A local estimate from Radon transform and stability of Inverse EIT with partial data

Transcription

1 A local estimate from Radon transform and stability of Inverse EIT with partial data Alberto Ruiz Universidad Autónoma de Madrid U. California, Irvine.June 2012 w/ P. Caro (U. Helsinki) and D. Dos Santos Ferreira (Paris 13) Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

2 The purpose of this talk is to address the following points To prove a local control of the L p norm of a function by its local Radon transform. Quantitative version of Helgason-Holmgren Thm. (Inspired by: DSF-K-Sj-U, Boman, Hörmander). To appy the above to prove stability for Calderón Inverse problem in two partial data contexts in which uniqueness holds true. (1) Bukhgeim and Uhlmann (Recovering a potential from partial Cauchy data 2002)(semiglobal) (2) Kenig, Sjöstrand and Uhlmann (The Calderón Problem for partial data 2007) Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

3 Description of Partial data: Calderón inverse problem. Let u be the solution of the Dirichlet BVP { (γ u) = 0 u Ω = f H 1 2 ( Ω) γ is a positive function of class C 2 on Ω. The Dirichlet-to-Neumann map: Λ γ f = γ ν u Ω, in Ω (0.1) where ν denotes the exterior normal derivative of u. Partial data: We assume F, G Ω. Data restricted to f s so that supp f B and measurements ν u restricted to F. After standard reduction to Schrödinger equation: ( + q)v = 0. Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

4 Bukhgeim and Uhlmann (light from infinity) Assume q 1, q 2 be two bounded potentials on Ω, suppose that 0 is neither a Dirichlet eigenvalue of the Schrödinger operator + q 1 nor of + q 2. Given a direction ξ S n 1. Consider the ξ-illuminated boundary and the ξ-shadowed boundary Ω (ξ) = {x Ω : ξ ν(x) 0} Ω + (ξ) = {x Ω : ξ ν(x) 0}. F, B two open neighborhoods in Ω, respectively of the sets Ω (ξ) and Ω + (ξ). If the two Dirichlet-to-Neumann maps with Dirichlet data f H 1/2 ( Ω) supported in B coincide on F Λ q1 f (x) = Λ q2 f (x), x F then the two potentials agree q 1 = q 2. Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

5 Kenig, Sjöstrand and Uhlmann, light from a point In this case consider x 0 out of Convex hull of Ω. The x 0 -illuminated boundary and the x 0 -shadowed boundary Ω (x 0 ) = {x Ω : (x x 0 ) ν(x) 0} Ω + (x 0 ) = {x Ω : (x x 0 ) ν(x) 0}. F, B two open neighborhoods in Ω, respectively of the sets Ω (ξ) and Ω + (ξ). If the two Dirichlet-to-Neumann maps with Dirichlet data f H 1/2 ( Ω) supported in B coincide on F Λ q1 f (x) = Λ q2 f (x), x F then the two potentials agree q 1 = q 2. Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

6 Previous results. Stability for partial data BU case H. Heck and J.N. Wang (2006) (Without the support constrain on Dirichlet data) L. Tzou (2008) (extends to the magnetic case, without the support constrain on Dirichlet data) Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

7 Previous results. Stability for partial data BU case H. Heck and J.N. Wang (2006) (Without the support constrain on Dirichlet data) L. Tzou (2008) (extends to the magnetic case, without the support constrain on Dirichlet data) KSU case Nachman and Street (2010) (Reconstruction of Radon-like integrals of the potential) Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

8 Our results: Stability in BU partial data Theorem Let Ω be a bounded open set in R n, n 3 with smooth boundary, Assume given an open set N in S n 1 and consider two open neighbourhoods F, B respectively of the front and back sets Ω (ξ) and Ω + (ξ) of Ω for any ξ N. Let q 1, q 2 be two allowable potentials on Ω, suppose that 0 is neither a Dirichlet eigenvalue of the Schrödinger operator + q 1 nor of + q 2, then q 1 q 2 L p C ( log log Λq1 Λ q2 B F ) λ/2 (0.2) Remarks: The norm Λ q1 Λ q2 B F (Nachman-Street) The allowable potentials: L W λ,p Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

