a local estimate from radon transform and stability of inverse eit with partial...

A local estimate from Radon transform and stability ofInverse 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 2012 1 / 31

The purpose of this talk is to address the following points

To prove a local control of the Lp norm of a function by its local Radontransform.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 twopartial data contexts in which uniqueness holds true.(1) Bukhgeim and Uhlmann (Recovering a potential from partial Cauchydata 2002)(semiglobal)(2) Kenig, Sjöstrand and Uhlmann (The Calderón Problem for partialdata 2007)


Description of Partial data:Calderón inverse problem.Let u be the solution of the Dirichlet BVP

∇ · (γ∇u) = 0 in Ωu|∂Ω = f ∈ H

12 (∂Ω)

(0.1)

γ is a positive function of class C2 on Ω.The Dirichlet-to-Neumann map:

Λγ f = γ∂νu|∂Ω,

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 ⊂ Band measurements ∂νu restricted to F.After standard reduction to Schrödinger equation: (−∆ + q)v = 0.


Bukhgeim and Uhlmann (light from infinity)

Assume q1, q2 be two bounded potentials on Ω, suppose that 0 is neither aDirichlet eigenvalue of the Schrödinger operator −∆ + q1 nor of −∆ + q2.Given a direction ξ ∈ Sn−1.Consider the ξ-illuminated boundary

∂Ω−(ξ) = x ∈ ∂Ω : ξ · ν(x) ≤ 0

and the ξ-shadowed boundary

∂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 ∈ H1/2(∂Ω)supported in B coincide on F

Λq1 f (x) = Λq2 f (x), x ∈ F

then the two potentials agree q1 = q2.Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June 2012 4 / 31

Kenig, Sjöstrand and Uhlmann, light from a point

In this case consider x0 out of Convex hull of Ω. The x0-illuminated boundary

∂Ω−(x0) = x ∈ ∂Ω : (x− x0) · ν(x) ≤ 0

and the x0-shadowed boundary

∂Ω+(x0) = x ∈ ∂Ω : (x− x0) · 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 ∈ H1/2(∂Ω)supported in B coincide on F

Λq1 f (x) = Λq2 f (x), x ∈ F

then the two potentials agree q1 = q2.


Previous results. Stability for partial data

BU case H. Heck and J.N. Wang (2006) (Without the support constrainon Dirichlet data)L. Tzou (2008) (extends to the magnetic case, without the supportconstrain on Dirichlet data)


Previous results. Stability for partial data

BU case H. Heck and J.N. Wang (2006) (Without the support constrainon Dirichlet data)L. Tzou (2008) (extends to the magnetic case, without the supportconstrain on Dirichlet data)

KSU caseNachman and Street (2010) (Reconstruction of Radon-like integrals ofthe potential)


Our results: Stability in BU partial data

TheoremLet Ω be a bounded open set in Rn, n ≥ 3 with smooth boundary, Assumegiven an open set N in Sn−1 and consider two open neighbourhoods F,Brespectively of the front and back sets Ω−(ξ) and Ω+(ξ) of ∂Ω for any ξ ∈ N.Let q1, q2 be two allowable potentials on Ω, suppose that 0 is neither aDirichlet eigenvalue of the Schrödinger operator −∆ + q1 nor of −∆ + q2,then

‖q1 − q2‖Lp ≤ C(log∣∣ log ‖Λq1 − Λq2‖B→F

∣∣)−λ/2 (0.2)

Remarks: The norm ‖Λq1 − Λq2‖B→F (Nachman-Street)The allowable potentials: L∞ ∩Wλ,p


Our results: Stability in KSU partial data

TheoremLet Ω be a bounded open set in Rn, n ≥ 3 with smooth boundary, Assumegiven an open set N in R3 and consider two open neighbourhoods F,Brespectively of the front and back sets Ω−(x0) and Ω+(x0) of ∂Ω for anyx0 ∈ N. Let q1, q2 be two allowable potentials on Ω, suppose that 0 is neithera Dirichlet eigenvalue of the Schrödinger operator −∆ + q1 nor of −∆ + q2,then

