Stability of the determination of a coefficient for the wave equation in an infinite wave guide

TL;DR

This paper establishes a Hölder stability estimate for determining an electric potential in an infinite wave guide from boundary measurements of the wave equation, extending results to measurements on a bounded subset of the boundary.

Contribution

It provides the first Hölder stability estimate for the inverse problem of determining a potential in an unbounded wave guide from boundary data.

Findings

Proves Hölder stability estimate for the inverse problem.

Extends stability results to measurements on a bounded boundary subset.

Demonstrates stability under certain potential gap conditions.

Abstract

We consider the stability in the inverse problem consisting in the determination of an electric potential $q$, appearing in a Dirichlet initial-boundary value problem for the wave equation $\partial_t^2u-\Delta u+q(x)u=0$ in an unbounded wave guide $\Omega=\omega\times\mathbb R$ with $\omega$ a bounded smooth domain of $\mathbb R^2$, from boundary observations. The observation is given by the Dirichlet to Neumann map associated to a wave equation. We prove a H\"older stability estimate in the determination of $q$ from the Dirichlet to Neumann map. Moreover, provided that the gap between two electric potentials rich its maximum in a fixed bounded subset of $\bar{\Omega}$, we extend this result to the same inverse problem with measurements on a bounded subset of the lateral boundary $(0,T)\times\partial\Omega$.