Free Boundary Regularity for Almost-Minimizers

TL;DR

This paper investigates the regularity of free boundaries for almost-minimizers of a specific variational functional, establishing uniform rectifiability and smoothness under certain conditions, despite the lack of PDE or monotonicity tools.

Contribution

It proves free boundary regularity and rectifiability for almost-minimizers, extending known results from minimizers to a broader class without relying on PDE or monotonicity formulas.

Findings

Free boundary is uniformly rectifiable under non-degeneracy conditions.

When $q_- ot eq 0$, free boundary regularity is established.

For $q_- ext{=0}$ and H"older continuous }q_+, ext{free boundary is } C^{1,eta}.

Abstract

In this paper we study the free boundary regularity for almost-minimizers of the functional \begin{equation*} J(u)=\int_{\mathcal O} |\nabla u(x)|^2 +q^2_+(x)\chi_{\{u>0\}}(x) +q^2_-(x)\chi_{\{u<0\}}(x)\ dx \end{equation*} where $q_\pm \in L^\infty(\mathcal O)$. Almost-minimizers satisfy a variational inequality but not a PDE or a monotonicity formula the way minimizers do (see [AC], [ACF], [CJK], [W]). Nevertheless we succeed in proving that, under a non-degeneracy assumption on $q_\pm$, the free boundary is uniformly rectifiable. Furthermore, when $q_-\equiv 0$, and $q_+$ is H\"older continuous we show that the free boundary is almost-everywhere given as the graph of a $C^{1,\alpha}$ function (thus extending the results of [AC] to almost-minimizers).