A minimization problem with free boundary related to a cooperative system

Abstract: 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.