Inverse Scattering with Partial data on Asymptotically Hyperbolic Manifolds

Transcription

1 Inverse Scattering with Partial data on Asymptotically Hyperbolic Manifolds Raphael Hora UFSC 29/04/2014 Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

2 Motivation: The BC Method Belishev and Kurylev ( 92) introduced a method that gives an efficient way to reconstruct a Riemannian manifold via the Dirichlet-to-Neumann operator. Given a compact Riemannian manifold (Ω, g) with boundary Ω, consider the problem u tt g u = 0 on Ω (0, T ), (1) with u t=0 = u t t=0 = 0 and u Ω [0,T ] = f. Let u = u f (x, t) be the solution. The Dirichlet-to-Neumann operator is defined by R T : f u ν Ω [0,T ], where u ν is the normal derivative of u. We have that the operator R2T is determined by Ω T = {x Ω : dist(x, Ω) < T }. Belishev and Kurylev ( 92) showed that the operator R 2T determines (Ω T, g), modulo isometries. Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

3 The Scattering Matrix It is known that in an asimptotically hyperbolic manifolds, if x is a boundary defining function, then f C ( X ) and λ R \ {0}, there is a unique u C ( X ) such that ) ( g n2 4 λ2 u = 0 is given by u x n/2+iλ u + j (y)x j + x n/2 iλ u j (y)x j, u 0 + = f. (2) j=0 j=0 The scattering matrix at energy λ 0 is the operator A(λ) : C ( X ) C ( X ), f u 0. (3) One can see that given f, the functions u ± j, j 1, are uniquely determined by f, A(λ)f and the Taylor series of h(x) on x = 0. Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

4 Dirichlet-to-Neumann and the Scattering Matrix If (X, g) is a compact manifold X with boundary, the Laplacian g is a elliptic operator, so given f C ( X ), there is an unique u C (X ) such that g u = 0 in X and u X = f. If (x, y) are normal coordinates on X, one can show that the Taylor series of u(x, y) on X = {x = 0} is determined by the first two terms, u(0, y) = f (y) and x u(0, y), and by the equation. Then, x u(0, y) is globally determined by the equation and f. The Dirichlet-to-Neumann operator is given by Λ g : C ( X ) C ( X ), where Λ g f = x u X. Thus, here, one can see the scattering matrix as a generalization of the Dirichlet-to-Neumann operator. Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

5 Our Problem Sá Barreto ( 05) showed that on AH manifolds, A(λ), λ R \ 0, determines (X, g) modulo certain difeomorphisms. H and Sá Barreto considered the same problem with partial data for A(λ) restrict to an open subset Γ of the boundary X. One can define the following operator: A Γ (λ)f = (A(λ)f ) Γ, f C 0 (Γ). H and Sá Barreto showed that given (X 1, g 1 ) and (X 2, g 2 ) and an open Γ X j, such that X j \ Γ doesn t have empty interior, j = 1, 2, and if A j,γ (λ), j = 1, 2, λ R \ {0}, satisfy A 1,Γ (λ) Γ = A 2,Γ (λ) Γ, λ R \ {0}, then there is a difeo. Ψ : X 1 X 2, smooth up to X 1, such that Ψ = Id on Γ and Ψ g 2 = g 1. (4) Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

6 Our Model Space We work on spaces which are modeled by the by the hyperbolic space B n+1 = {z R n+1 : z < 1} with the metric g 0 = 4dz 2 (1 z 2 ) 2. If one uses polar coordinates, r = z and θ = z/ z, and take x = 1 r 1 + r, then on [0, 1) S n, the metric g 0 is given by g 0 = dx 2 x 2 + (1 x 2 ) 4 dθ 2 x 2. Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

7 Asymptotically Hyperbolic Manifolds A smooth manifold (X, g) with boundary X is called asymptotically hyperbolic if H = x 2 g (5) is smooth on X and non-degenerate on X, where x C (X ) is a boundary defining function. Joshi and Sá Barreto ( 99) showed that if g satisfies (5) above, then there is ɛ > 0 and a unique product structure X [0, ɛ) X such that g = dx 2 x 2 h(x, y, dy) + x 2, h 0 = h(0, y, dy). (6) We also assume that dx H = 1 on X, so that the sectional curvature goes to 1 at infinity. Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

8 Idea of the Proof The case where the Dirichlet-to-Neumann is known only on part of the boundary, the reconstruction of a compact manifold was done by Kurylev and Lassas ( 00), using a modified version of the boundary control method of Belishev and Kurylev. As done by Sá Barreto ( 05), H and Sá Barreto adapt the methods of boundary control of Belishev, Kurylev and Lassas, and a unique continuation result of Tataru ( 95), to this setting, using a dynamic definition of the scattering matrix via the radiation fields of Friedlander. Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

