Regularity of almost minimizers with free boundary

Abstract

In this paper we study the local regularity of almost minimizers of the functional $$\begin{aligned} J(u)=\int _\Omega |\nabla u(x)|^2 +q^2_+(x)\chi _{\{u>0\}}(x) +q^2_-(x)\chi _{\{u<0\}}(x) \end{aligned}$$ where \(q_\pm \in L^\infty (\Omega )\). 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.