The limit as $p\to \infty$ in a two-phase free boundary problem for the $p$-Laplacian

Abstract

In this paper, we study the limit as $p$ goes to infinity of a minimizer of a variational problem that is a two-phase free boundary problem of phase transition for the $p$-Laplacian. Under a geometric compatibility condition, we prove that this limit is a solution of a free boundary problem for the $\infty$-Laplacian. When the compatibility condition does not hold, we prove that there still exists a uniform limit that is a solution of a minimization problem for the Lipschitz constant. Moreover, we provide, in the latter case, an example that shows that the free boundary condition can be lost in the limit.