Determining the potential in a wave equation without a geometric condition. Extension to the heat equation

TL;DR

This paper establishes a logarithmic stability estimate for inverse potential problems in wave and heat equations without geometric constraints, improving previous results and extending methods between equations.

Contribution

It introduces a new stability estimate that removes geometric conditions and adapts techniques from wave to heat equations for inverse problems.

Findings

Logarithmic stability estimate for wave equation inverse problem

Extension of analysis to heat equation inverse coefficient problem

Improved results over previous stability estimates

Abstract

We prove a logarithmic stability estimate for the inverse problem of determining the potential in a wave equation from boundary measurements obtained by varying the first component of the initial condition. The novelty of the present work is that no geometric condition is imposed to the sub-boundary where the measurements are made. Our results improve those obtained by the first and second authors in [2]. We also show how the analysis for the wave equation can be adapted to an inverse coefficient problem for the heat equation