Lipschitz Stability of an Inverse Problem of Transmission Waves with Variable Jumps

TL;DR

This paper establishes Lipschitz stability for an inverse transmission wave problem with a discontinuous coefficient across an interface, using a novel Carleman inequality tailored for variable jumps.

Contribution

It introduces a new Carleman inequality for variable coefficients with jumps, extending stability results to more general transmission wave problems.

Findings

Lipschitz stability in recovering the zeroth-order coefficient.

Development of a new Carleman inequality for discontinuous coefficients.

Extension of inverse problem results to coefficients with variable jumps.

Abstract

This article studies an inverse problem for a transmission wave equation, a system where the main coefficient has a variable jump across an internal interface given by the boundary between two subdomains. The main result obtains Lipschitz stability in recovering a zeroth-order coefficient in the equation. The proof is based on the Bukhgeim-Klibanov method and uses a new one-parameter global Carleman inequality, specifically constructed for the case of a variable main coefficient which is discontinuous across a strictly convex interface. In particular, our hypothesis allows the main coefficient to vary smoothly within each subdomain up to the interface, thereby extending the preceding literature on the subject.