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Journal of Convex Analysis 18 (2011), No. 4, 1013--1024
Copyright Heldermann Verlag 2011

Symmetry in Multi-Phase Overdetermined Problems

Ceni Babaoglu
Dept. of Mathematics, Faculty of Science and Letters, Istanbul Technical University, 34469 Maslak-Istanbul, Turkey

Henrik Shahgholian
Dept. of Mathematics, Royal Institute of Technology, 10044 Stockholm, Sweden


We prove symmetry for a multi-phase overdetermined problem, with nonlinear governing equations. The most simple form of our problem (in the two-phase 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 multi-phase analog of this problem.

Keywords: Symmetry, overdetermined problems, multi-phases, viscosity solutions, Green's function.

MSC: 35R35, 35B06

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