Variational problems with long-range interaction

Abstract

We consider a class of variational problems for densities that repel each other at distance. Typical examples are given by the Dirichlet functional and the Rayleigh functional \[ D(\mathbf{u}) = \sum_{i=1}^k \int_{\Omega} |\nabla u_i|^2 \quad \text{or} \quad R(\mathbf{u}) = \sum_{i=1}^k \frac{\int_{\Omega} |\nabla u_i|^2}{\int_{\Omega} u_i^2} \] minimized in the class of $H^1(\Omega,\mathbb{R}^k)$ functions attaining some boundary conditions on $\partial \Omega$, and subjected to the constraint \[ \mathrm{dist} (\{u_i > 0\}, \{u_j > 0\}) \ge 1 \qquad \forall i \neq j. \] 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 $\partial \{\sum_{i=1}^k u_i > 0\}$.