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Fractional Calderón problem on a closed Riemannian manifold

Ali Feizmohammadi

Abstract

Given a fixed $α\in (0,1)$, we study the inverse problem of recovering the isometry class of a smooth closed and connected Riemannian manifold $(M,g)$, given the knowledge of a source-to-solution map for the fractional Laplace equation $(-Δ_g)^αu=f$ on the manifold subject to an arbitrarily small observation region $\mathcal O$ where sources can be placed and solutions can be measured. This can be viewed as a non-local analogue of the well known anisotropic Calderón problem that is concerned with the limiting case $α=1$. While the latter problem is widely open in dimensions three and higher, we solve the non-local problem in broad geometric generality, assuming only a local property on the a priori known observation region $\mathcal O$ while making no geometric assumptions on the inaccessible region of the manifold, namely $M\setminus \mathcal O$. Our proof is based on discovering a hidden connection to a variant of Carlson's theorem in complex analysis that allows us to reduce the non-local inverse problem to the Gel'fand inverse spectral problem.

Fractional Calderón problem on a closed Riemannian manifold

Abstract

Given a fixed , we study the inverse problem of recovering the isometry class of a smooth closed and connected Riemannian manifold , given the knowledge of a source-to-solution map for the fractional Laplace equation on the manifold subject to an arbitrarily small observation region where sources can be placed and solutions can be measured. This can be viewed as a non-local analogue of the well known anisotropic Calderón problem that is concerned with the limiting case . While the latter problem is widely open in dimensions three and higher, we solve the non-local problem in broad geometric generality, assuming only a local property on the a priori known observation region while making no geometric assumptions on the inaccessible region of the manifold, namely . Our proof is based on discovering a hidden connection to a variant of Carlson's theorem in complex analysis that allows us to reduce the non-local inverse problem to the Gel'fand inverse spectral problem.

Paper Structure

This paper contains 8 sections, 11 theorems, 130 equations.

Table of Contents

  1. Introduction and the main result
  2. Fractional Laplace equation
  3. The inverse problem and our main result
  4. Motivation
  5. Previous literature
  6. Analytic interpolation in the complex plane
  7. Proof of the main theorem via a standard reduction to the Gel'fand inverse spectral problem
  8. A variant of Carlson's theorem with faster than exponential growth along the real axis

Key Result

theorem 4

Let $\alpha \in (0,1)$. For $j=1,2$, let $(M_j,g_j)$ be a smooth closed and connected Riemannian manifold with dimension $n\geqslant 2$ and let $\mathcal{O}_j\subset M_j$ be a smooth submanifold of codimension zero with a smooth boundary. Assume that $(\mathcal{O}_j,g_{j}|_{\mathcal{O}_j})$, $j=1,2$ Suppose that Then there exists a diffeomorphism $\Phi:M_1\to M_2$ that fixes the set $\mathcal{O}_

Theorems & Definitions (27)

  • Definition 1: Source-to-Solution map
  • Definition 2
  • Definition 3
  • theorem 4
  • Remark 5
  • Proposition 6
  • Lemma 7
  • proof
  • Remark 8
  • Lemma 9
  • ...and 17 more