‖q1 − q2‖Lp(G) ≤ C(log∣∣ log ‖Λq1 − Λq2‖B→F

∣∣)−λ/2 (0.3)

where G is an open neighborhood in R3of the penumbra region F ∩ B

Remarks: The norm ‖Λq1 − Λq2‖B→F

(After F. J. Chung, A Partial Data Result for the Magnetic SchrodingerInverse Problem. Norm H1/2(∂Ω)→ H−1/2(∂Ω))


Steps in the proof

Step 1 Stability from DtoN map to partial Radon transform.

Step 2 Stability from partial Radon transform to potential


Second step: from limited angle-distance Radon data→local values of the function

Quantitave Helgason-Holmgren Theorem:Limited data:

Distance

I = s ∈ R : |s| < α (0.4)

Angle (ω0 ∈ Sn−1)

Γ = ω ∈ Sn−1 : (ω · ω0)2 > 1− β2 (0.5)

= ω ∈ Sn−1 : d(ω0, ω) < arcsinβ. (0.6)

Dependence domain

E =

x ∈ Rn : ω · (x− y0) = s, s ∈ I, ω ∈ Γ.


Allowable potentials

Let p, 1 ≤ p <∞, and λ, 0 < λ < 1/p, two functions q1, q2 satisfy thefollowing conditions:

(a) 1Eqj ∈ X ∩ L∞(Rn) for j = 1, 2,

‖qj‖L∞(E) + ‖1Eqj‖X < M.

X-norm q ∈ L1(Rn) and∫R

(1 + |s|)n ‖R0q(s, )‖L1(Sn−1) ds <∞.

(b) (λ, p, p)-Besov regularity on the dependence domain∫Rn

‖1Eqj − (1Eqj)(· − y)‖pLp(Rn)

|y|n+λp dy < Mp

for j = 1, 2.


Support condition

y0 ∈ supp (q1 − q2)

supp (q1 − q2) ⊂ x ∈ Rn : (x− y0) · ω0 ≤ 0

.


Notation:Ry0q(s, ω) is the y0-centered Radon transform, integral of q inH = x ∈ Rn : ω · (x− y0) = s.

Constant

C = CnM max(

1, |G|1p

)(1 + α−n + β−n + αλ

)(0.8)


Sketch of proof 1: uniqueness theorem

AssumeRy0(q1 − q2)|I×Γ = 0.Then such (y0, ω0) does not exits:

1 Microlocal Helgason’s theorem:

(y0, ω0) /∈ WFa(q1 − q2)

2 Microlocal Holmgren’s theorem (Hörmander):From the conditions y0 ∈ supp (q1 − q2) andsupp (q1 − q2) ⊂ x ∈ Rn : (x− y0) · ω0 ≤ 0 one has

(y0, ω0) ∈ WFa(q1 − q2)


1.Quantitative Microlocal Helgason’s theorem.Sketch ofproof

The WFa(u) is characterized be the exponential decay of the wave packettransform:For any u ∈ S ′(Rn) define the Wave packet (Segal-Bargmann) transform of ufor ζ ∈ Cn as

T u(ζ) = 〈u, e−12h (ζ−)2〉.

Where 〈, 〉 stands for the duality between S ′(Rn) and S(Rn) –the class ofsmooth rapidly decreasing functions.


Step 1:From Radon to wave packet

Proposition

Consider q ∈ Aλ(Rn), y0 ∈ Rn and ω0 ∈ Sn−1. Let Rq be a positive constantsuch that supp q ⊂ x ∈ Rn : |x− y0| ≤ Rq. Given α > 0 and β ∈ (0, 1]consider the sets

I = s ∈ R : |s| < α, Γ = ω ∈ Sn−1 : |ω · ω0|2 > 1− β2.