9 Our results: Stability in KSU partial data Theorem Let Ω be a bounded open set in R n, n 3 with smooth boundary, Assume given an open set N in R 3 and consider two open neighbourhoods F, B respectively of the front and back sets Ω (x 0 ) and Ω + (x 0 ) of Ω for any x 0 N. Let q 1, q 2 be two allowable potentials on Ω, suppose that 0 is neither a Dirichlet eigenvalue of the Schrödinger operator + q 1 nor of + q 2, then q 1 q 2 L p (G) C ( log log Λ q1 Λ q2 B F ) λ/2 (0.3) where G is an open neighborhood in R 3 of the penumbra region F B Remarks: The norm Λ q1 Λ q2 B F (After F. J. Chung, A Partial Data Result for the Magnetic Schrodinger Inverse Problem. Norm H 1/2 ( Ω) H 1/2 ( Ω)) Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

10 Steps in the proof tep 1 Stability from DtoN map to partial Radon transform. tep 2 Stability from partial Radon transform to potential Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

11 Second step: from limited angle-distance Radon data local values of the function Quantitave Helgason-Holmgren Theorem: Limited data: Distance Angle (ω 0 S n 1 ) Dependence domain I = {s R : s < α} (0.4) Γ = {ω S n 1 : (ω ω 0 ) 2 > 1 β 2 } (0.5) = {ω S n 1 : d(ω 0, ω) < arcsin β}. (0.6) E = { x R n : ω (x y 0 ) = s, s I, ω Γ }. Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

12 Allowable potentials Let p, 1 p <, and λ, 0 < λ < 1/p, two functions q 1, q 2 satisfy the following conditions: (a) 1 E q j X L (R n ) for j = 1, 2, q j L (E) + 1 Eq j X < M. X-norm q L 1 (R n ) and (1 + s ) n R 0 q(s, ) L 1 (S n 1 ) R ds <. (b) (λ, p, p)-besov regularity on the dependence domain for j = 1, 2. 1 E q j (1 E q j )( y) R n y n+λp p L p (R n ) dy < M p Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

13 Support condition y 0 supp (q 1 q 2 ) supp (q 1 q 2 ) {x R n : (x y 0 ) ω 0 0}. Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

14 Theorem1. Quantitave H-H Theorem Let M 1 be constant. Given y 0 R n, ω 0 S n 1, α > 0 and β (0, 1] Let q i be allowable potentials and assume the support condition holds. Then there exists a positive constant C = C(M, G, α, β), such that q 1 q 2 L p (G) C ( ) λ, log (1 + s ) n R y0 (q 1 q 2 )(s, ) L 1 (Γ) ds 2 I (0.7) where { } G = x R n α : x y 0 <. 8 cosh(8π/β) Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

15 Notation: R y0 q(s, ω) is the y 0 -centered Radon transform, integral of q in H = {x R n : ω (x y 0 ) = s}. Constant ( ) ( C = C n M max 1, G 1 p 1 + α n + β n + α λ) (0.8) Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

16 Sketch of proof 1: uniqueness theorem Assume R y0 (q 1 q 2 ) I Γ = 0. Then such (y 0, ω 0 ) does not exits: 1 Microlocal Helgason s theorem: (y 0, ω 0 ) / WF a (q 1 q 2 ) 2 Microlocal Holmgren s theorem (Hörmander): From the conditions y 0 supp (q 1 q 2 ) and supp (q 1 q 2 ) {x R n : (x y 0 ) ω 0 0} one has (y 0, ω 0 ) WF a (q 1 q 2 ) Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

17 1.Quantitative Microlocal Helgason s theorem.sketch of proof The WF a (u) is characterized be the exponential decay of the wave packet transform: For any u S (R n ) define the Wave packet (Segal-Bargmann) transform of u for ζ C n as T u(ζ) = u, e 1 2h (ζ )2. Where, stands for the duality between S (R n ) and S(R n ) the class of smooth rapidly decreasing functions. Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

