In this paper, we study the static Born–Infeld equation -div(∇u1-|∇u|2)=∑k=1nakδxkinRN,lim|x|→∞u(x)=0,where N≥ 3 , ak∈ R for all k= 1 , ⋯ , n, xk∈ RN are the positions of the point charges, possibly non-symmetrically distributed, and δxk is the Dirac delta distribution centered at xk. For this problem, we give explicit quantitative sufficient conditions on ak and xk 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 xk’s depending on the sign of charges ak’s. For every m∈ N, we also consider the approximated problem -∑h=1mαhΔ2hu=∑k=1nakδxkinRN,lim|x|→∞u(x)=0where 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.

On the Born–Infeld equation for electrostatic fields with a superposition of point charges / Bonheure D.; Colasuonno F.; Foldes J.. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - STAMPA. - 198:3(2019), pp. 749-772. [10.1007/s10231-018-0796-y]

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

Colasuonno F.
;
2019

Abstract

In this paper, we study the static Born–Infeld equation -div(∇u1-|∇u|2)=∑k=1nakδxkinRN,lim|x|→∞u(x)=0,where N≥ 3 , ak∈ R for all k= 1 , ⋯ , n, xk∈ RN are the positions of the point charges, possibly non-symmetrically distributed, and δxk is the Dirac delta distribution centered at xk. For this problem, we give explicit quantitative sufficient conditions on ak and xk 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 xk’s depending on the sign of charges ak’s. For every m∈ N, we also consider the approximated problem -∑h=1mαhΔ2hu=∑k=1nakδxkinRN,lim|x|→∞u(x)=0where 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.
2019
On the Born–Infeld equation for electrostatic fields with a superposition of point charges / Bonheure D.; Colasuonno F.; Foldes J.. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - STAMPA. - 198:3(2019), pp. 749-772. [10.1007/s10231-018-0796-y]
Bonheure D.; Colasuonno F.; Foldes J.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/771355
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