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

  • Denis Bonheure
  • Francesca ColasuonnoEmail author
  • Juraj Földes


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.


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

Mathematics Subject Classification

35B40 35B65 35J62 35Q60 78A30 



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.”


  1. 1.
    Acerbi, E., Mingione, G.: Regularity results for a class of functionals with non-standard growth. Arch. Ration. Mech. Anal. 156(2), 121–140 (2001)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baroni, P.: Riesz potential estimates for a general class of quasilinear equations. Calc. Var. Partial Diff. Equ. 53(3–4), 803–846 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bartnik, R., Simon, L.: Spacelike hypersurfaces with prescribed boundary values and mean curvature. Commun. Math. Phys. 87(1), 131–152 (1982)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bonheure, D., D’Avenia, P., Pomponio, A.: On the electrostatic Born–Infeld equation with extended charges. Commun. Math. Phys. 346(3), 877–906 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bonheure, D., De Coster, C., Derlet, A.: Infinitely many radial solutions of a mean curvature equation in Lorentz–Minkowski space. Rend. Istit. Mat. Univ. Trieste 44, 259–284 (2012)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Born, M., Infeld, L.: Foundations of the new field theory. Proc. R. Soc. Lond. Ser. A 144(852), 425–451 (1934)CrossRefGoogle Scholar
  7. 7.
    Brezis, H., Sibony, M.: Équivalence de deux inéquations variationnelles et applications. Arch. Ration. Mech. Anal. 41(4), 254–265 (1971)CrossRefGoogle Scholar
  8. 8.
    Caffarelli, L.A., Friedman, A.: The free boundary for elastic-plastic torsion problems. Trans. Am. Math. Soc. 252, 65–97 (1979)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cellina, A.: On the regularity of solutions to the plastoelasticity problem. Adv. Calc. Var. (2017).
  10. 10.
    Colasuonno, F., Squassina, M.: Eigenvalues for double phase variational integrals. Ann. Mat. Pura Appl. (4) 195(6), 1917–1959 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cupini, G., Marcellini, P., Mascolo, E.: Existence and regularity for elliptic equations under \( p, q \)-growth. Adv. Diff. Equ. 19(7–8), 693–724 (2014)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ecker, K.: Area maximizing hypersurfaces in Minkowski space having an isolated singularity. Manuscripta Math. 56(4), 375–397 (1986)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics. Vol. 2: Mainly Electromagnetism and Matter. Addison-Wesley Publishing Co., Inc., Reading (1964)zbMATHGoogle Scholar
  14. 14.
    Fortunato, D., Orsina, L., Pisani, L.: Born–Infeld type equations for electrostatic fields. J. Math. Phys. 43(11), 5698–5706 (2002)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 editionzbMATHGoogle Scholar
  16. 16.
    Kichenassamy, S., Véron, L.: Singular solutions of the \(p\)-laplace equation. Math. Ann. 275(4), 599–615 (1986)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kiessling, M.K.-H.: On the quasi-linear elliptic pde \(-\nabla \cdot (\nabla u/\sqrt{1-|\nabla u|^2})= 4\pi \sum _k a_k\delta _{s_k}\) in physics and geometry. Commun. Math. Phys. 314(2), 509–523 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001)zbMATHGoogle Scholar
  19. 19.
    Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12(11), 1203–1219 (1988)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Martínez, S., Wolanski, N.: A minimum problem with free boundary in Orlicz spaces. Adv. Math. 218(6), 1914–1971 (2008)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Mihăilescu, M.: Classification of isolated singularities for nonhomogeneous operators in divergence form. J. Funct. Anal. 268(8), 2336–2355 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Serrin, J.: Singularities of solutions of nonlinear equations. Proc. Symp. App. Math 17, 68–88 (1965)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Szulkin, A.: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 3(2), 77–109 (1986)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Treu, G., Vornicescu, M.: On the equivalence of two variational problems. Calc. Var. Partial Diff. Equ. 11(3), 307–319 (2000)MathSciNetCrossRefGoogle Scholar

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© 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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