Lipschitz regularity of almost minimizers in one-phase problems driven by the $p$-Laplace operator

TL;DR

This paper proves that nonnegative almost minimizers of a nonlinear free boundary functional involving the p-Laplace operator are Lipschitz continuous for certain p values, advancing understanding of regularity in free boundary problems.

Contribution

It establishes Lipschitz regularity of almost minimizers in one-phase free boundary problems driven by the p-Laplace operator, for a range of p values.

Findings

Almost minimizers are Lipschitz continuous for p > max{2n/(n+2), 1}.

Regularity result applies to nonlinear free boundary problems involving the p-Laplace operator.

Provides a foundation for further regularity analysis in nonlinear free boundary problems.

Abstract

We prove that, given~$p>\max\left\{\frac{2n}{n+2},1\right\}$, the nonnegative almost minimizers of the nonlinear free boundary functional $$ J_p(u,\Omega):=\int_{\Omega}\Big( |\nabla u(x)|^p+\chi_{\{u>0\}}(x)\Big)\,dx$$ are Lipschitz continuous.