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

In this work we establish the optimal Lipschitz regularity for non-negative almost minimizers of the one-phase Bernoulli-type functional $${\mathcal{J}}_{\mathrm{G}}(u,\mathrm{\Omega }):={\int }_{\mathrm{\Omega }}(\mathrm{G}(\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid })+{\chi }_{\{u>0\}})dx$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded domain and $\mathrm{G}:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is a Young function with ${\mathrm{G}}^{\mathrm{\prime }}=g$ 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.