Stability estimates for an inverse Steklov problem in a class of hollow spheres

TL;DR

This paper investigates the inverse Steklov problem on hollow spheres with warped product metrics, establishing uniqueness and stability estimates for the warping function based on the Steklov spectrum, including symmetric cases.

Contribution

It provides the first stability estimates for the inverse Steklov problem on hollow spheres, including symmetric cases and links to the Calderón problem.

Findings

Unique determination of the warping function near the boundary from spectral data.

Log-type stability estimates for symmetric warping functions.

Extension of stability results to the Calderón problem.

Abstract

In this paper, we study an inverse Steklov problem in a class of n-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. Precisely, we aim at studying the continuous dependence of the warping function dening the warped product with respect to the Steklov spectrum. We first show that the knowledge of the Steklov spectrum up to an exponential decreasing error is enough to determine uniquely the warping function in a neighbourhood of the boundary. Second, when the warping functions are symmetric with respect to 1/2, we prove a log-type stability estimate in the inverse Steklov problem. As a last result, we prove a log-type stability estimate for the corresponding Calder{\'o}n problem.