Regularity for almost minimizers with free boundary

TL;DR

This paper proves that almost minimizers of a free boundary functional are locally Lipschitz continuous, establishing optimal regularity despite the lack of PDE or monotonicity properties.

Contribution

It demonstrates the local Lipschitz regularity of almost minimizers for a free boundary problem, extending regularity results beyond minimizers.

Findings

Almost minimizers are locally Lipschitz.

Regularity holds despite absence of PDE or monotonicity formula.

Results apply to functionals with bounded coefficients.

Abstract

In this paper we study the local regularity of almost minimizers of the functional \begin{equation*} J(u)=\int_\Omega |\nabla u(x)|^2 +q^2_+(x)\chi_{\{u>0\}}(x) +q^2_-(x)\chi_{\{u<0\}}(x) \end{equation*} where $q_\pm \in L^\infty(\Omega)$. Almost minimizers do not satisfy a PDE or a monotonicity formula like minimizers do (see \cite{AC}, \cite{ACF}, \cite{CJK}, \cite{W}). Nevertheless we succeed in proving that they are locally Lipschitz, which is the optimal regularity for minimizers.