Determining a boundary coefficient in a dissipative wave equation: Uniqueness and directional lipschitz stability

TL;DR

This paper investigates the inverse problem of identifying a boundary damping coefficient in a dissipative wave equation, establishing uniqueness at the origin and a Lipschitz stability estimate through linearization.

Contribution

It introduces a novel approach linking boundary coefficient determination to a refined unique continuation property and provides stability estimates at the origin.

Findings

Uniqueness of the boundary coefficient at the origin under specific initial conditions.

Lipschitz directional stability estimate established at the origin.

Connection between initial condition choice and unique continuation property.

Abstract

We are concerned with the problem of determining the damping boundary coefficient appearing in a dissipative wave equation from a single boundary measurement. We prove that the uniqueness holds at the origin provided that the initial condition is appropriately chosen. We show that the choice of the initial condition leading to uniqueness is related to a fine version of unique continuation property for elliptic operators. We also establish a Lipschitz directional stability estimate at the origin, which is obtained by a linearization process.