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

DA SILVA, JOÃO VITOR; ANALÍA SILVA; HERNÁN VIVAS

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS

AMER INST MATHEMATICAL SCIENCES

Lugar: springfield; Año: 2024 vol. 44 p. 1555 - 1586

1078-0947

In this work we establish the optimal Lipschitz regularity for non-negative almost minimizers of the one-phase Bernoulli-type functional$$mathcal{J}_{mathrm{G}}(u,Omega) defeq int_Omega left(mathrm{G}(|abla u|)+chi_{{u>0}}ight),dx$$where $Omega subset mathbb{R}^n$ is a bounded domain and $mathrm{G}: [0, infty) o [0, infty) $ is a Young function with $mathrm{G}^{prime}=g$ satisfying the Lieberman´s classical conditions. %Nonetheless, we dropped the usual assumption of $Delta^{prime}$ on $g$, which makes our results applicable for a wide class of operators with Orlicz-Sobolev type structure.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.