Variational Problems with Long-Range Interaction
Résumé
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(u) = k i=1 ˆΩ |∇u i | 2 or R(u) = k i=1 ´Ω |∇u i | 2 ´Ω u 2 i minimized in the class of H 1 (Ω, R k) functions attaining some boundary conditions on ∂Ω, and subjected to the constraint dist({u i > 0}, {u j > 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 ∂{ k i=1 u i > 0}.
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)