9 Spectral Theory on AH Manifolds Mazzeo and Melrose ( 87) showed that the spectrum of g, σ( g ), consists of a discrete set σ pp ( g ) and a absolutely continuous spectrum σ ac ( g ) which satisfies [ ) n 2 σ ac ( g ) = 4,, σ pp ( g ) ) (0, n2. 4 This gives a decomposition of L 2 (X ), L 2 (X ) = L 2 pp(x ) L 2 ac(x ). Let u be the solution of the wave equation ) (D 2t g + n2 u(t, z) = 0 on R + X, 2 u(0, z) = f 1 (z), D t u(0, z) = f 2 (z), f 1, f 2 C 0 ( X ). (7) Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

10 Energy Spaces This equation has an conserved energy given by ) E(u, t u)(t) = ( du(t) 2 n2 X 4 u(t) 2 + t u(t) 2 dvol g, ) E(u, t u)(0) = E(f 1, f 2 ) = ( df 1 2 n2 4 f f 2 2 dvol g. We define the energy space by X (8) H E (X ) = {(f 1, f 2 ) : f 1, f 2 L 2 (X ), df 1 L 2 (X ) e E(f 1, f 2 ) < } and the projection P ac : L 2 (X ) L 2 ac(x ) N f f f, φ j φ j, j=1 where {φ j, 1 j N} are the eigenfunctions of g. Let E ac (X ) = P ac (H E (X )). Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

11 Friedlander s Radiation Fields Given (f 1, f 2 ) C 0 (X ) E ac (X ), Sá Barreto ( 05) showed V + (x, s, y) = x n/2 u(s log x, x, y) C 0 ([0, ɛ) R s X ), (9) and the forward radiation field is defined by R + : C 0 ( X ) C 0 ( X ) C (R X ), R + (f 1, f 2 )(s, y) = D s V + (0, s, y) = lim x 0 x n/2 D s u(s log x, x, y). (10) Similarly, if one considers u for t < 0, V (x, s, y) = x n/2 u(s + log x, x, y) C ([0, ɛ) x R s X ), R (f 1, f 2 )(s, y) = D s V (0, s, y). (11) Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

12 The Scattering Operator Sá Barreto ( 05) showed that R ± extend to unitary operators R ± : E ac (X ) L 2 (R X ), which are translation representations of the wave group, U(f 1, f 2 )(t) = (u(t), t u(t)), in the theory of Lax and Phillips, i.e., R ± (U(T )(f 1, f 2 ))(s, y) = R ± (f 1, f 2 )(s + T, y). (12) We define the scattering operator by S : L 2 (R X ) L 2 (R X ), S = R + R, (13) which is unitary and commutes with translations. Sá Barreto showed that it is equivalent to define the scattering matriz, A(λ), by A(λ) = FSF 1, (14) where F is the Fourier transform in s. Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

13 A Support Theorem for the Dirichlet-to-Neumann Operator First we stablish a relation between the support of R + (0, f ) and the support of f. It is interesting to see the analogy with the Dirichlet-to-Neumann operator on compact manifolds. If follows from the finite speed of propagation of the wave equation that if Γ X is open and u(t, z) is a soltuion of the Dirichlet problem ( 2 t + g )u = 0 on ( T, T ) X u(0, z) = 0, t u(0, z) = f (z) C ( X ), u(t, z) ( T,T ) X = 0, then ν u(t, z) ( T0,T 0 ) X = 0, if t (0, T ), z Γ and d g (Supp f, Γ) < t. The converse of this result is also valid and follows from a theorem of Tataru ( 95). Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

14 A Support Theorem for the Scattering Matrix Sá Barreto ( 05) proved the following. Theorem (Sá Barreto). A function f L 2 ac(x ) satisfies R + (0, f ) = 0 for all s s 0 << 0 and y X if and only if f (x, y) = 0 if x e s 0, y X. H and Sá Barreto ( 13) proved a similar result with partial data. Theorem (H e Sá Barreto). Let Γ X be open and s 0 R. A function f L 2 ac(x ) satisfies R + (0, f ) = 0 for all s s 0 and y Γ if and only if there exists ɛ > 0 such that f = 0 a.e. on the set D s0 (Γ) = {z X : ω = (α, y ) with 0 < α < ɛ e s 0 e y Γ, d g (z, ω) ( e s 0 ) < log }, α where d g denotes the distance with respect to the metric g. Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

15 The set D s0 It is interesting to see that D s0 (Γ) is an union of open ( balls with center at e (α, y ), with 0 < α < ɛ, and y s 0 ) Γ, and radii log. α Lax and Phillips ( 82) showed a similar theorem in the hyperbolic setting. Here, ( this open set corresponds ) to the ball D(α) with center at 1 2 es 0 (1 + α 2 e 2s 0 ), y and radius 1 2 es 0 (1 α 2 e 2s 0 ). Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

16 The Control Space One of the key ideas of the general case is to show that the range of the radiation fields M ± = R ± (0, L 2 ac(x )) = {R ± (0, f ) : f L 2 ac(x )} are closed subspaces of L 2 (R X ), and determined by the scattering operator. We define similar spaces for functions with support in R Γ. First L 2 (R Γ) = {F R Γ : F L 2 (R X )}, and S R Γ : L 2 (R Γ) L 2 (R Γ), F (SF ) R Γ. H and Sá Barreto proved that given an open Γ X, X \ Γ, the set M ± = {R ± (0, f ) R Γ : f L 2 ac(x )}, with norm N defined by N (R ± (0, f ) R Γ ) = f L 2 (X ) is a Hilbert space determined by S R Γ. Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

