Global minimizers of the two-phase Bernoulli problem with the $p$-Laplace operator

Abstract

In this paper, we study the classification of Lipschitz global solutions for a two-phase $p$-Laplace Bernoulli problem. Specifically, we focus on the scenario where the \textit{interior} two-phase points of the global solution are non-empty. Our results show that the expected $C^{1,η}$ regularity holds in a suitable neighborhood of certain two-phase points, which we refer to to as \textit{regular} two-phase points.