Title: Non-symmetric solutions to an overdetermined problem for the Helmholtz equation in the plane Show Chinese title

Title: 平面上Helmholtz方程的超定问题的非对称解

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]. Show Chinese abstract

Abstract: 在本文中，我们构造了光滑有界区域$\Omega \subset \mathbb R^2$，除了圆之外，对于某些常数$\lambda,b,c \ne 0$，超定问题$$ \left\{ \begin{alignedat}{2} \Delta u + \lambda u &= 0 &\qquad& \text{ in } \Omega, \newline u &= b &\qquad& \text{ on } \partial \Omega, \newline \frac{\partial u}{\partial n} &= c &\qquad& \text{ on } \partial \Omega \end{alignedat} \right. $$有解。 这些似乎是 Willms 和 Gladwell [WG94] 的一个猜想的第一个反例。