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

TL;DR

This paper proves optimal Lipschitz regularity for almost-minimizers of a vectorial Alt-Caffarelli functional in Orlicz spaces, extending previous results to more general growth conditions and providing universal gradient estimates.

Contribution

It establishes Lipschitz regularity for vectorial almost-minimizers in Orlicz spaces, generalizing prior p-Laplacian results and introducing new boundary regularity techniques.

Findings

Optimal Lipschitz continuity in the interior of the domain.

Universal gradient bounds in non-coincidence sets.

Extension of regularity results to Orlicz growth conditions.

Abstract

For a fixed constant $\lambda > 0$ and a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$ with $n \geq 2$, we establish that almost-minimizers (functions satisfying a sort of variational inequality) of the Alt-Caffarelli type functional \[ \mathcal{J}_G({\bf v};\Omega) \coloneqq \int_\Omega \left(\sum_{i=1}^mG\big(|\nabla v_i(x)|\big) + \lambda \chi_{\{|{\bf v}|>0\}}(x)\right) dx , \] where ${\bf v} = (v_1, \dots, v_m)$ and $m \in \mathbb{N}$, exhibit optimal Lipschitz continuity on compact subsets of $\Omega$, 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.