A remark on an overdetermined problem in Riemannian Geometry

TL;DR

This paper investigates overdetermined boundary value problems involving the p-Laplacian on Riemannian manifolds with isoparametric distance functions, proving that certain boundary conditions imply the domain is a geodesic ball.

Contribution

It establishes a rigidity result for overdetermined p-Laplacian problems on Riemannian manifolds with isoparametric distance functions, extending classical symmetry results.

Findings

Domains with specified boundary conditions are geodesic balls.

Results apply to rotationally symmetric metrics on Euclidean space.

Provides conditions under which overdetermined problems imply symmetry.

Abstract

Let $(M,g)$ be a Riemannian manifold with a distinguished point $O$ and assume that the geodesic distance $d$ from $O$ is an isoparametric function. Let $\Omega\subset M$ be a bounded domain, with $O \in \Omega$, and consider the problem $\Delta_p u = -1$ in $\Omega$ with $u=0$ on $\partial \Omega$, where $\Delta_p$ is the $p$-Laplacian of $g$. We prove that if the normal derivative $\partial_{\nu}u$ of $u$ along the boundary of $\Omega$ is a function of $d$ satisfying suitable conditions, then $\Omega$ must be a geodesic ball. In particular, our result applies to open balls of $\mathbb{R}^n$ equipped with a rotationally symmetric metric of the form $g=dt^2+\rho^2(t)\,g_S$, where $g_S$ is the standard metric of the sphere.