Boundary Reconstruction for the Anisotropic Fractional Calder\'on Problem

TL;DR

This paper demonstrates how to reconstruct the boundary metric in the anisotropic fractional Calderón problem using boundary measurements, extending the problem into a degenerate elliptic setting and building on prior localization techniques.

Contribution

It introduces a boundary reconstruction method for the anisotropic fractional Calderón problem via an extension approach, advancing previous inverse boundary value problem techniques.

Findings

Boundary metric can be reconstructed from source-to-solution data.

Extension into the upper half plane facilitates the reconstruction process.

The approach generalizes previous methods to anisotropic fractional settings.

Abstract

In this article, we provide a boundary reconstruction result for the anisotropic fractional Calder\'on 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.