The limit as p → ∞ in a two-phase free boundary problem for the p-Laplacian

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 infinity-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.