Lipschitz regularity of almost-minimizers in one-phase problems with generalized Orlicz growth

Abstract

The optimal local Lipschitz regularity for scalar almost-minimizers of Alt-Caffarelli-type functionals $$ \mathcal{F}({v}; \Omega) = \int_\Omega \varphi(x,\left|\nabla v(x) \right|)+ \lambda \chi_{\{{v} >0\}} (x) \, \mathrm{d}x\,, $$ with growth function $\varphi$ a generalized Orlicz function, is established.