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Symmetry and rigidity results for Serrin's overdetermined type problems in weighted Riemannian manifolds

Laura Accornero, Giulio Ciraolo

Abstract

We study Serrin's overdetermined boundary value problems in bounded domains on weighted Riemannian manifolds. When the closure of the domain is compact, we establish a rigidity result that characterizes both the solution and the geometry of the ambient manifold. We further address the case of domains with non-compact closure for manifolds conformally equivalent to the Euclidean space, possibly degenerating or becoming singular at a point, where both the weight and the conformal factor are radial functions.

Symmetry and rigidity results for Serrin's overdetermined type problems in weighted Riemannian manifolds

Abstract

We study Serrin's overdetermined boundary value problems in bounded domains on weighted Riemannian manifolds. When the closure of the domain is compact, we establish a rigidity result that characterizes both the solution and the geometry of the ambient manifold. We further address the case of domains with non-compact closure for manifolds conformally equivalent to the Euclidean space, possibly degenerating or becoming singular at a point, where both the weight and the conformal factor are radial functions.

Paper Structure

This paper contains 11 sections, 13 theorems, 422 equations.

Table of Contents

  1. Introduction
  2. The weighted setting and two motivating examples
  3. Main results
  4. Notations
  5. The compact case
  6. Preliminary results
  7. Proof of \ref{['main']}
  8. The non-compact case
  9. Preliminary results
  10. Proof of Theorem \ref{['main_nc']}
  11. Failure of the rigidity result: an example

Key Result

Theorem 1.1

Let $(M,g,e^{-f}dV_g)$ be a weighted, Riemannian manifold of dimension $n$, with $e^{-f}\in C^2(M)$. Let $\Omega$ be a bounded, open and connected set in $(M,g,e^{-f}dV_g)$ with boundary $\partial\Omega$ of class $C^{1}$. Assume that If $u$ is a smooth solution to then $u$ is radial and $\Omega$ is a metric ball $B_R^g(p)$ isometric to a Euclidean ball, $\beta=1$, $e^{-f}$ is constant and $u$ is

Theorems & Definitions (36)

  • Example 1
  • Example 2
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • ...and 26 more