Then, there exists a positive constant C such that

|T q(ζ)| ≤ C

hn2(|Im ζ|n + αn + Rn

q) e1

2h |Im ζ|2

×[‖Ry0q‖L1(I×Γ) + ‖q‖L∞(Rn) (e−

12h

α24 + e−

12h

γ2β2

16 )],

for all h ∈ (0, 1], ζ ∈ Cn such that |Re ζ − y0| < α/2, |Im ζ| ≥ γ > 0 and|ω0 · θ|2 > 1− β2/4 with θ = |Im ζ|−1Im ζ.


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 heatequation. Need a priori Besov regularity on q.


The key Proposition

Proposition

Under the same notation and assumptions of the Theorem. There exists apositive constant C such that

e−1

2h |Im ζ|2 |T q(ζ)| < CMq((2αβ

)n + αn + Rnq) ‖Ry0q‖κL1(I×Γ) , (0.9)

with

κ :=1

4(

cosh(

8πβ

))2 , h :=α2

8| log ‖Ry0q‖L1(I×Γ) |,

for all ζ ∈ Cn such that

|Re ζ − y0| <α

8 cosh(

8πβ

) , |Im ζ| < 2α(4− β2)1/2 .


Proof of the Key proposition

Estimates of the wave packet transform:(K1)

|T q(ζ)| ≤ (2πh)n2 ‖q‖L∞(Rn) e

12h |Im ζ|2 ,

(K2)|T q(ζ)| ≤ (2πh)

n2 ‖q‖L∞(Rn) e

12h |Im ζ|2e−

12h |ω0·(Re ζ−y0)|2 ,

for all ζ ∈ Cn such that ω0 · (Re ζ − y0) ≥ 0 (follows from the hyperplanebeing supporting )(K3)

|T q(ζ)| ≤ C

hn2(|Im ζ|n + αn + Rn

q) e1

2h |Im ζ|2

×[‖Ry0q‖L1(I×Γ) + ‖q‖L∞(Rn) e−

12h

α24

],

for all h ∈ (0, 1], ζ ∈ Cn such that |Re ζ − y0| < α/2, |Im ζ| ≥ γ > 0 and|ω0 · θ|2 > 1− β2/4 with θ = |Im ζ|−1Im ζ. (Follows from quantitativeHelgason-Holmgren with γ = 2α/β)


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 thatF(z) < (min0, Re z)2, for all z ∈ C. Assume that F(z) < −2a2 for z ∈ Rsuch that |Im z| ≥ b. Then

F(z) < −12

2a2

cosh(π b

a

) min

(1

cosh(π b

a

) , 13

),

for

|Im z| < b, |Re z| < a2

min

(1

cosh(π b

a

) , 13

).


Proof of lemma

Take the subharmonic function

G(x + iy) = 2a2 cosh(πa y)

cosh(πa b) sin

(πa

(x + δ))

+ F(x + iy)− δ2,

for

δ = min

(a

cosh(π b

a

) , a3

)Check that G < 0 on the boundary of [−δ, a− δ]× [−b, b].The estimate follows from the maximun principle when we restrict to thevalues |x| < δ/2.


Step 2b.The backward estimate

The key point is the fact that the Segal-Bargmann transform restricted to realvalues is a convolution with the Gaussian. Backward estimate for the heatequation h = time

Lemma

Consider q ∈ Lp(Rn) and G an open set in Rn such that

supp q ∩ G 6= ∅.

Assume that there exists λ ∈ (0, 1) such that

Lq :=

(∫Rn

‖q− q(· − y)‖pLp(Rn)

|y|n+λp dy

)1/p

< +∞.

