On a free boundary value problem for the anisotropic N-Laplace operator on an N−dimensional ring domain

Abstract

Abstract In this paper we are going to investigate a free boundary value problem for the anisotropic N-Laplace operator on a ring domain Ω : = Ω 0 \ Ω ¯ 1 ⊂ 𝕉 N \Omega : = {\Omega _0}\backslash {\bar \Omega _1} \subset {\mathbb{R}^N} , N ≥ 2. Our aim is to show that if the problem admits a solution in a suitable weak sense, then the underlying domain Ω is a Wulff shaped ring. The proof makes use of a maximum principle for an appropriate P-function, in the sense of L.E. Payne and some geometric arguments involving the anisotropic mean curvature of the free boundary.