Regularity of almost minimizers with free boundary

In this paper we study the local regularity of almost minimizers of the functional J(u)=∫Ω|∇u(x)|2+q+2(x)χ{u>0}(x)+q-2(x)χ{u<0}(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} J(u)=\int _\Omega |\nabla u(x)|^2 +q^2_+(x)\chi _{\{u>0\}}(x) +q^2_-(x)\chi _{\{u<0\}}(x) \end{aligned}$$\end{document}where q±∈L∞(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_\pm \in L^\infty (\Omega )$$\end{document}. Almost minimizers do not satisfy a PDE or a monotonicity formula like minimizers do (see Alt and Caffarelli, in J Reine Angew Math, 325:105–144, 1981; Alt et al., in Trans Am Math Soc 282:431–461, 1984; Caffarelli et al., in Global energy minimizers for free boundary problems and full regularity in three dimensions. In: Non-compact Problems at the Intersection of Geometry, Analysis, and Topology, vol. 8397. Contemporary Mathematics, vol. 350. American Mathematical Society, Providence, 2004; DeSilva and Jerison, in J Reine Angew Math 635:121, 2009). Nevertheless we succeed in proving that they are locally Lipschitz, which is the optimal regularity for minimizers.