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.

Variational Problems with Long-Range Interaction

Soave, Nicola;NABAIS TAVARES, HUGO RICARDO;Terracini, Susanna;Zilio, Alessandro
2018

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 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.
Analysis; Mathematics (miscellaneous); Mechanical Engineering
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11311/1052440
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