18 Step 1:From Radon to wave packet Proposition Consider q A λ (R n ), y 0 R n and ω 0 S n 1. Let R q be a positive constant such that supp q {x R n : x y 0 R q }. Given α > 0 and β (0, 1] consider the sets I = {s R : s < α}, Γ = {ω S n 1 : ω ω 0 2 > 1 β 2 }. Then, there exists a positive constant C such that T q(ζ) C ( Im ζ n + α n + R n q) e 1 Im h n ζ 2 2h 2 [ ] R y0 q L 1 (I Γ) + q L (R n ) (e 1 α 2 2h 4 + e 1 γ 2 β 2 2h 16 ), for all h (0, 1], ζ C n such that Re ζ y 0 < α/2, Im ζ γ > 0 and ω 0 θ 2 > 1 β 2 /4 with θ = Im ζ 1 Im ζ. Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

19 Step 2 Step 2a (Key step) We tray to remove the condition Im ζ γ > 0. Try to go from the wave packets transform of q to the heat evolution of q (t = h) by allowing Im ζ = 0 Step 2b. Estimate the backward initial value problem for the heat equation. Need a priori Besov regularity on q. Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

20 The key Proposition Proposition Under the same notation and assumptions of the Theorem. There exists a positive constant C such that e 1 2h Im ζ 2 T q(ζ) < CM q (( 2α β )n + α n + R n q) R y0 q κ L 1 (I Γ), (0.9) with κ := 1 ( ( )) 2, h := 4 cosh 8π β α 2 8 log R y0 q L 1 (I Γ), for all ζ C n such that Re ζ y 0 < α 8 cosh ( 8π β ), Im ζ < 2α (4 β 2 ) 1/2. Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

21 Proof of the Key proposition Estimates of the wave packet transform: (K1) T q(ζ) (2πh) n 2 q L (R n ) e 1 2h Im ζ 2, (K2) T q(ζ) (2πh) n 2 q L (R n ) e 1 2h Im ζ 2 e 1 2h ω 0 (Re ζ y 0 ) 2, for all ζ C n such that ω 0 (Re ζ y 0 ) 0 (follows from the hyperplane being supporting ) (K3) T q(ζ) C h n 2 ( Im ζ n + α n + R n q) e 1 Im ζ 2 2h [ R y0 q L 1 (I Γ) + q L (R n ) e 1 2h ] α 2 4, for all h (0, 1], ζ C n such that Re ζ y 0 < α/2, Im ζ γ > 0 and ω 0 θ 2 > 1 β 2 /4 with θ = Im ζ 1 Im ζ. (Follows from quantitative Helgason-Holmgren with γ = 2α/β) Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

22 Barrier on a rectangle Lemma Let a, b be positive constants. Consider R = {z C : Re z < a, Im z < b + ε}, for some ε > 0. Let F be a sub-harmonic function in R such that F(z) < (min{0, Re z}) 2, for all z C. Assume that F(z) < 2a 2 for z R such that Im z b. Then ( ) F(z) < 1 2a 2 2 cosh ( 1 π b ) min a cosh ( π b a), 1, 3 for ( ) Im z < b, Re z < a 2 min 1 cosh ( π a), 1 b. 3 Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

23 Proof of lemma Take the subharmonic function for G(x + iy) = 2a 2 cosh ( π a y) ( π ) cosh ( π sin a b) a (x + δ) + F(x + iy) δ 2, ( ) a δ = min cosh ( π b a), a 3 Check that G < 0 on the boundary of [ δ, a δ] [ b, b]. The estimate follows from the maximun principle when we restrict to the values x < δ/2. Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

24 End of proof of Key step α Choose h := 2, to obtain an appropriate negative 8 log R y0 q L 1 (I Γ) exponential in (K3). Let us denote z = ω 0 (ζ y 0 ) C and write ζ = (z + ω 0 y 0 )ω 0 + w with w C n such that Re w ω 0 = Im w ω 0 = 0. Use the Lemma for the function Φ(z) = Re z 2 Im z 2 + 2h log T q((z + ω 0 y 0 )ω 0 + w) ( ) e 1 Im w 2 2h +2h log CM q (ρ n + α n + R n, q) in the rectangle a = α/4, b = 2α/β Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

25 Step 2b.The backward estimate The key point is the fact that the Segal-Bargmann transform restricted to real values is a convolution with the Gaussian. Backward estimate for the heat equation h = time Lemma Consider q L p (R n ) and G an open set in R n such that supp q G. Assume that there exists λ (0, 1) such that L q := ( q q( y) R n y n+λp p L p (R n ) dy ) 1/p < +. Then, there exists a positive constant C, only depending on n, such that ) q L p (G) (h C n 2 T q L p (G) + L qh λ 2, Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

