Published Paper

Inserted: 22 jul 2024
Last Updated: 22 aug 2024

Journal: Journal of Differential Equations
Volume: 412
Pages: 447-473
Year: 2024
Doi: https://doi.org/10.1016/j.jde.2024.08.040

Abstract:

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

\[ J(\mathbf{v}; D) = \int_D \sum_{i=1}^{m}
\nabla v_i(x)
^p+ \lambda \chi_{\{
\mathbf{v}
>0\}} (x) \, dx, \qquad 1<p<\infty, \]

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 $\mathbf{v}$. This approach simultaneously yields alternative proof for the optimal local Lipschitz regularity for the interior case.