On the electrostatic Born-Infeld equation with extended charges

International audienceIn this paper, we deal with the electrostatic Born-Infeld equation (BI) − div ∇φ 1 − |∇φ| 2 = ρ, x ∈ R N , lim |x|→∞ φ(x) = 0, 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 smootheness 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 [18]. Other models are discussed, as for instance a system arising from the coupling of the nonlinear Klein-Gordon equation with the Born-Infeld theory