Lipschitz regularity of weakly coupled vectorial almost-minimizers for Alt-Caffarelli functionals in Orlicz spaces

Abstract

For a fixed constant $\hat{I}»>0$ and a bounded Lipschitz domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ with $n\ge 2$, we establish that almost-minimizers (functions satisfying a sort of variational inequality) of the Alt-Caffarelli type functional \[ \mathcal{J}_G({\bf v};Ω) \coloneqq \int_Ω\left(\sum_{i=1}^mG\big(|\nabla v_i(x)|\big) + λχ_{\{|{\bf v}|>0\}}(x)\right) dx , \] where $\mathbf{v}=(v_1,\dots ,v_m)$ and $m\in \mathbb{N}$, exhibit optimal Lipschitz continuity on compact subsets of $\hat{I}\text{c◯}$, where $G$ is a Young function satisfying specific growth conditions. Furthermore, we obtain universal gradient estimates for non-negative almost-minimizers in the interior of non-coincidence sets. %{\color{blue}Furthermore, under the additional convexity assumption on $G$, we address the problem of boundary Lipschitz regularity for $v$ by adopting a fundamentally different analytical approach.} Notably, this method also provides an alternative proof of the optimal local Lipschitz regularity in the domain's interior. Our work extends the recent regularity results for weakly coupled vectorial almost-minimizers for the $p$-Laplacian addressed in \cite{BFS24}, and even the scalar case treated in \cite{daSSV}, \cite{DiPFFV24} and \cite{PelegTeix24}, thereby providing new insights and approaches applicable to a variety of non-linear one or two-phase free boundary problems with non-standard growth.