Variational Problems with Long-Range Interaction

Abstract

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 functional

$$\begin{aligned} D(\mathbf{u}) = \sum_{i=1}^k \int_{\Omega} |\nabla u_i|^2 \quad \text{or} \quad R(\mathbf{u}) = \sum_{i=1}^k \frac{\int_{\Omega} |\nabla u_i|^2}{\int_{\Omega} {u_i^2}}, \end{aligned}$$

minimized in the class of ${H^1(\Omega,\mathbb{R}^k)}$ functions attaining some boundary conditions on ∂Ω, and subjected to the constraint

$$\begin{aligned} \text{dist} (\{u_i > 0\}, \{u_j > 0\}) \ge 1 \quad \forall i \neq j. \end{aligned}$$

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 ${\partial \{\sum_{i=1}^k u_i > 0\}}$.