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

Abstract

In this note we construct smooth bounded domains $\Omega \subset {\mathbb{R}}^2$, other than disks, for which the overdetermined problem $$\{\begin{array}{rlrl}\Delta u+\lambda u& =0& & \text{ in }\Omega ,\\ u& =b& & \text{ on }\partial \Omega ,\\ \frac{\partial u}{\partial n}& =c& & \text{ on }\partial \Omega \end{array}$$ 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].

arXiv comment11 pages, 2 figures