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 Höder regularity characterization via Campanato-type estimates in the context of Orlicz modulars, which is new for such a class of non-standard growth functionals.

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