26 First step: from DtoN map to Radon transform The uniqueness Results. Locally vanishing Radon Transform. Stability: Hunk and Wang: Log-log type stability (log stable recovery of the Fourier tranform +log stable continuation of analytic functions) The Carleman estimates with signed partition of the boundary allow to partially recover the 2-planes X-ray transform in some sets I Γ which are neighborhood of (y 0, ω 0 ) from the partial Dirichlet to Neuman map in a logarithmic stable way (following the approach of DSF, Kenig, Sjoestrand and Uhlmann). This allows to control the norm of q 1 q 2 in a neighborhood of y 0. In the case of BU ( semiglobal ) one can estimate q 1 q 2 L p, since we have the control of R y0 (q 1 q 2 ) L 1 ((0, ) Γ). In KSU We have the Radon transform if n = 3. Giving us the control on q 1 q 2 at points for which we control the Radon transform (the penumbra boundary) Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

27 From partial data to Radon transform We reduce to Schrödinger + q equation in the standard way. Use of DSF- K-S-U approach. Existence of Calderón- Fadeev type solutions. Use of the Carleman estimate with splitting in the illuminated and shadowed regions. Theorem ( [BU] and [NS]) Theorem For τ >> 1 sufficiently large and g C (R n 2 ) and (ξ + iζ) 2 = 0, there exists a unique solution w τ H(Ω; ) of the equation ( + q)w τ = 0 in Ω, such that tr 0 w τ H( Ω) E (B) and which can be written as w τ (x) = e τ(ξ+iζ) x (g(x ) + R(τ, x)) where R(τ, ) L 2 (Ω) C 1 τ ( qg L 2 (Ω) + τ 1/2 g L 2 (Ω)). The same is true by changing τ by τ and B by F. Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

28 The spaces of solutions in H(Ω; ) Traces (in H( Ω) H 1/2 ( Ω)) Integral formula by using extensions of data (Λ q2 Λ q1 )(φ)ψ = P q1 (φ)(q 1 q 2 )P q2 (ψ). (0.10) Difference of DtoN maps Lemma Let q j, j = 1, 2 be L potentials so that 0 is not a Dirichlet eigenvalue of + q j in Ω. Then Λ q2 Λ q1 extends to a continuous map H( Ω) H( Ω). Ω Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

29 2-plane Radon transform estimate Proposition For any g C (R n 2 ) there exits C > 0 which only depends on Ω and the a priori bound of q j L, such that sup ξ N,ζ ξ g(x ) [ξ,ζ] q(x + tξ + sζ)dtdsdx R 2 (0.11) C(τ 1/2 + e cτ Λ q1 Λ q2 B F ) g H 2 (Ω) (0.12) By choosing appropriate g = g(rη) for any η [ξ, ζ] sup ξ N sup η ξ g(r)rq(r, η)dr (0.13) R C(τ 1/2 + e cτ Λ q1 Λ q2 B F ) g H 2 (R) (0.14) Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

30 L 1 norm of Radon transform Interpolation with (1 + τ 2 ) (n 1)/2 Rq(, η)ˆ(τ) 2 dτdσ(η) q L 2 S n 1 R gives: Theorem (Stability of Radon transform) ( M ( Rq(r, η) 2 dr R ) n+3 4 dσ(η) ) 2 n+3 (0.15) C(τ 1/2 + e cτ Λ q1 Λ q2 B F ) n+1 n+3. (0.16) Where M S n 1 M = ξ N [ξ], Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

31 Looking for the supporting plane H in quantitative Helgason-Holmgren Let us take η M and by translation there exist r R and y 0 supp q such that H(s, η) contains y 0 and it is a supporting hyperplane for supp q. We are now in the hypothesis of the previous theorem, since M is a neighborhood of η in the sphere and we can control the Radon transform por ω M and s R. Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

32 Open problem: Cloaking of 0 s????? Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

33 Open problem: Cloaking of 0 s????? 60 =06 Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31

34 Open problem: Cloaking of 0 s????? 60 =06 FELICIDADES GUNTHER Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June / 31