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

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 B1 are universally Lipschitz continuous in B1/2, 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.