Symmetry for a general class of overdetermined elliptic problems

Abstract

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^N$, and let $u\in C^1(\overline{\Omega })$ be a weak solution of the following overdetermined BVP: $-\nabla (g(|\nabla u|)|\nabla u{|}^{-1}\nabla u)=f(|x|,u)$, $u>0$ in $\Omega$ and $u(x)=0,|\nabla u(x)|=\lambda (|x|)$ on $\partial \Omega$, where $g\in C([0,+\infty ))\cap C^1((0,+\infty ))$ with $g(0)=0$, $g^{\prime }(t)>0$ for $t>0$, $f\in C([0,+\infty ))\times [0,+\infty ))$, $f$ is nonincreasing in $|x|$, $\lambda \in C([0,+\infty ))$ and $\lambda$ is positive and nondecreasing. We show that $\Omega$ is a ball and $u$ satisfies some "local" kind of symmetry. The proof is based on the method of continuous Steiner symmetrization.

arXiv comment16 pages