On the Electrostatic Born–Infeld Equation with Extended Charges

Abstract

Abstract. In this paper, we deal with the electrostatic Born-Infeld equationwhere 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 [17]. Other models are discussed, as for instance a system arising from the coupling of the nonlinear Klein-Gordon equation with the Born-Infeld theory.

On the Regularity of the Minimizer of the Electrostatic Born–Infeld Energy

“…From a physical point of view this means that Maxwell's model violates the principle of finiteness of the energy. To avoid this phenomenon, Born [11,12] and later on Born and Infeld [13,14], proposed a new model based on the modification of Maxwell's Lagrangian density (we refer to [9,Sect. 1] and the references therein for more details).…”

“…We point out that, in our context, the notion of critical point in weak sense is equivalent to ask that u ρ is a minimum for the functional I ρ (see [9,Sect. 2]), and, if ρ is a distribution, the weak formulation of (1.4) extends to any test function ψ ∈ C ∞ c (R N ).…”

Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions

“…We are interested in radial, C 2 (Ω) solutions of (1.1), thus writing, with the usual abuse of notation, u(x) = u(r) for r = |x|. The nonlinear differential operator appearing in (1.1) is usually meant as a mean curvature operator in the Lorentz-Minkowski space and it is of interest in Differential Geometry and General Relativity [3,26,27]; it also appears in the nonlinear theory of Electromagnetism, where it is usually referred to as Born-Infeld operator [9,10,13]. In recent years, it has become very popular among specialists in Nonlinear Analysis, and various existence/multiplicity results for the associated boundary value problems are available, both in the ODE and in the PDE case, possibly in a non-radial setting (see, among others, [1,2,4,5,6,7,17,18,19,20,23,24,30] and the references therein).…”

On the speed rate of convergence of solutions to conservation laws with nonlinear diffusions

“…Regarding the unbounded and singular case (1.5), the nonlinear differential operator (1.9) div ∇u 1 − |∇u| 2 appears in many applications and it is usually meant as mean curvature operator in the Lorentz-Minkowski space. It is of interest in differential geometry, general relativity and appears in the nonlinear theory of electromagnetism, where it is usually referred to as Born-Infeld operator; for details see, among others, [4], [7] and references therein. Recently, the equation div ∇u…”