Free Boundary Regularity for Almost-Minimizers

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)χ_{\{u>0\}}(x) +q^2_-(x)χ_{\{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 Hölder continuous we show that the free boundary is almost-everywhere given as the graph of a $C^{1,\hat{I}\pm }$ function (thus extending the results of [AC] to almost-minimizers).

70 pages. Revised as per referee's comments