The Calder\'on problem for space-time fractional parabolic operators with variable coefficients

TL;DR

This paper establishes the unique recovery of potential functions in variable coefficient fractional parabolic operators using novel Carleman estimates and Runge approximation, advancing inverse problem theory for fractional PDEs.

Contribution

It introduces a new Carleman estimate for variable coefficient extension operators and applies it to prove unique identifiability of coefficients in fractional parabolic equations.

Findings

Unique recovery of potential function q from exterior data.

Development of a new Carleman estimate for variable coefficient operators.

Extension of results to fractional parabolic operators with drift.

Abstract

We study an inverse problem for variable coefficient fractional parabolic operators of the form $(\partial_t -\operatorname{div}(A(x) \nabla_x)^s + q(x,t)$ for $s\in(0,1)$ and show the unique recovery of $q$ from exterior measured data. Similar to the fractional elliptic case, we use Runge type approximation argument which is obtained via a global weak unique continuation property. The proof of such a unique continuation result involves a new Carleman estimate for the associated variable coefficient extension operator. In the latter part of the work, we prove analogous unique determination results for fractional parabolic operators with drift.