On inverse problems for a strongly damped wave equation on compact manifolds

TL;DR

This paper investigates inverse problems for a strongly damped wave equation on compact manifolds, demonstrating unique metric determination from boundary and source-to-solution data, using spectral analysis and Laplace transforms.

Contribution

It establishes unique recovery of the manifold's metric from partial boundary data and source-to-solution maps, extending inverse problem theory to damped wave equations on manifolds.

Findings

Unique metric determination from source-to-solution map on closed manifolds.

Unique metric determination from partial Dirichlet-to-Neumann map on manifolds with boundary.

Single measurement suffices for metric recovery in both cases.

Abstract

We consider a strongly damped wave equation on compact manifolds, both with and without boundaries, and formulate the corresponding inverse problems. For closed manifolds, we prove that the metric can be uniquely determined, up to an isometry, from the knowledge of the source-to-solution map. Similarly, for manifolds with boundaries, we prove that the metric can be uniquely determined, up to an isometry, from partial knowledge of the Dirichlet-to-Neumann map. The key point is to retrieve the spectral information of the Laplace-Beltrami operator, from the Laplace transform of the measurements. Further we show that the metric can be determined up to an isometry, using a single measurement in both scenarios.