The limit as $p \to \infty$ in a two-phase free boundary problem for the $p$-Laplacian
Julio Daniel Rossi, Peiyong Wang · Interfaces and Free Boundaries Mathematical Analysis Computation and Applications · 2016
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.