Stability estimates in inverse problems for the Schr\"odinger and wave equations with trapping

TL;DR

This paper establishes Hölder stability estimates for inverse boundary value problems on certain Riemannian manifolds, including cases with trapped geodesics, for determining potentials from Dirichlet-to-Neumann maps.

Contribution

It extends stability results to manifolds with trapped geodesics, allowing infinite-length trapped geodesics, which was not previously addressed.

Findings

Hölder stability estimates are proven for inverse problems on manifolds with trapping.

The results apply to all negatively curved manifolds with strictly convex boundary.

The approach accommodates geodesics that are trapped inside the manifold.

Abstract

For a class of Riemannian manifolds with boundary that includes all negatively curved manifolds with strictly convex boundary, we establish H\"older type stability estimates in the geometric inverse problem of determining the electric potential or the conformal factor from the Dirichlet-to-Neumann map associated with the Schr\"odinger equation and the wave equation. The novelty in this result lies in the fact that we allow some geodesics to be trapped inside the manifold and have infinite length.