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

For a given constant $\lambda > 0$ and a bounded Lipschitz domain $D \subset \mathbb{R}^n$ ($n \geq 2$), we establish that almost-minimizers of the functional $$J(\mathbf{v}; D) = \int_D \sum_{i=1}^{m} \left|\nabla v_i(x) \right|^p+ \lambda \chi_{\{\left|\mathbf{v} \right|>0\}} (x) \, dx, \qquad 1<p<\infty,$$ where $\mathbf{v} = (v_1, \cdots, v_m)$, and $m \in \mathbb{N}$, exhibit optimal Lipschitz continuity in compact sets of $D$. Furthermore, assuming $p \geq 2$ 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.