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

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