ (Ω)l 1, h C (Ω) l L r (Ω) l with dh =, δh = s.t. ω = δα + dβ + h δα = δα, dβ = dβ, h = h. (ii) ω W s,r (Ω) l, s 1 α X r d (Ω)l+1 W s+1,r (Ω) l+1, β V r δ (Ω)l 1 W s+1,r (Ω) l 1, h C (Ω) W s,r (Ω) l with α W s+1,r + β W s+1,r + h W s,r C ω W s,r.

13 Corollary 3.2. (L r -Helmholtz-Weyl decomposition in 3D-domains) Ω R 3 : bounded domain, Ω C, 1 < r <. (i) u L r (Ω), α W 1,r (Ω) (scalar potential), β W 1,r (Ω): div β =, β ν Ω = (vector potential), h C ( Ω): div h =, rot h =, h ν Ω = (harmonic vector) s.t. u = α + rot β + h. (ii) u L r (Ω), β W 1,r (Ω), (scalar potential) α W 1,r (Ω) with div α =, α ν Ω = (vector potential), h C ( Ω) with div h =, rot h =, h ν Ω = (harmonic vector) s.t. u = β + rot α + h. Proof. Use Theorem 3.1 with g = (δ ij ) 1 i,j 3 the Euclidean metric. (i) n = 3, l = 2 (ii) n = 3, l = 1

14 Lemma 3.3. ( Ω, g):compact n-dimensional Riemannian manifold, Ω C. 1 < r <, l = 1,, n 1. (1) ω L r (Ω) l, α X r d (Ω)l+1 s.t. with (δα, δψ) = (ω, δψ), Ψ X r d (Ω)l+1 α W 1,r C ω r. ω W s,r (Ω) l, s 1 α X r d (Ω)l+1 W s+1,r (Ω) l+1 with (2) ω L r (Ω) l, β V r δ (Ω)l 1 s.t. with α W s+1,r C ω W s,r. (dβ, dψ) = (ω, dψ), ψ V r δ (Ω)l 1 β W 1,r C ω r. ω W s,r (Ω) l, s 1, β V r δ (Ω)l 1 W s+1,r (Ω) l 1 with β W s+1,r C ω W s,r.

15 Proof of Lemma 3.3. (1) X X r d (Ω)l+1, Y X r d (Ω)l+1, a(, ) : X Y C a(α, Ψ) (δα, δψ), α X, Ψ Y. u W s,r C( du W s 1,r + δu W s 1,r + u r + νu W s 1/r,r ( Ω) ), s 1 for u W s,r (Ω) l+1 (see e.g., Georgesgue) N X = {α X; a(α, Ψ) =, Ψ Y } N Y = {Ψ Y ; a(α, Ψ) =, α X} {H C ( Ω) l+1 ; dh =, δh = in Ω, νh = on Ω} X har (Ω) l+1 dim.x l+1 har (Ω) <. (Assumption (ii)) Variational inequality in X r d (Ω)l+1 α W 1,r C sup Ψ X r d (Ω)l+1 (δα, δψ) Ψ W 1,r + N i=1 (α, Ψ i ) for α X r d (Ω)l+1, where X har (Ω) l+1 = Span.{Ψ 1,, Ψ N }. (Assumption (iii)), 1 < r <

16 ω L r (Ω) l : given, Define F ω Y by F ω (Ψ) = (ω, δψ) for Ψ Y. F ω N Y, i.e., F ω(φ) =, Φ N Y = X har (Ω) l+1 & F ω Y ω r. Theorem α X r d (Ω)l+1 s.t. a(α, Ψ) = F ω (Ψ) (δα, δψ) = (ω, δψ), Ψ X r d (Ω)l+1. ω W s,r (Ω) l+1, s 1 α X r d (Ω)l+1 is characterized by the equations α = dω in Ω, dα = in Ω, να = on Ω. Agmon-Douglis-Nirenberg α X r d (Ω)l+1 W s+1,r (Ω) l+1. (2) The proof of (2) is similar to that of (1).

17 Assumption. (i) (bi-continuity) M > s.t. a(u, ϕ) M u X ϕ Y, u X, ϕ Y ; (ii) (decomposition of direct sums) N X {u X; a(u, ϕ) =, ϕ Y }, N Y {ϕ Y ; a(u, ϕ) =, u X} R X : closed subspace in X, R Y : closed subspace in Y s. t. X = N X R X (direct sum), Y = N Y R Y (direct sum). (iii) (variational inequalities) C > s.t. u X C ϕ Y C ( sup ϕ Y ( sup u X a(u, ϕ) + P X u X ϕ Y a(u, ϕ) + P Y ϕ X u X ) ), u X,, ϕ Y, P X : X N X, P Y : Y N Y : projections.

18 Theorem. (generalization of the Lax-Milgram theorem) (X, X ): Banach space, (Y, Y ): reflexive Banach space a(, ) : X Y C: bilinear form satisfying the Assumption. F N Y, i.e., F Y with F (φ) =, φ N Y, w X such that a(w, ϕ) = F (ϕ), ϕ Y with w X C F Y,

19 Proof of Theorem. decomposition (R X, X ), (R Y, Y ); Banach spaces T : R X R Y T w, ψ a(w, ψ), w R X, ψ R Y,, : duality pairing between R Y and R Y. bi-continuity T B(R X, R Y ) Claim 1. Range(T ) is closed in R Y. Use the variational inequality for a(, ) on X. Claim 2. Range(T ) = R Y Use Y = Y and the variational inequality for a(, ) on Y. f R Y, w R X s.t. T w = f, i.e., a(w, ψ) = f(ψ), ψ R Y.

20 Claim 3. F N Y, i.e., F Y, F (φ) =, φ N Y Indeed, a(w, ϕ) = F (ϕ), ϕ Y with w X C F Y. a(w, ϕ) = a(w, (1 P Y )ϕ) (P Y : Y N Y = {φ Y ; a(u, φ) =, u X}) = F ((1 P Y )ϕ) (Note that (1 P Y )ϕ R Y ) = F (ϕ) (Note that P Y ϕ N Y and F (P Y ϕ) = ) variational inequality for a(, ) on X w X C ( sup ϕ Y = C sup ϕ Y = C sup ϕ Y = C F Y ) a(w, ϕ) + P X w X ϕ Y a(w, ϕ) (Note that w R X P X w = ) ϕ Y F (ϕ) ϕ Y

21 Thank you for your attention!