A MINIMIZATION PROBLEM WITH FREE BOUNDARY RELATED TO A COOPERATIVE SYSTEM

We study the minimum problem for the functional integral(Omega)(vertical bar del u vertical bar(2) + Q(2) chi({vertical bar u vertical bar>0}))dx with the constraint u(i) >= 0 for i = 1,... , m, where Omega subset of R-n is a bounded domain and u = (u(1),... , u(m)) is an element of H-1 (Omega;R-m). First we derive the Euler equation satisfied by each component. Then we show that the noncoincidence set {vertical bar u vertical bar > 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 boolean AND partial derivative{vertical bar u vertical bar> 0}.