Inverse Conductivity Problem for a Parabolic Equation using a Carlemen Estimate with one Observation

TL;DR

This paper establishes a stability result for identifying a diffusion coefficient in a heat equation using a Carleman estimate, a Poincaré estimate, and a boundary observation, advancing inverse problem techniques.

Contribution

It introduces a novel stability analysis for the inverse conductivity problem in parabolic equations with a single boundary observation using Carleman estimates.

Findings

Proves stability of the diffusion coefficient reconstruction

Develops a new Carleman estimate for the heat equation

Achieves results with minimal boundary observations

Abstract

For the heat equation in a bounded domain we give a stability result for a smooth diffusion coefficient. The key ingredients are a global Carleman-type estimate, a Poincar\'e-type estimate and an energy estimate with a single observation acting on a part of the boundary.