Stable determination of the potential for the Helmholtz equation in the high frequency limit from boundary measurements

TL;DR

This paper proves a triple logarithmic stability estimate for determining the potential in a Helmholtz equation from boundary measurements at high frequencies, assuming boundary knowledge of the potential in dimensions three and higher.

Contribution

It introduces a new stability estimate for inverse boundary value problems for the Helmholtz equation in the high frequency regime, under boundary potential knowledge.

Findings

Established a triple logarithmic stability estimate for the potential

Proved the estimate for the interior impedance problem

Applied the results to high frequency inverse problems

Abstract

We establish a triple logarithmic stability estimate of determining the potential in a Helmholtz equation from a partial Dirichlet-to-Neumann map in the high frequency limit. This estimate is proved under the assumption that the potential is known near the boundary of a domain when the dimension is greater than or equal to $3$. In addition, we show a triple logarithmic stability for an interior impedance problem.