# Equilibrium measures and equilibrium potentials in the Born-Infeld model

Bonheure, Denis; D'Avenia, Pietro; Pomponio, Alessio; Reichel, Wolfgang

Abstract:
In this paper, we consider the electrostatic Born-Infeld model \begin{equation*} \tag{$\mathcal{BI}$} \left\{ \begin{array}{rcll}
-\operatorname{div}\left(\displaystyle\frac{\nabla \phi}{\sqrt{1-|\nabla
\phi|^2}}\right)&=& \rho & \hbox{in }\mathbb{R}^N, \\[6mm]
\displaystyle\lim_{|x|\to \infty}\phi(x)&=& 0 \end{array} \right.
\end{equation*} where $\rho$ is a charge distribution on the boundary of a bounded domain $\Omega\subset \mathbb{R}^N$. We are interested in its equilibrium measures, i.e. charge distributions which minimize the electrostatic energy of the corresponding potential among all possible distributions with fixed total charge. We prove existence of equilibrium measures and we show that the corresponding equilibrium potential is unique and constant in $\overline \Omega$. Furthermore, for smooth domains, we obtain the uniqueness of the equilibrium measure, we give its precise expression, and we verify that the equilibrium potential solves $\mathcal{BI}$. Finally we characterize balls in $\mathbb{R}^N$ as the unique sets among all bounded $C^{2,\alpha}$-domains $\Omega$ for which the equilibrium d ... mehr

 Zugehörige Institution(en) am KIT Institut für Analysis (IANA)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht Jahr 2018 Sprache Englisch Identifikator ISSN: 2365-662X URN: urn:nbn:de:swb:90-869045 KITopen ID: 1000086904 Verlag KIT, Karlsruhe Umfang 23 S. Serie CRC 1173 ; 2018/29 Schlagworte Born-Infeld model, nonlinear electromagnetism, equilibrium measure, equilibrium potential
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