# Overdetermined problems with possibly degenerate ellipticity, a geometric approach

created by gazzola on 04 Jul 2005
modified by fragala on 14 Dec 2006

[BibTeX]

Published Paper

Inserted: 4 jul 2005
Last Updated: 14 dec 2006

Journal: Math. Zeitschrift
Volume: 254
Pages: 117-132
Year: 2006

Abstract:

Given an open bounded connected subset $\Omega$ of ${\mathbb R}^n$, we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary data to the elliptic equation $-{\rm div}(A( \nabla u )\nabla u)=1$ in $\Omega$. We prove that, if this problem admits a solution in a suitable weak sense, then $\Omega$ is a ball. This is obtained under fairly general assumptions on $\Omega$ and $A$. In particular, $A$ may be degenerate and no growth condition is required. Our method of proof is quite simple. It relies on a maximum principle for a suitable $P$-function, combined with some geometric arguments involving the mean curvature of $\partial\Omega$.