Boundary deformation techniques for Neumann problems for the Helmholtz equation

TL;DR

This paper develops boundary deformation methods to solve Neumann problems for the Helmholtz equation with rough potentials, analyzing eigenvalue dependence on boundary shape and enabling solutions for any energy level.

Contribution

It introduces boundary deformation techniques for Helmholtz Neumann problems, linking eigenvalue behavior to domain shape and facilitating solutions across energy levels.

Findings

Eigenvalues depend continuously on boundary deformation.

Domains can be chosen to solve the Neumann problem at any energy.

Method applicable to inverse scattering problems with partial data.

Abstract

We adapt boundary deformation techniques to solve a Neumann problem for the Helmholtz equation with rough electric potentials in bounded domains. In particular, we study the dependance of Neumann eigenvalues of the perturbed Laplacian with respect to boundary deformation, and we illustrate how to find a domain in which the Neumann problem can be solved for any energy, if there is some freedom in the choice of the domain. This work is motivated by a Runge approximation result in the context of an inverse problem in point-source scattering with partial data.