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

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 (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 Martínez--Wolanski \cite{MW08} and da Silva \textit{et al.} \cite{daSSV2024}.

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