Uniqueness for the anisotropic fractional conductivity equation

TL;DR

This paper establishes the unique recovery of anisotropic fractional conductivity matrices from boundary data, advancing inverse problem theory for nonlocal operators with anisotropic properties.

Contribution

It introduces a novel nonlocal operator based on vector calculus, extending techniques from isotropic fractional elasticity to anisotropic conductivities, and proves uniqueness in the inverse problem.

Findings

Unique determination of anisotropic conductivity matrices from boundary data.

Extension of isotropic fractional elasticity techniques to anisotropic case.

Addresses a key open problem related to the classical anisotropic Calderón problem.

Abstract

In this paper we study an inverse problem for fractional anisotropic conductivity. Our nonlocal operator is based on the well-developed theory of nonlocal vector calculus, and differs substantially from other generalizations of the classical anisotropic conductivity operator obtained spectrally. We show that the anisotropic conductivity matrix can be recovered uniquely from fractional Dirichlet-to-Neumann data up to a natural gauge. Our analysis makes use of techniques recently developed for the study of the isotropic fractional elasticity equation, and generalizes them to the case of non-separable, anisotropic conductivities. The motivation for our study stems from its relation to the classical anisotropic Calder\'on problem, which at the time of writing is one of the main open problems in the field.