Symmetry in Multi-Phase Overdetermined Problems

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.