Boundary regularity of weakly coupled vectorial almost-minimizers for Alt-Caffarelli functionals with non-standard growth

TL;DR

This paper proves that weakly coupled vectorial almost-minimizers for a class of non-standard growth Alt-Caffarelli functionals are Lipschitz continuous up to the boundary, extending regularity results to more general non-linear free boundary problems.

Contribution

It extends regularity results for weakly coupled vectorial almost-minimizers to functionals with non-standard growth, broadening the scope of free boundary problem analysis.

Findings

Establishes optimal Lipschitz regularity up to the boundary.

Extends regularity results to non-standard growth functionals.

Provides new methods applicable to non-linear free boundary problems.

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 (up-to-the boundary) Lipschitz continuity, where $G$ is a $\mathcal{N}$-function satisfying specific growth conditions. Our work extends the recent regularity results for weakly coupled vectorial almost-minimizers for the $p$-Laplacian addressed in \cite{BFS24}, thereby providing new insights and approaches applicable to a wide class of non-linear one or two-phase free boundary problems with non-standard growth. Our findings remain novel and significant even in the scalar setting and for minimizers of the type considered by Mart\'{i}nez--Wolanski \cite{MW08} and da Silva \textit{et al.} \cite{daSSV2024}.