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

### Julio D. Rossi

Universidad de Buenos Aires, Argentina### Peiyong Wang

Wayne State University, Detroit, USA

## 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.

## Cite this article

Julio D. Rossi, Peiyong Wang, The limit as $p \to \infty$ in a two-phase free boundary problem for the $p$-Laplacian. Interfaces Free Bound. 18 (2016), no. 1, pp. 115–135

DOI 10.4171/IFB/359