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

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Abstract

We consider the electrostatic Born–Infeld energy

$$\int_{\mathbb{R}^N}\left(1-{\sqrt{1-|\nabla u|^2}}\right)\, {\rm d}x -\int_{\mathbb{R}^N}\rho u\, {\rm d}x,$$

where \({\rho \in L^{m}(\mathbb{R}^N)}\) is an assigned charge density, \({m \in [1,2_*]}\), \({2_*:=\frac{2N}{N+2}}\), \({N\geq 3}\). We prove that if \({\rho \in L^q(\mathbb{R}^N) }\) for \({q > 2N}\), the unique minimizer \({u_\rho}\) is of class \({W_{loc}^{2,2}(\mathbb{R}^N)}\). Moreover, if the norm of \({\rho}\) is sufficiently small, the minimizer is a weak solution of the associated PDE

$$\label{eq:BI-abs}-\operatorname{div}\left(\displaystyle\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)= \rho \quad\hbox{in }\mathbb{R}^N,\quad \quad \quad \mathcal{(BI)}$$

with the boundary condition \({\lim_{|x|\to\infty}u(x)=0}\), and it is of class \({C^{1,\alpha}_{loc}(\mathbb{R}^N)}\) for some \({\alpha \in (0,1)}\).

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Acknowledgements

Research partially supported by the project ERC Advanced Grant 2013 n. 339958 Complex Patterns for Strongly Interacting Dynamical Systems COMPAT, by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) by FNRS (PDR T.1110.14F and MIS F.4508.14) and by ARC AUWB-2012-12/17-ULB1- IAPAS

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Correspondence to Denis Bonheure.

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Communicated by G. Dal Maso

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Bonheure, D., Iacopetti, A. On the Regularity of the Minimizer of the Electrostatic Born–Infeld Energy. Arch Rational Mech Anal 232, 697–725 (2019). https://doi.org/10.1007/s00205-018-1331-4

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  • DOI: https://doi.org/10.1007/s00205-018-1331-4

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