Runge Approximation and Stability Improvement for a Partial Data Calder\'on Problem for the Acoustic Helmholtz Equation

TL;DR

This paper investigates the Runge approximation properties and stability improvements for the high-frequency partial data Calderón problem related to the acoustic Helmholtz equation, emphasizing frequency-dependent unique continuation estimates.

Contribution

It provides new quantitative Runge approximation results and stability enhancement techniques for the acoustic Helmholtz equation in high-frequency regimes.

Findings

Quantitative Runge approximation properties established.

Stability improvements achieved in high-frequency limit.

Frequency dependence of approximation results analyzed.

Abstract

In this article, we discuss quantitative Runge approximation properties for the acoustic Helmholtz equation and prove stability improvement results in the high frequency limit for an associated partial data inverse problem modelled on \cite{AU04, KU19}. The results rely on quantitative unique continuation estimates in suitable function spaces with explicit frequency dependence. We contrast the frequency dependence of interior Runge approximation results from non-convex and convex sets.