Lipschitz regularity of almost minimizers in one-phase problems driven by the $p$-Laplace operator

Abstract

We prove that, given~$p>\max \left\{\frac{2n}{n+2},1\right\}$, the nonnegative almost minimizers of the nonlinear free boundary functional $$J_p(u,\Omega ):={\int }_{\Omega }(|\nabla u(x){|}^p+{\chi }_{\{u>0\}}(x))dx$$ are Lipschitz continuous.