
Journal of Convex Analysis 18 (2011), No. 4, 10131024 Copyright Heldermann Verlag 2011 Symmetry in MultiPhase Overdetermined Problems Ceni Babaoglu Dept. of Mathematics, Faculty of Science and Letters, Istanbul Technical University, 34469 MaslakIstanbul, Turkey ceni@itu.edu.tr Henrik Shahgholian Dept. of Mathematics, Royal Institute of Technology, 10044 Stockholm, Sweden henriksh@math.kth.se [Abstractpdf] We prove symmetry for a multiphase overdetermined problem, with nonlinear governing equations. The most simple form of our problem (in the twophase case) is as follows: For a bounded $C^1$ domain $\Omega \subset \mathbb{R}^n$ ($n\geq 2$) let $u^+$ be the Green's function (for the $p$Laplace operator) with pole at some interior point (origin, say), and $u^$ the Green's function in the exterior with pole at infinity. If for some strictly increasing function $F(t)$ (with some growth assumption) the condition $ \partial_\nu u^+ = F(\partial_\nu u^)$ holds on the boundary $\partial \Omega$, then $\Omega$ is necessarily a ball. We prove the more general multiphase analog of this problem. Keywords: Symmetry, overdetermined problems, multiphases, viscosity solutions, Green's function. MSC: 35R35, 35B06 [ Fulltextpdf (136 KB)] for subscribers only. 