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 $-\Delta_{g_\mathcal{M}} u = f(r)$ 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 $\mathbb{S}^N$.