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

TL;DR

The paper addresses the problem of establishing Lipschitz regularity for weakly coupled vectorial almost-minimizers of Alt-Caffarelli-type functionals with non-standard (Orlicz) growth in bounded Lipschitz domains. It develops a g-harmonic approximation framework together with Campanato-type estimates in Orlicz–Sobolev spaces and employs scaling and compactness arguments within the Lieberman class $\mathcal{G}(\delta,g_0)$ to prove universal local Lipschitz bounds and gradient estimates inside non-coincidence sets, with Hölder continuity results extending to the gradient away from the free boundary under strengthened hypotheses. The results extend scalar and vectorial Alt-Caffarelli-type theories to non-polynomial growth scenarios, connecting to prior work on $p$-Laplacian and non-standard growth functionals. They provide a robust framework for quasi-minimizers and free boundary problems in non-homogeneous growth settings, with potential applications to a broad class of one- and two-phase problems in Orlicz spaces.

Abstract

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

Theorems & Definitions (31)

• Definition 1.1: $\mathcal{N}$-function
• Definition 1.2: Non-degenerate classes - BM
• Theorem 1.1: Local Hölder regularity of almost-minimizers
• Theorem 1.2: $C^{1, \alpha}$ of almost-minimizers
• Theorem 1.3: Local Lipschitz regularity of almost-minimizers
• Corollary 1.1: Lipschitz regularity for minimizers
• Definition 2.1: Orlicz space
• Definition 2.2: $\Delta_2$ and $\nabla_2-$condition
• Proposition 2.1: daSSV
• Theorem 2.1: daSSV