A limit case in non-isotropic two-phase minimization problems driven by $p$-Laplacians

Abstract

In this work we study a minimization problem with two-phases where in each phase region the problem is ruled by a quasi-linear elliptic operator of $p-$Laplacian type. The problem in its variational form is as follows:

Here we minimize among all admissible functions $v$ in an appropriate Sobolev space with a prescribed boundary datum $v=g$ on $\partial \Omega$. First, we show existence of a minimizer, prove some properties, and provide an example for non-uniqueness. Moreover, we analyze the limit case where $p$ and $q$ go to infinity, obtaining a limiting free boundary problem governed by the $\infty$-Laplacian operator. Consequently, Lipschitz regularity for any limiting solution is obtained. Finally, we establish some weak geometric properties for solutions.