Overdetermined problems for the rotationally invariant Poisson equation in model manifolds

Abstract

We present rigidity results for overdetermined problems associated to the rotationally invariant Poisson equation $-\hat{I}{”}_{g_{\mathcal{M}}}u=f(r)$ 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 \[ \begin{array}{ll} -Δ_{g_\mathcal{M}} {u}=f(r) &\mbox{in $\hat{I}\text{c◯}$}, u=φ(r) &\mbox{on $\partial \hat{I}\text{c◯}$}, \frac{\partial u}{\partial ν} = κ(r) &\mbox{on $\partial \hat{I}\text{c◯}$}, \end{array} \] where $\hat{I}\text{c◯}\subset \mathcal{M}$ is a bounded domain containing the point $O\in \mathcal{M}$ corresponding to $r=0$, $\hat{I}½$ is the exterior unit normal vector on $\partial \hat{I}\text{c◯}$, and $f$, $\ddot{I}†$, $\hat{I}º$ 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 $\hat{I}\text{c◯}\setminus {\overline{B}}_{R_0}(O)$, 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 $f$, $\ddot{I}†$ and $\hat{I}º$ implying that the solution $u$ is radial and $\hat{I}\text{c◯}$ is a geodesic ball centered at $O$. Our results apply in particular to the three space forms ${\mathbb{R}}^N$, ${\mathbb{H}}^N$ and ${\mathbb{S}}^N$.