Serrin's over-determined Problem on Riemannian Manifolds

Abstract

Let $(\mathcal{M},g)$ be a compact Riemannian manifold of dimension $N$, $N\ge 2$. In this paper, we prove that there exists a family of domains $({\Omega }_{\epsilon }{)}_{\epsilon \in (0,{\epsilon }_0)}$ and functions $u_{\epsilon }$ such that $-{\Delta }_g{u}_{\epsilon }=1\text{ in }{\Omega }_{\epsilon },u_{\epsilon }=0\text{ on }\partial {\Omega }_{\epsilon },g({\nabla }_g{u}_{\epsilon },{\nu }_{\epsilon })=-\frac{\epsilon }{N}\text{ on }\partial {\Omega }_{\epsilon },$ where ${\nu }_{\epsilon }$ is the unit outer normal of $\partial {\Omega }_{\epsilon }$. The domains ${\Omega }_{\epsilon }$ are smooth perturbations of geodesic balls of radius $\epsilon$ centered at some point $p_0$. If, in addition, $p_0$ is a non-degenerate critical point of the scalar curvature of $g$ then, the family $(\partial {\Omega }_{\epsilon }{)}_{\epsilon \in (0,{\epsilon }_0)}$ constitutes a smooth foliation of a neighborhood of $p_0$. By considering a family of domains ${\Omega }_{\epsilon }$ in which the above overdetermined system is satisfied, we also prove that if this family converges to some point $p_0$ in a suitable sense as $\epsilon \to 0$, then $p_0$ is a critical point of the scalar curvature. A Taylor expansion of he energy rigidity for the torsion problem is also given.

40 pages