Then, there exists a positive constant C, only depending on n, such that

‖q‖Lp(G) ≤ C(

h−n2 ‖T q‖Lp(G) + Lqh

λ2

),

for all h ∈ (0, 1].Alberto Ruiz ( Universidad Autónoma de Madrid) U. California, Irvine.June 2012 24 / 31

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 theFourier tranform +log stable continuation of analytic functions)The Carleman estimates with signed partition of the boundary allow topartially recover the 2-planes X-ray transform in some sets I × Γ which areneighborhood of (y0, ω0) from the partial Dirichlet to Neuman map in alogarithmic stable way (following the approach of DSF, Kenig, Sjoestrand andUhlmann).This allows to control the norm of q1 − q2 in a neighborhood of y0. In thecase of BU ( semiglobal ) one can estimate ‖q1 − q2‖Lp , since we have thecontrol of ‖Ry0(q1 − q2)‖L1((0,∞)×Γ).In KSU We have the Radon transform if n = 3. Giving us the control onq1 − q2 at points for which we control the Radon transform (the penumbraboundary)


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 estimatewith splitting in the illuminated and shadowed regions. Theorem ( [BU] and[NS])

TheoremFor τ >> 1 sufficiently large and g ∈ C∞(Rn−2) and (ξ + iζ)2 = 0, thereexists a unique solution wτ ∈ H(Ω; ∆) of the equation (−∆ + q)wτ = 0 in Ω,such that tr0wτ ∈ H(∂Ω) ∩ E ′(B) and which can be written as

wτ (x) = eτ(ξ+iζ)·x(g(x′′) + R(τ, x))

where‖R(τ, ·)‖L2(Ω) ≤ C

1τ

(‖qg‖L2(Ω) + τ 1/2‖∆g‖L2(Ω)).

The same is true by changing τ by −τ and B by F.


The spaces of solutions in H(Ω; ∆)Traces (inH(∂Ω) ⊂ H−1/2(∂Ω))

Integral formula by using extensions of data

(Λq2 − Λq1)(φ)ψ =∫

ΩPq1(φ)(q1 − q2)Pq2(ψ). (0.10)

Difference of DtoN maps

LemmaLet qj , j = 1, 2 be L∞ potentials so that 0 is not a Dirichlet eigenvalue of−∆ + qj in Ω. Then Λq2 − Λq1 extends to a continuous mapH(∂Ω)→ H(∂Ω)∗.


2-plane Radon transform estimate

Proposition

For any g ∈ C∞(Rn−2) there exits C > 0 which only depends on Ω and the apriori bound of ‖qj‖L∞ , such that

supξ∈N,ζ∈ξ⊥

∫[ξ,ζ]⊥

g(x′′)∫

R2q(x′′ + tξ + sζ)dtdsdx′′ (0.11)

≤ C(τ−1/2 + ecτ‖Λq1 − Λq2‖B→F)‖g‖H2(Ω) (0.12)

By choosing appropriate g = g(rη) for any η ∈ [ξ, ζ]⊥

supξ∈Nsupη∈ξ⊥∫

Rg(r)Rq(r, η)dr (0.13)

≤ C(τ−1/2 + ecτ‖Λq1 − Λq2‖B→F)‖g‖H2(R) (0.14)


L1 norm of Radon transform

Interpolation with∫Sn−1

∫R(1 + τ 2)(n−1)/2|Rq(·, η) (τ)|2dτdσ(η) ≤ ‖q‖L2

gives:

Theorem (Stability of Radon transform)

(∫M

(∫R|Rq(r, η)|2dr

) n+34

dσ(η)

) 2n+3

(0.15)

≤ C(τ−1/2 + ecτ‖Λq1 − Λq2‖B→F)n+1n+3 . (0.16)

Where M ⊂ Sn−1

M = ∪ξ∈N [ξ]⊥,


Looking for the supporting plane H in quantitativeHelgason-Holmgren

Let us take η ∈ M and by translation there exist r ∈ R and y0 ∈ supp q suchthat H(s, η) contains y0 and it is a supporting hyperplane for supp q. We arenow in the hypothesis of the previous theorem, since M is a neighborhood ofη in the sphere and we can control the Radon transform por ω ∈ M and s ∈ R.


Open problem: Cloaking of 0’s

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FELICIDADES GUNTHER


a local estimate from radon transform and stability of inverse eit with partial...

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