A free boundary problem with multiple boundaries for a general class of anisotropic equations

Abstract In this paper we are going to investigate a free boundary problem for a class of anisotropic equations on a multiply-connected domain Ω ⊂ 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. This result represents the anisotropic extension of a result obtained by Payne and Philippin (1991) [13] . For the proof, we make use of a maximum principle for an appropriate P -function, a Rellich type identity, an anisotropic form of Minkowski inequality for convex sets and some geometric arguments involving the anisotropic mean curvatures of the free boundaries.