Regularity in the two-phase Bernoulli problem for the $p$-Laplace operator

TL;DR

This paper proves regularity properties of minimizers and free boundaries in the two-phase Bernoulli problem for the p-Laplacian, establishing flatness decay and Lipschitz regularity through viscosity solutions and a dichotomy argument.

Contribution

It demonstrates that minimizers are viscosity solutions and establishes $C^{1,eta}$ regularity of the free boundary for the p-Laplacian two-phase problem.

Findings

Minimizers of the ACF functional are viscosity solutions.

Flatness decay at free boundary points is established.

Minimizers are Lipschitz continuous.

Abstract

We show that any minimizer of the well-known ACF functional (for the $p$-Laplacian) is a viscosity solution. This allows us to establish a uniform flatness decay at the two-phase free boundary points to improve the flatness, that boils down to $C^{1,\eta}$ regularity of the flat part of the free boundary. This result, in turn, is used to prove the Lipschitz regularity of minimizers by a dichotomy argument.