Lipschitz regularity of a weakly coupled vectorial almost-minimizers for the $p$-Laplacian

For a given constant $\lambda >0$ and a bounded Lipschitz domain $D\subset {\mathbb{R}}^n$ ($n\ge 2$), we establish that almost-minimizers of the functional $$J(\mathbf{v};D)={\int }_D\sum _{i=1}^m{\mid \mathrm{\nabla }{v}_i(x)\mid }^p+\lambda {\chi }_{\{\mid \mathbf{v}\mid >0\}}(x)dx,1<p<\mathrm{\infty },$$ where $\mathbf{v}=(v_1,\cdots ,v_m)$, and $m\in \mathbb{N}$, exhibit optimal Lipschitz continuity in compact sets of $D$. Furthermore, assuming $p\ge 2$ and employing a distinctly different methodology, we tackle the issue of boundary Lipschitz regularity for $v$. This approach simultaneously yields alternative proof for the optimal local Lipschitz regularity for the interior case.