17 First Step of the Proof First, we construct an isometry (as in the main theorem) between neighborhoods of Γ. Then we apply Kurylev and Lassas result to show that the difeomorphism between the two neighborhoods can be extended to a difeomorphism between the two manifolds. Our first result is the following. Proposition. Suppose that (X 1, g 1 ), (X 2, g 2 ) and Γ satisfy the hypothesis of the main theorem. Let R j,± (s, y, x, y ) be the respective radiation fields acting on pairs of the form (0, f ), f L 2 ac(x ). Then there is an ɛ > 0 such that on [0, ɛ) Γ, h 1 (x, y, dy) = h 2 (x, y, dy) and R 1,± (s, y, x, y ) = R 2,± (s, y, x, y ), if y, y Γ, x < ɛ. Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

18 Proof of the Proposition The key is to understand the effect of the following orthogonal projections at the initial data P ± x 1 : M ± (Γ) M ± x 1 (Γ), where M + x 1 (Γ) = {F M + (Γ) : F (s, y) = 0, s log x 1 }, M x 1 (Γ) = {F M (Γ) : F (s, y) = 0, s log x 1 }. It follows from the support theorem that M + x 1 (Γ) = {R + (0, h) R Γ : h L 2 ac(x ), h(z) = 0, z D log x1 (Γ)}, M x 1 (Γ) = {R (0, h) R Γ : h L 2 ac(x ), h(z) = 0, z D log x1 (Γ)}, Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

19 Relation between P ± x 1 and R ± Lemma. Let χ x1 be the characteristic function of X x1 = X \ D log x1 (Γ). Then there is an ɛ 0 such that if ɛ < ɛ 0, for all f L 2 ac(x ) there exists α(x 1, f ), which is a linear function of f, such that N Px 1 (R (0, f ) R Γ ) = R 0, χ x1 f α j (x 1, f )φ j. R Γ Now consider T (x 1 )f = j j=1 α j (x 1, f )φ j. Lemma. Let F M (Γ), F = R (0, f ) R Γ with F and f smooth. Then there exists an ɛ > 0 such that for all x 1 (0, ɛ), and for s log x 1 small enough, R + R 1 (P x 1 F )(s, y) = 1 2 x n/2 1 [(Id T (x 1 ))f ](x 1, y) h 1/4 (x 1, y) h 1/4 (0, y) (s log x 1) termos suaves. Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

20 Difeomorphism between Neighborhoods The fact that the metrics are the same on certain coordinates imply that there are neighborhoods V j,ɛ X j, j = 1, 2, of Γ and smooth difeomorphisms Ψ j,ɛ : Γ [0, ɛ) V j,ɛ X j such that Then, the map defined by Ψ ɛ = Ψ 1,ɛ Ψ 1 2,ɛ satisfies Ψ 1,ɛ(g 1 V1,ɛ ) = Ψ 2,ɛ(g 2 V2,ɛ ). Ψ ɛ : V 1,ɛ V 2,ɛ, Ψ ɛ(g 2 V2,ɛ ) = g 1 V1,ɛ, and Ψ ɛ = Id on Γ. Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

21 Last Step Note that X j,ɛ = X j \ {x j < ɛ} are compact manifolds with boundary and there are open subsets Γ 1,ɛ X 1,ɛ, Γ 2,ɛ X 2,ɛ which are identified by difeomorphisms Ψ j,ɛ with Γ {ɛ} = Γ ɛ. To use result of Kurylev and Lassas, we have to show that (X j,ɛ, g j ), j = 1, 2, have the same spectral data on Γ ɛ. Then we have that for every ɛ > 0, there is a smooth difeo. Ψ ɛ such that Ψ ɛ : X 1,ɛ X 2,ɛ, Ψ ɛ Γ = Id, Ψ ɛ(g 2 X2,ɛ ) = g 1 X1,ɛ. We can extend this map Ψ ɛ to a smooth difeo. Ψ which satisfies the result of our theorem. Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

22 End of the Proof ( n ) Let E j 2 + iλ, y, z = R j,+ (s, y, z), then ( n ) ( n ) E iλ, y, z = E iλ, y, z, y Γ, z = (x, y) [0, ɛ) Γ. Using armungents of Melrose ( 95), Sá Barreto showed that for all λ 0, the functions given by ( n v j (z, λ) = 2 + iλ, y, z) φ(y )dvol h0 (y ), j = 1, 2, φ C ( X j ), X j E j form a dense subset on the set of the solutions of ) ( gj λ 2 n2 u = 0 on X j,ɛ, j = 1, 2, 4 on the Sobolev space H k (X j,ɛ ), k 2. It follows from what we showed above that v 1 (x, y, λ) = v 2 (x, y, λ), (x, y) (0, ɛ) Γ, λ R \ {0}. Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23

23 THANK YOU! Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/ / 23