Variational Problems with Long-Range Interaction

We consider a class of variational problems for densities that repel each other at a distance. Typical examples are given by the Dirichlet functional and the Rayleigh functionalD(u)=∑i=1k∫Ω|∇ui|2orR(u)=∑i=1k∫Ω|∇ui|2∫Ωui2,minimized in the class of H1(Ω , Rk) functions attaining some boundary conditions on ∂Ω, and subjected to the constraint dist({ui>0},{uj>0})≥1∀i≠j.For these problems, we investigate the optimal regularity of the solutions, prove a free-boundary condition, and derive some preliminary results characterizing the free boundary ∂{∑i=1kui>0}.