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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 distribution is a constant multiple of the surface measure on $\partial \Omega$. ... mehr

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DOI: 10.5445/IR/1000086904
Veröffentlicht am 24.10.2018
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsjahr 2018
Sprache Englisch
Identifikator ISSN: 2365-662X
urn:nbn:de:swb:90-869045
KITopen-ID: 1000086904
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 23 S.
Serie CRC 1173 ; 2018/29
Schlagwörter Born-Infeld model, nonlinear electromagnetism, equilibrium measure, equilibrium potential
Nachgewiesen in arXiv
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