Regularity of Lipschitz free boundaries in two-phase problems for the p-Laplace operator

Abstract

In this paper we study the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator and we prove, in particular, that Lipschitz free boundaries are C 1 , γ -smooth for some γ ∈ ( 0 , 1 ) . As part of our argument, and which is of independent interest, we establish a Hopf boundary type principle for non-negative p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain.