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On the Electrostatic Born–Infeld Equation with Extended Charges

par Bonheure, Denis ;d’Avenia, Pietro;Pomponio, Alessio
Référence Communications in Mathematical Physics, 346, 3, page (877-906)
Publication Publié, 2016-09

Article révisé par les pairs

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Résumé :

In this paper, we deal with the electrostatic Born–Infeld equation (Forumala Presented.)where ρ is an assigned extended charge density. We are interested in the existence and uniqueness of the potential ϕ and finiteness of the energy of the electrostatic field - ∇ ϕ. We first relax the problem and treat it with the direct method of the Calculus of Variations for a broad class of charge densities. Assuming ρ is radially distributed, we recover the weak formulation of (BI) and the regularity of the solution of the Poisson equation (under the same smoothness assumptions). In the case of a locally bounded charge, we also recover the weak formulation without assuming any symmetry. The solution is even classical if ρ is smooth. Then we analyze the case where the density ρ is a superposition of point charges and discuss the results in (Kiessling, Commun Math Phys 314:509–523, 2012). Other models are discussed, as for instance a system arising from the coupling of the nonlinear Klein–Gordon equation with the Born–Infeld theory.