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Boundary Reconstruction for the Anisotropic Fractional Calderón Problem

Xiaopeng Cheng, Angkana Rüland

TL;DR

The paper proves boundary reconstruction for the anisotropic fractional Calderón problem via an extension approach, showing that the measurement metric on the boundary region can be recovered from source-to-solution or solution-to-source data without apriori boundary information. By constructing highly oscillatory, localized approximate solutions to a degenerate elliptic PDE with weight $x_{n+1}^{1-2s}$ and analyzing the resulting Dirichlet-to-Neumann and Neumann-to-Dirichlet symbols, the authors extract boundary symbols that encode metric components and establish their recovery on the measurement set. The results extend from Euclidean settings to smooth Riemannian manifolds, yielding constructive boundary reconstruction theorems and enabling a broader equivalence between local and nonlocal Calderón problems without apriori boundary data. Consequently, the work strengthens the links between interior and boundary reconstructions in anisotropic fractional Calderón problems and solidifies the role of extension techniques in geometric inverse problems. The findings have implications for recovering geometric information from partial, nonlocal measurements in anisotropic media.

Abstract

In this article, we provide a boundary reconstruction result for the anisotropic fractional Calderón problem and its associated degenerate elliptic extension into the upper half plane. More precisely, considering the setting from \cite{FGKU21}, we show that the metric on the measurement set can be reconstructed from the source-to-solution data. To this end, we rely on the approach by Brown \cite{B01} in the framework developed in \cite{NT01} (see also \cite{KY02}) after localizing the problem by considering it through an extension perspective.

Boundary Reconstruction for the Anisotropic Fractional Calderón Problem

TL;DR

The paper proves boundary reconstruction for the anisotropic fractional Calderón problem via an extension approach, showing that the measurement metric on the boundary region can be recovered from source-to-solution or solution-to-source data without apriori boundary information. By constructing highly oscillatory, localized approximate solutions to a degenerate elliptic PDE with weight and analyzing the resulting Dirichlet-to-Neumann and Neumann-to-Dirichlet symbols, the authors extract boundary symbols that encode metric components and establish their recovery on the measurement set. The results extend from Euclidean settings to smooth Riemannian manifolds, yielding constructive boundary reconstruction theorems and enabling a broader equivalence between local and nonlocal Calderón problems without apriori boundary data. Consequently, the work strengthens the links between interior and boundary reconstructions in anisotropic fractional Calderón problems and solidifies the role of extension techniques in geometric inverse problems. The findings have implications for recovering geometric information from partial, nonlocal measurements in anisotropic media.

Abstract

In this article, we provide a boundary reconstruction result for the anisotropic fractional Calderón problem and its associated degenerate elliptic extension into the upper half plane. More precisely, considering the setting from \cite{FGKU21}, we show that the metric on the measurement set can be reconstructed from the source-to-solution data. To this end, we rely on the approach by Brown \cite{B01} in the framework developed in \cite{NT01} (see also \cite{KY02}) after localizing the problem by considering it through an extension perspective.

Paper Structure

This paper contains 23 sections, 20 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Main ideas
  3. Boundary reconstruction for a class of degenerate elliptic PDE
  4. Returning to the manifold setting
  5. Applications
  6. The anisotropic fractional Calderón problem without apriori information in the measurement domain
  7. Relating the local and nonlocal Calderón problems without apriori information on the metric on the measurement domain
  8. Relation to the literature
  9. Outline of the article
  10. Preliminaries
  11. Sets and approximation
  12. Function spaces and well-posedness
  13. The Solution-to-Source Problem -- Boundary Reconstruction for the Anisotropic Extension Problem with Generalized Dirichlet-to-Neumann Data
  14. Construction of approximate solutions -- preliminaries
  15. Construction of approximate solutions -- expansion and conclusion
  16. ...and 8 more sections

Key Result

Theorem 1

Let $s\in (0,1)$, assume that $(M_1,g_1)$, $(M_2, g_2)$ are smooth, closed, connected Riemannian manifolds of dimension $n\geq 2$. Let $(O_1,g_1|_{O})\subset (M_1,g_1)$ and $(O_2,g_2|_{O})\subset (M_2,g_2)$ be open, non-empty sets such that $(O_1,g_1|_{O})=(O_2,g_2|_{O})=(O,g)$ for some known metric

Theorems & Definitions (38)

  • Theorem 1: Theorem 1 in FGKU21
  • Theorem 2: Source-to-solution measurements and boundary reconstruction
  • Corollary 1.1
  • Theorem 3: Solution-to-source measurements and boundary reconstruction
  • Corollary 1.2
  • Theorem \ref{thm:bdry_reconstr_ext_nonloc_sol}**: Whole space PDE setting
  • Corollary 1.3
  • Remark 1.4
  • Theorem \ref{thm:bdry_reconstr_ext_nonloc_source}*: Neumann-to-Dirichlet data and boundary reconstruction
  • Theorem \ref{thm:bdry_reconstr_ext_nonloc_sol}*: Dirichlet-to-Neumann data and boundary reconstruction
  • ...and 28 more