Lipschitz regularity of solutions to two-phase $p$-Laplacian free boundary problems with right hand side

TL;DR

This paper establishes the optimal Lipschitz regularity of viscosity solutions for two-phase p-Laplacian free boundary problems with non-zero right hand side, extending known results even in the homogeneous case.

Contribution

It proves the local Lipschitz continuity for solutions with right hand side and introduces new regularity results applicable to viscosity solutions in this context.

Findings

Proves Lipschitz regularity for solutions with right hand side

Establishes H"older continuity for broader problem classes

Results are new even for homogeneous problems

Abstract

We prove the local Lipschitz continuity of viscosity solutions for two-phase free boundary problems for the $p$-Laplacian with non-zero right hand side, where $p\in (1,\infty)$. This is the optimal regularity for the problem. We also obtain the local H\"older continuity for a larger class of problems. The results introduced here are new even in the homogeneous situation, that is, when the right hand side is zero. Our work applies to merely viscosity solutions, which allows a wide applicability.