Overdetermined problems for the rotationally invariant Poisson equation in model manifolds

TL;DR

This work analyzes overdetermined elliptic problems for the rotationally invariant Poisson equation $-\Delta u = f(r)$ on model manifolds, establishing Serrin-type rigidity results that force the domain to be a geodesic ball and the solution to be radial under explicit relations between boundary data and the radial forcing. The authors introduce radial flux functions $v$ and $w$ to express compatibility conditions and derive a pair of comparison-based, ODE-driven arguments that yield radial symmetry, both for the interior Dirichlet problem and the exterior Bernoulli problem, in general model manifolds and specifically in space forms. Key contributions include precise necessary-and-sufficient criteria for radiality (via zeros of $v-\kappa$ and $w-\kappa$) and comprehensive extensions to annular domains with boundary data, along with explicit radial solution formulas and monotonicity-based criteria. The results broaden Serrin-type rigidity to rotationally symmetric geometries beyond Euclidean space, providing a unified framework for rigidity in $\mathbb R^N$, $\mathbb H^N$, and $\mathbb S^N$, with potential applications to capacity problems and geometric analysis on model manifolds.

Abstract

We present rigidity results for overdetermined problems associated to the rotationally invariant Poisson equation $-Δ_{g_\mathcal{M}} u = f(r)$ 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 \[ \begin{array}{ll} -Δ_{g_\mathcal{M}} {u}=f(r) &\mbox{in $Ω$}, u=\varphi(r) &\mbox{on $\partial Ω$}, \frac{\partial u}{\partial ν} = κ(r) &\mbox{on $\partial Ω$}, \end{array} \] where $Ω\subset \mathcal{M}$ is a bounded domain containing the point $O \in \mathcal{M}$ corresponding to $r = 0$, $ν$ is the exterior unit normal vector on $\partial Ω$, and $f$, $\varphi$, $κ$ 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 $Ω\setminus \overline B_{R_0}(O)$, 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 $κ$ implying that the solution $u$ is radial and $Ω$ is a geodesic ball centered at $O$. Our results apply in particular to the three space forms $\mathbb{R}^N$, $\mathbb{H}^N$ and $\mathbb{S}^N$.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• proof : Proof of Theorem \ref{['th1']}