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

TL;DR

This paper proves the optimal local Lipschitz regularity for almost-minimizers of a class of functionals with generalized Orlicz growth, extending regularity results to more general growth conditions.

Contribution

It establishes the Lipschitz regularity for almost-minimizers in one-phase problems with generalized Orlicz growth, a significant extension of existing regularity theory.

Findings

Proves optimal Lipschitz regularity for almost-minimizers.

Extends regularity results to generalized Orlicz growth functions.

Provides a framework for analyzing one-phase free boundary problems.

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.