Overdetermined problems for the rotationally invariant Poisson equation in model manifolds

TL;DR

This paper establishes conditions under which solutions to overdetermined Poisson problems on rotationally symmetric manifolds are radially symmetric, extending classical symmetry results to curved spaces like hyperbolic, spherical, and Euclidean geometries.

Contribution

It provides new rigidity results for overdetermined boundary value problems on model manifolds with rotational symmetry, including conditions ensuring solutions are radially symmetric and domains are geodesic balls.

Findings

Solutions are radially symmetric under specified conditions.

Domains must be geodesic balls centered at the pole.

Results apply to Euclidean, hyperbolic, and spherical spaces.

Abstract

We present rigidity results for overdetermined problems associated to the rotationally invariant Poisson equation $-\Delta_{g_\mathcal{M}} u = f(r)$ in a model manifold $\mathcal{M} = [0,S) \times_h \mathbb S^{N-1}$ with warping function $h$. The variable $r$ ranges in the interval $[0,S)$, whose endpoint $S$ is positive and possibly infinite. The first part of the paper deals with the problem \[ \begin{array}{ll} -\Delta_{g_\mathcal{M}} {u}=f(r) &\mbox{in $\Omega$}, u=\varphi(r) &\mbox{on $\partial \Omega$}, \frac{\partial u}{\partial \nu} = \kappa(r) &\mbox{on $\partial \Omega$}, \end{array} \] where $\Omega \subset \mathcal{M}$ is a bounded domain containing the point $O \in \mathcal{M}$ corresponding to $r = 0$, $\nu$ is the exterior unit normal vector on $\partial \Omega$, and $f$, $\varphi$, $\kappa$ are three prescribed functions. In the second part of the paper, we consider a similar overdetermined problem for the exterior Bernoulli problem in a domain $\Omega \setminus \overline B_{R_0}(O)$, where $B_{R_0}(O)$ denotes the geodesic ball centered at $O$ with radius $R_0$, within the class of functions that vanish on $\partial B_{R_0}(O)$. In both cases, we give conditions on $f$, $\varphi$ and $\kappa$ implying that the solution $u$ is radial and $\Omega$ is a geodesic ball centered at $O$. Our results apply in particular to the three space forms $\mathbb{R}^N$, $\mathbb{H}^N$ and $\mathbb{S}^N$.