Overdetermined elliptic problems in nontrivial contractible domains of the sphere

TL;DR

This paper demonstrates the existence of nontrivial contractible domains on the sphere where overdetermined elliptic problems have solutions, challenging the generalization of Serrin's theorem from Euclidean space to spherical geometry.

Contribution

It constructs specific contractible domains on the sphere supporting solutions to overdetermined elliptic problems, showing limitations of Euclidean symmetry results in spherical settings.

Findings

Existence of solutions in perturbed spherical domains

Counterexample to Euclidean overdetermined problem generalization

Domains are perturbations of the sphere minus a small geodesic ball

Abstract

In this paper, we prove the existence of nontrivial contractible domains $\Omega\subset\mathbb{S}^{d}$, $d\geq2$, such that the overdetermined elliptic problem \begin{equation*} \begin{cases} -\varepsilon\Delta_{g} u +u-u^{p}=0 &\mbox{in $\Omega$, } u>0 &\mbox{in $\Omega$, } u=0 &\mbox{on $\partial\Omega$, } \partial_{\nu} u=\mbox{constant} &\mbox{on $\partial\Omega$, } \end{cases} \end{equation*} admits a positive solution. Here $\Delta_{g}$ is the Laplace-Beltrami operator in the unit sphere $\mathbb{S}^{d}$ with respect to the canonical round metric $g$, $\varepsilon>0$ is a small real parameter and $1<p<\frac{d+2}{d-2}$ ($p>1$ if $d=2$). These domains are perturbations of $\mathbb{S}^{d}\setminus D,$ where $D$ is a small geodesic ball. This shows in particular that Serrin's theorem for overdetermined problems in the Euclidean space cannot be generalized to the sphere even for contractible domains.