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Journal Articles Archive for Rational Mechanics and Analysis Year : 2017

Variational Problems with Long-Range Interaction

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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}.
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hal-03884821 , version 1 (05-12-2022)



Nicola Soave, Hugo Tavares, Susanna Terracini, Alessandro Zilio. Variational Problems with Long-Range Interaction. Archive for Rational Mechanics and Analysis, 2017, 228 (3), pp.743-772. ⟨10.1007/s00205-017-1204-2⟩. ⟨hal-03884821⟩
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