Symmetry in Multi-Phase Overdetermined Problems


Babaoglu C., Shahgholian H.

JOURNAL OF CONVEX ANALYSIS, vol.18, no.4, pp.1013-1024, 2011 (Journal Indexed in SCI) identifier

  • Publication Type: Article / Article
  • Volume: 18 Issue: 4
  • Publication Date: 2011
  • Title of Journal : JOURNAL OF CONVEX ANALYSIS
  • Page Numbers: pp.1013-1024

Abstract

In this paper 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 of R-n (n >= 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 derivative(v)u(+) = F(partial derivative(v)u(-)) holds on the boundary partial derivative Omega, then Omega is necessarily a ball. We prove the more general multi-phase analog of this problem.