The Calder\'{o}n problem for nonlocal parabolic operators: A new reduction from the nonlocal to the local

TL;DR

This paper establishes a novel connection between nonlocal and local parabolic inverse problems, enabling the recovery of coefficients in nonlocal equations through local Dirichlet-to-Neumann data, applicable in any spatial dimension.

Contribution

It introduces a new reduction method linking nonlocal and local Calderón problems for parabolic equations, extending previous results with different techniques across all spatial dimensions.

Findings

The nonlocal Dirichlet-to-Neumann map determines the local one.

The method applies to any spatial dimension n ≥ 2.

Extends previous results with a new approach.

Abstract

In this article, we investigate the Calder\'on problem for nonlocal parabolic equations, where we are interested to recover the leading coefficient of nonlocal parabolic operators. The main contribution is that we can relate both (anisotropic) variable coefficients local and nonlocal Calder\'on problem for parabolic equations. More concretely, we show that the (partial) Dirichlet-to-Neumann map for the nonlocal parabolic equation determines the (full) Dirichlet-to-Neumann map for the local parabolic equation. This article extends our earlier results [LLU22] by using completely different methods. Moreover, the results hold for any spatial dimension $n\geq 2$.