Calculus of Variations and Geometric Measure Theory
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I. FragalĂ  - F. Gazzola - B. Kawohl

Overdetermined problems with possibly degenerate ellipticity, a geometric approach

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


Published Paper

Inserted: 4 jul 2005
Last Updated: 14 dec 2006

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


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 $\partial\Omega$.


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