A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem

TL;DR

This paper establishes a global Carleman estimate for a transmission wave equation with discontinuous coefficients, enabling unique and stable recovery of a stationary potential from a single boundary measurement.

Contribution

It introduces a novel Carleman inequality for transmission wave equations with convex inner domains and applies it to solve an inverse problem with minimal measurement data.

Findings

Proved a global Carleman inequality under convexity and coefficient conditions.

Achieved uniqueness and Lipschitz stability for the inverse problem.

Demonstrated potential for minimal measurement-based inverse problem solutions.

Abstract

We consider a transmission wave equation in two embedded domains in $R^2$, where the speed is $a1 > 0$ in the inner domain and $a2 > 0$ in the outer domain. We prove a global Carleman inequality for this problem under the hypothesis that the inner domain is strictly convex and $a1 > a2$ . As a consequence of this inequality, uniqueness and Lip- schitz stability are obtained for the inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data and discontinuous principal coefficient from a single time-dependent Neumann boundary measurement.