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 ) where q ± ∈ L ∞ ( Ω ) . 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.