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

For a fixed constant $\lambda >0$ and a bounded Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with $n\ge 2$, we establish that almost-minimizers (functions satisfying a sort of variational inequality) of the Alt-Caffarelli type functional $${\mathcal{J}}_G(\mathbf{v};\mathrm{\Omega }){\int }_{\mathrm{\Omega }}\left(\sum _{i=1}^{m}G\right(\mathrm{\mid }\mathrm{\nabla }{v}_i(x)\mathrm{\mid })+\lambda {\chi }_{\{\mathrm{\mid }\mathbf{v}\mathrm{\mid }>0\}}(x))dx,$$ where $\mathbf{v}=(v_1,\dots ,v_m)$ 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 $\mathit{\text{et}}\mathit{\text{al.}}$ \cite{daSSV2024}.