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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Approximation of the equilibrium distribution
by distributions of equal point charges
with minimal energy


Authors: J. Korevaar and M. A. Monterie
Journal: Trans. Amer. Math. Soc. 350 (1998), 2329-2348
MSC (1991): Primary {31B15; Secondary 31B05, 31B10, 31B25}
MathSciNet review: 1473445
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Abstract: Let $\omega $ denote the classical equilibrium distribution (of total charge $1$) on a convex or $C^{1,\alpha }$-smooth conductor $K$ in $\mathbb{R}^{q}$ with nonempty interior. Also, let $\omega _{N}$ be any $N$th order ``Fekete equilibrium distribution'' on $K$, defined by $N$ point charges $1/N$ at $N$th order ``Fekete points''. (By definition such a distribution minimizes the energy for $N$-tuples of point charges $1/N$ on $K$.) We measure the approximation to $\omega $ by $\omega _{N}$ for $N \to \infty $ by estimating the differences in potentials and fields,

\begin{equation*}U^{\omega }-U^{\omega _{N}}\quad \text{\rm and}\quad {\mathcal{E}}^{\omega }-{\mathcal{E}}^{\omega _{N}},\end{equation*}

both inside and outside the conductor $K$. For dimension $q \geq 3$ we obtain uniform estimates ${\mathcal{O}}(1/N^{1/(q-1)})$ at distance $\geq \varepsilon >0$ from the outer boundary $\Sigma $ of $K$. Observe that ${\mathcal{E}}^{\omega }=0$ throughout the interior $\Omega $ of $\Sigma $ (Faraday cage phenomenon of electrostatics), hence ${\mathcal{E}}^{\omega _{N}}={\mathcal{O}}(1/N^{1/(q-1)})$ on the compact subsets of $\Omega $. For the exterior $\Omega ^{\infty }$ of $\Sigma $ the precise results are obtained by comparison of potentials and energies. Admissible sets $K$ have to be regular relative to capacity and their boundaries must allow good Harnack inequalities. For the passage to interior estimates we develop additional machinery, including integral representations for potentials of measures on Lipschitz boundaries $\Sigma $ and bounds on normal derivatives of interior and exterior Green functions. Earlier, one of us had considered approximations to the equilibrium distribution by arbitrary distributions $\mu _{N}$ of equal point charges on $\Sigma $. In that context there is an important open problem for the sphere which is discussed at the end of the paper.


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Additional Information

J. Korevaar
Affiliation: Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, Netherlands
Email: korevaar@wins.uva.nl

M. A. Monterie
Affiliation: Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, Netherlands

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02187-4
PII: S 0002-9947(98)02187-4
Keywords: Capacity, capacity-regular sets, electrostatic fields, energies, equilibrium distributions, Fekete points, Green functions and their gradients, harmonic functions, harmonic measure, Harnack-type inequalities, integral representations, Kelvin transform, level surfaces, Lipschitz domains, potentials
Received by editor(s): April 1, 1996
Article copyright: © Copyright 1998 American Mathematical Society