# On a two-phase free boundary problem ruled by the infinity Laplacian

## TL;DR

This paper studies a two-phase free boundary problem governed by the infinity Laplacian, proving that solutions are Lipschitz continuous and introducing a novel method applicable to similar problems.

## Contribution

It establishes optimal regularity for solutions and employs a new use of the Ishii-Lions' method as a surrogate for monotonicity formulas.

## Key findings

- Bounded viscosity solutions are Lipschitz continuous in the domain.
- The method is applicable to related free boundary problems.
- Optimal regularity for the problem is achieved.

## Abstract

In this paper we consider a two-phase free boundary problem ruled by the infinity Laplacian. Our main result states that bounded viscosity solutions in $B_1$ are universally Lipschitz continuous in $B_{1/2}$, which is the optimal regularity for the problem. We make a new use of the Ishii-Lions' method, which works as a surrogate for the lack of a monotonicity formula and is bound to be applicable in related problems.