# Non-symmetric solutions to an overdetermined problem for the Helmholtz equation in the plane

## TL;DR

This paper constructs non-symmetric bounded domains in the plane where the Helmholtz equation's overdetermined boundary value problem admits solutions, challenging previous conjectures about symmetry.

## Contribution

It provides the first known counterexamples to a conjecture that solutions only exist in symmetric domains like disks.

## Key findings

- Existence of solutions in non-disk domains for the Helmholtz overdetermined problem
- Counterexamples to Willms and Gladwell's conjecture
- New insights into boundary value problems for PDEs

## Abstract

In this note we construct smooth bounded domains $\Omega \subset \mathbb R^2$, other than disks, for which the overdetermined problem $$ \left\{ \begin{alignedat}{2} \Delta u + \lambda u &= 0 &\qquad& \text{ in } \Omega, \newline u &= b &\qquad& \text{ on } \partial \Omega, \newline \frac{\partial u}{\partial n} &= c &\qquad& \text{ on } \partial \Omega \end{alignedat} \right. $$ has a solution for some constants $\lambda,b,c \ne 0$. These appear to be the first counterexamples to a conjecture of Willms and Gladwell [WG94].