Inverse parabolic problems of determining functions with one spatial-component independence by Carleman estimate

TL;DR

This paper develops a Carleman estimate-based method to solve inverse problems for parabolic equations, specifically determining coefficients independent of one spatial variable using boundary data, with stability results.

Contribution

It introduces a new Carleman estimate approach to establish stability in inverse parabolic problems with coefficients independent of one spatial component.

Findings

Conditional stability estimates for the inverse coefficient problem

Extension of results to inverse source problems

Application of Carleman estimates to boundary data-based inverse problems

Abstract

For an initial-boundary value problem for a parabolic equation in the spatial variable $x=(x_1,.., x_n)$ and time $t$, we consider an inverse problem of determining a coefficient which is independent of one spatial component $x_n$ by extra lateral boundary data. We apply a Carleman estimate to prove a conditional stability estimate for the inverse problem. Also we prove similar results for the corresponding inverse source problem.