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)=âi=1kâ«Î©|âui|2orR(u)=âi=1kâ«Î©|âui|2â«Î©ui2,minimized in the class of H1(Ω , Rk) functions attaining some boundary conditions on âΩ, and subjected to the constraint dist(ui>0,uj>0)â¥1âiâ 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 ââi=1kui>0.
Variational Problems with Long-Range Interaction
Soave, Nicola;NABAIS TAVARES, HUGO RICARDO;Terracini, Susanna;Zilio, Alessandro
2018-01-01
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)=âi=1kâ«Î©|âui|2orR(u)=âi=1kâ«Î©|âui|2â«Î©ui2,minimized in the class of H1(Ω , Rk) functions attaining some boundary conditions on âΩ, and subjected to the constraint dist(ui>0,uj>0)â¥1âiâ 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 ââi=1kui>0.File in questo prodotto:
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