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Pré-Publication, Document De Travail Année : 2014

Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere

Résumé

We study the Hamiltonian of a two-dimensional Coulomb system of n repelling points confined by an external field verifying the weak growth assumption due to Hardy and Kuijlaars. We prove an asymptotic expansion (as n goes to infinity) for the minimum of this Hamiltonian using a Gamma-Convergence's method of Sandier and Serfaty and depending on the minimum of a ''renormalized energy" W introduced in their previous work. We connect our result with the next-order term for optimal logarithmic energy on the unit sphere to prove the conjecture of Rakhmanov, Saff and Zhou about the existence of this term for which we find an upper bound. Finally we prove the equivalence between the conjecture of Brauchart, Hardin and Saff about the value of this coefficient and the conjecture of the global minimality of the triangular lattice for W among configurations of fixed average density.
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Dates et versions

hal-00979926 , version 1 (17-04-2014)
hal-00979926 , version 2 (21-04-2014)
hal-00979926 , version 3 (19-01-2015)
hal-00979926 , version 4 (28-10-2015)
hal-00979926 , version 5 (17-04-2018)

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Laurent Bétermin. Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere. 2014. ⟨hal-00979926v3⟩
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