Regularity for the two phase singular perturbation problems

TL;DR

This paper establishes that BMO gradient bounds lead to Lipschitz regularity in two-phase singular perturbation problems, with applications to combustion models involving the p-Laplacian, using a weak energy identity approach.

Contribution

It introduces a new method linking BMO estimates to Lipschitz regularity for two-phase problems, applicable to broader solution classes.

Findings

BMO gradient estimates imply Lipschitz regularity of limits.

The approach characterizes free boundary points intrinsically.

Method applicable to general solution classes.

Abstract

We prove that an a priori BMO gradient estimate for the two phase singular perturbation problem implies Lipschitz regularity for the limits. This problem arises in the mathematical theory of combustion where the reaction-diffusion is modelled by the $p$-Laplacian. A key tool in our approach is the weak energy identity. Our method proves a natural and intrinsic characterization of the free boundary points and can be applied to more general classes of solutions.