Local analytic regularity in the linearized Calderón problem

Sjöstrand, Johannes; Uhlmann, Günther

doi:10.2140/apde.2016.9.515

Cited by 14 publications

(17 citation statements)

References 21 publications

(24 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let P : C ∞ (∂Ω) → C ∞ (Ω) be the Poisson operator for the boundary value problem in (1.2), so that u h (x) = Pϕ h (x). The first step in the proof of Theorem 1 amounts to understanding the microlocal structure of the Poisson operator, P following the analysis in [SU16].…”

Section: Motivating Example Of Fbi Transformsmentioning

confidence: 99%

“…We sometimes write S m,k cla = S m,k cla (T * M). Following [SU16], we also define the notion of a homogeneous analytic symbol of order k and write a ∈ S k ha provided that there exist holomorphic functions a k on a fixed complex conic neighborhood of T * M \ {0} homogeneous of degree k in ξ so that there exists C 0 > 0 so that…”

Section: Eigenfunction Mass Microlocalizationmentioning

confidence: 99%

See 1 more Smart Citation

Pointwise Bounds for Steklov Eigenfunctions

Galkowski

Toth

2018

J Geom Anal

View full text Add to dashboard Cite

Let (Ω, g) be a compact, real-analytic Riemannian manifold with real-analytic boundary ∂Ω. The harmonic extensions of the boundary Dirchlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfuntions decay exponentially into the interior in terms of the Dirichlet-to-Neumann eigenvalues and give a sharp rate of decay to first order at the boundary. The proof uses the Poisson representation for the Steklov eigenfunctions combined with sharp h-microlocal concentration estimates for the boundary Dirichlet-to-Neumann eigenfunctions near the cosphere bundle S * ∂Ω. These estimates follow from sharp estimates on the concentration of the FBI transforms of solutions to analytic pseudodifferential equations P u = 0 near the characteristic set {σ(P ) = 0}.

show abstract

Section: Motivating Example Of Fbi Transformsmentioning

confidence: 99%

Section: Eigenfunction Mass Microlocalizationmentioning

confidence: 99%

Pointwise Bounds for Steklov Eigenfunctions

Galkowski

Toth

2018

J Geom Anal

View full text Add to dashboard Cite

show abstract

“…The argument involves analytic microlocal analysis and Kashiwara's watermelon theorem. This result has been extended in [SU16] to the Calderón problem linearized at a real-analytic potential with measurements on a real-analytic part of the boundary. We remark that corresponding results are open for the nonlinear Calderón problem if n ≥ 3 (see the survey [KS14]).…”

Section: The Linearized Calderón Problemmentioning

confidence: 92%

The Linearized Calderón Problem in Transversally Anisotropic Geometries

Ferreira

Kurylev

Lassas

et al. 2018

International Mathematics Research Notices

View full text Add to dashboard Cite

In this article we study the linearized anisotropic Calderón problem. In a compact manifold with boundary, this problem amounts to showing that products of harmonic functions form a complete set. Assuming that the manifold is transversally anisotropic, we show that the boundary measurements determine an FBI type transform at certain points in the transversal manifold. This leads to proving a uniqueness result for transversal singularities in the linearized problem. The method requires a geometric condition on the transversal manifold related to pairs of intersecting geodesics, but it does not involve the geodesic X-ray transform which has limited earlier results on this problem.

show abstract

“…Our strategy will be as follows. In Section 2 we formulate our main theorems in terms of a class of integral identities analogous to the linearised Calderón problem [7,8,11,23]. Starting from Section 3, our method will begin to differ from [11].…”

Section: Introductionmentioning

confidence: 99%

Semilinear Calderón Problem on Stein Manifolds With Kähler Metric

Tzou

2020

Bull. Aust. Math. Soc.

View full text Add to dashboard Cite

We extend existing methods which treat the semilinear Calderón problem on a bounded domain to a class of complex manifolds with Kähler metric. Given two semilinear Schrödinger operators with the same Dirchlet-to-Neumann data, we show that the integral identities that appear naturally in the determination of the analytic potentials are enough to deduce uniqueness on the boundary up to infinite order. By exploiting the assumed complex structure, this information allows us to apply the method of stationary phase and recover the potentials in the interior as well.

show abstract

Local analytic regularity in the linearized Calderón problem

Abstract: International audienceWe show that the linearized local Dirichlet-to-Neumann map at a real-analytic potential for measurements made at an analytic open subset of the boundary is injective

Cited by 14 publications

References 21 publications

Pointwise Bounds for Steklov Eigenfunctions

Pointwise Bounds for Steklov Eigenfunctions

The Linearized Calderón Problem in Transversally Anisotropic Geometries

Semilinear Calderón Problem on Stein Manifolds With Kähler Metric

Contact Info

Product

Resources

About