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

domain

\Omega \subset \mathbb{R}^n Ω ⊂ R n

(

n\geq 2 n ≥ 2

) let

u^+ u +

be the Green's function (for the

p p

-Laplace operator) with pole at some interior point (origin, say), and

u^- u −

the Green's function in the exterior with pole at infinity. If for some strictly increasing function

F(t) F ( t )

(with some growth assumption) the condition

\partial_\nu u^+ = F(\partial_\nu u^-) ∂ ν u + = F ( ∂ ν u − )

holds on the boundary

\partial \Omega ∂ Ω

, then

\Omega Ω

is necessarily a ball. We prove the more general multi-phase analog of this problem.