Nonsymmetric sign-changing solutions to overdetermined elliptic problems in bounded domains

TL;DR

This paper demonstrates the existence of nonsymmetric, sign-changing solutions to overdetermined elliptic problems in bounded domains, challenging the classical symmetry result by Serrin which requires positive solutions.

Contribution

It introduces sign-changing solutions in non-ball domains, showing positivity is essential for symmetry in Serrin's overdetermined problems.

Findings

Existence of sign-changing solutions in non-spherical domains

Positivity of solutions is necessary for symmetry results

Uses bifurcation theory to construct solutions

Abstract

In 1971 J. Serrin proved that, given a smooth bounded domain $\Omega \subset \mathbb{R}^N$ and $u$ a positive solution of the problem: \begin{equation*} \begin{array}{ll} -\Delta u = f(u) &\mbox{in $\Omega$, } u =0 &\mbox{on $\partial\Omega$, } \partial_{\nu} u =\mbox{constant} &\mbox{on $\partial\Omega$, } \end{array} \end{equation*} then $\Omega$ is necessarily a ball and $u$ is radially symmetric. In this paper we prove that the positivity of $u$ is necessary in that symmetry result. In fact we find a sign-changing solution to that problem for a $C^2$ function $f(u)$ in a bounded domain $\Omega$ different from a ball. The proof uses a local bifurcation argument, based on the study of the associated linearized operator.