Lipschitz regularity of almost minimizers in a Bernoulli problem with non-standard growth

Abstract

In this work we establish the optimal Lipschitz regularity for non-negative almost minimizers of the one-phase Bernoulli-type functional $$ \mathcal{J}_{\mathrm{G}}(u,\Omega) := \int_\Omega \left(\mathrm{G}(|\nabla u|)+\chi_{\{u>0\}}\right)\,dx $$ where $\Omega \subset \mathbb{R}^n$ is a bounded domain and $\mathrm{G}: [0, \infty) \to [0, \infty) $ is a Young function with $\mathrm{G}^{\prime}=g$ satisfying the Lieberman's classical conditions. Moreover, of independent mathematical interest, we also address a H\"{o}der regularity characterization via Campanato-type estimates in the context of Orlicz modulars, which is new for such a class of non-standard growth functionals.