Quadrature domains for the Helmholtz equation with applications to non-scattering phenomena

TL;DR

This paper introduces quadrature domains for the Helmholtz equation, demonstrating their existence, implementing partial balayage, and applying these concepts to inverse scattering problems, revealing non-scattering domains with inward cusps at any positive frequency.

Contribution

It is the first to define and establish existence of quadrature domains for the Helmholtz equation and to connect these to inverse scattering applications.

Findings

Existence of quadrature domains for the Helmholtz equation.

Implementation of the partial balayage procedure.

Existence of non-scattering domains with inward cusps at any positive frequency.

Abstract

In this paper, we introduce quadrature domains for the Helmholtz equation. We show existence results for such domains and implement the so-called partial balayage procedure. We also give an application to inverse scattering problems, and show that there are non-scattering domains for the Helmholtz equation at any positive frequency that have inward cusps.