  We study the minimum problem for the functional $\int_{\Omega}\bigl( \vert
\nabla \mathbf{u} \vert^{2} + Q^{2}\chi_{\{\vert \mathbf{u}\vert>0\}} \bigr)dx$
with the constraint $u_i\geq 0$ for $i=1,\cdots,m$ where
$\Omega\subset\mathbb{R}^{n}$ is a bounded domain and
$\mathbf{u}=(u_1,\cdots,u_m)\in H^{1}(\Omega;\mathbb{R}^{m})$.
  Using an array of technical tools, from geometric analysis for the free
boundaries, we reduce the problem to its scalar counterpart and hence conclude
similar results as that of scalar problem. This can also be seen as the most
novel part of the paper, that possibly can lead to further developments of free
boundary regularity for systems.
