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Fractional anisotropic Calderón problem with external data

Ali Feizmohammadi, Tuhin Ghosh, Katya Krupchyk, Angkana Rüland, Johannes Sjöstrand, Gunther Uhlmann

TL;DR

This work resolves the fractional anisotropic Calderón problem with external data for smooth Riemannian metrics on $\mathbb{R}^n$ that agree with the Euclidean metric outside a compact set, showing that the partial exterior Dirichlet-to-Neumann data determines the metric up to a diffeomorphism fixing the exterior. Two complementary proofs are developed: a heat–semigroup/pseudodifferential approach that leverages Vishik–Eskin estimates to recover the exterior heat kernel and then the metric, and a degenerate elliptic extension approach based on the Caffarelli–Silvestre framework that recasts the problem as a weighted local PDE in the half-space and establishes uniqueness through extension- and energy-based arguments. The paper also clarifies the natural gauge obstruction and provides precise mapping properties for the fractional Laplace–Beltrami operator, including a thorough treatment of exterior Dirichlet problems, Vishik–Eskin estimates, and extension problems. Collectively, the results advance nonlocal inverse problems in noncompact settings, yielding global metric recovery from exterior measurements and enriching the theoretical toolbox with two robust, independent methodologies.

Abstract

In this paper, we solve the fractional anisotropic Calderón problem with external data in the Euclidean space, in dimensions two and higher, for smooth Riemannian metrics that agree with the Euclidean metric outside a compact set. Specifically, we prove that the knowledge of the partial exterior Dirichlet--to--Neumann map for the fractional Laplace-Beltrami operator, given on arbitrary open nonempty sets in the exterior of the domain in the Euclidean space, determines the Riemannian metric up to diffeomorphism, fixing the exterior. We provide two proofs of this result: one relies on the heat semigroup representation of the fractional Laplacian and a pseudodifferential approach, while the other is based on a variable-coefficient elliptic extension interpretation of the fractional Laplacian.

Fractional anisotropic Calderón problem with external data

TL;DR

This work resolves the fractional anisotropic Calderón problem with external data for smooth Riemannian metrics on that agree with the Euclidean metric outside a compact set, showing that the partial exterior Dirichlet-to-Neumann data determines the metric up to a diffeomorphism fixing the exterior. Two complementary proofs are developed: a heat–semigroup/pseudodifferential approach that leverages Vishik–Eskin estimates to recover the exterior heat kernel and then the metric, and a degenerate elliptic extension approach based on the Caffarelli–Silvestre framework that recasts the problem as a weighted local PDE in the half-space and establishes uniqueness through extension- and energy-based arguments. The paper also clarifies the natural gauge obstruction and provides precise mapping properties for the fractional Laplace–Beltrami operator, including a thorough treatment of exterior Dirichlet problems, Vishik–Eskin estimates, and extension problems. Collectively, the results advance nonlocal inverse problems in noncompact settings, yielding global metric recovery from exterior measurements and enriching the theoretical toolbox with two robust, independent methodologies.

Abstract

In this paper, we solve the fractional anisotropic Calderón problem with external data in the Euclidean space, in dimensions two and higher, for smooth Riemannian metrics that agree with the Euclidean metric outside a compact set. Specifically, we prove that the knowledge of the partial exterior Dirichlet--to--Neumann map for the fractional Laplace-Beltrami operator, given on arbitrary open nonempty sets in the exterior of the domain in the Euclidean space, determines the Riemannian metric up to diffeomorphism, fixing the exterior. We provide two proofs of this result: one relies on the heat semigroup representation of the fractional Laplacian and a pseudodifferential approach, while the other is based on a variable-coefficient elliptic extension interpretation of the fractional Laplacian.

Paper Structure

This paper contains 36 sections, 53 theorems, 401 equations.

Table of Contents

  1. Introduction and statement of results
  2. Previous literature
  3. Organization of the paper
  4. The fractional Laplace--Beltrami operator and the exterior Dirichlet problem
  5. The fractional Laplace--Beltrami operator on the Euclidean space as a pseudodifferential operator
  6. Some facts about Sobolev spaces
  7. The fractional Laplacian and the heat semigroup
  8. Solvability of the exterior Dirichlet problem for fractional Laplacians
  9. Solvability of the fractional Poisson equation
  10. Mapping properties of the fractional Laplace--Beltrami operator
  11. The Vishik--Eskin estimates for the fractional Laplace--Beltrami operator
  12. Proof of Theorem \ref{['thm_main']}. The heat semigroup and pseudodifferential approach
  13. An auxiliary unique continuation type result
  14. From the partial exterior Dirichlet--to--Neumann map to the full exterior Dirich-let--to--Neumann map
  15. From the full exterior Dirichlet--to--Neumann map to the exterior source--to--solution map
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $\alpha \in (0,1)$. Let $g_1$ and $g_2$ be $C^\infty$ Riemannian metrics on $\mathbb{R}^n$, $n \geq 2$, which agree with the Euclidean metric outside of a compact set. Let $\Omega \subset \mathbb{R}^n$ be a bounded open set with $C^\infty$ boundary such that $\Omega_e$ is connected, and let $W_1 Assume furthermore that for all $f \in C^\infty_0(W_1)$. Then there exists a $C^\infty$ diffeomorp

Theorems & Definitions (107)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Remark 1.4: On a PDE version of Theorem \ref{['thm_main']}
  • Theorem 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Proposition 2.6
  • ...and 97 more