A minimization problem with free boundary related to a cooperative system

Abstract

We study the minimum problem for the functional $${\int }_{\Omega }\left(\right|\nabla \mathbf{u}{|}^2+Q^2{\chi }_{\left\{\right|\mathbf{u}|>0\}})dx$$ with the constraint $u_i\ge 0$ for $i=1,\dots ,m$, where $\Omega \subset {\mathbb{R}}^n$ is a bounded domain and $\mathbf{u}=(u_1,\dots ,u_m)\in H^1(\Omega ;{\mathbb{R}}^m)$. First we derive the Euler equation satisfied by each component. Then we show that the noncoincidence set $\left\{\right|u|>0\}$ is (locally) nontangentially accessible. Having this, we are able to establish sufficient regularity of the force term appearing in the Euler equations and derive the regularity of the free boundary $\Omega \cap \partial \left\{\right|u|>0\}$.