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 functionalminimized in the class of H 1 (Ω, R k ) functions attaining some boundary conditions on ∂Ω, and subjected to the constraintFor these problems, we investigate the optimal regularity of the solutions, prove a free-boundary condition, and derive some preliminary results characterizing the free boundary ∂{ k i=1 u i > 0}.

Free boundary problems with long-range interactions: uniform Lipschitz estimates in the radius

“…Much less is known in the nonlocal case r > 0. In a joint paper with S. Terracini [28] (see Theorem 1.2 and Theorem 1.3-(3), (6) therein), we have shown the following properties.…”

“…The approach used both in the local [6,13,33] and in the nonlocal case [28] consists in studying the following relaxed formulation of c r in terms of measurable functions rather than sets:…”

“…, Ω 2,r ) achieves c r , which coincides with cr , and satisfies conditions (1)- (3). Under an additional regularity assumption of the free boundary ∂Ω i , we have also derived a free boundary condition, satisfied by the eigenfunctions of the optimal partitions (see [28,Theorem 1.6]). The validity of such a condition remains a crucial open problem in the general setting for optimal partition problems with a distance constraint.…”

“…Observe that, for each r > 0 fixed, Lipschitz regularity is proved via a barrier argument, which is possible due to the exterior sphere condition (see [28,Theorem 3.4]). However the barrier used depends on the radius, and the argument breaks down as r → 0 + .…”

“…As for the optimal partition problems, the local case r = 0 is essentially understood (see [7,33]), while for the nonlocal one r > 0, studied in [28], there are still many open questions. However, local and nonlocal cases share the same optimal regularity for the minimizers: if u r is a minimizer of h r , then it is locally Lipschitz continuous in Ω, both for r = 0 and r > 0.…”