*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
$\partial\Omega$.

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