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On the Born–Infeld equation for electrostatic fields with a superposition of point charges

  • Denis Bonheure
  • Francesca Colasuonno
  • Juraj Földes
Article
  • 33 Downloads

Abstract

In this paper, we study the static Born–Infeld equation
$$\begin{aligned} -\mathrm {div}\left( \frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) =\sum _{k=1}^n a_k\delta _{x_k}\quad \text{ in } \mathbb R^N,\qquad \lim _{|x|\rightarrow \infty }u(x)=0, \end{aligned}$$
where \(N\ge 3\), \(a_k\in \mathbb R\) for all \(k=1,\dots ,n\), \(x_k\in \mathbb R^N\) are the positions of the point charges, possibly non-symmetrically distributed, and \(\delta _{x_k}\) is the Dirac delta distribution centered at \(x_k\). For this problem, we give explicit quantitative sufficient conditions on \(a_k\) and \(x_k\) to guarantee that the minimizer of the energy functional associated with the problem solves the associated Euler–Lagrange equation. Furthermore, we provide a more rigorous proof of some previous results on the nature of the singularities of the minimizer at the points \(x_k\)’s depending on the sign of charges \(a_k\)’s. For every \(m\in \mathbb N\), we also consider the approximated problem
$$\begin{aligned} -\sum _{h=1}^m\alpha _h\Delta _{2h}u=\sum _{k=1}^n a_k\delta _{x_k}\quad \text{ in } \mathbb R^N, \qquad \lim _{|x|\rightarrow \infty }u(x)=0 \end{aligned}$$
where the differential operator is replaced by its Taylor expansion of order 2m (see (2.1)). It is known that each of these problems has a unique solution. We study the regularity of the approximating solution, the nature of its singularities, and the asymptotic behavior of the solution and of its gradient near the singularities.

Keywords

Born–Infeld equation Nonlinear electromagnetism Mean curvature operator in the Lorentz–Minkowski space Inhomogeneous quasilinear equation 

Mathematics Subject Classification

35B40 35B65 35J62 35Q60 78A30 

Notes

Acknowledgements

The authors thank Maria Colombo for a fruitful discussion and for pointing to us the reference [2]. The authors acknowledge the support of the projects MIS F.4508.14 (FNRS) & ARC AUWB-2012-12/17-ULB1- IAPAS. F. Colasuonno was partially supported by the INdAM - GNAMPA Project 2017 “Regolarità delle soluzioni viscose per equazioni a derivate parziali non lineari degeneri.”

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Département de MathématiqueUniversité Libre de BruxellesBruxellesBelgique
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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