Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. D.R. Adams and L.I. Hedberg, Function spaces and potential theory, Grundlehren Math. Wiss., vol. 314, Springer-Verlag, Berlin, 1996. MR 97j:46024
  • 2. B.E.J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), 275-288. MR 57:6470
  • 3. -, On the distribution of Fekete points, Duke Math. J. 45 (1978), 537-542. MR 80c:31003
  • 4. T. Erber and G.M. Hockney, Complex systems: Equilibrium configurations of $N$ equal charges on a sphere $(2\leq N\leq 112)$, Publication 95-075-T, Fermi National Accelerator Laboratory, 1995.
  • 5. D.M. Eydus, Estimates on the derivatives of Green's function, (Russian), Dokl. Akad. Nauk SSSR 106 (1956), 207-209. MR 17:960
  • 6. M. Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z. 17 (1923), 228-249.
  • 7. O. Frostman, Potentiel d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions, Dissertation, Lunds Univ. Mat. Sem. 3 (1935), 1-118, Zbl 13-063.
  • 8. R.M. Gabriel, An extended principle of the maximum for harmonic functions in 3 dimensions, J. London Math. Soc. 30 (1955), 388-401. MR 17:358
  • 9. -, A result concerning convex level surfaces of 3-dimensional harmonic functions, J. London Math. Soc. 32 (1957), 286-294. MR 19:848a
  • 10. -, Further results concerning the level surfaces of the Green's function for a 3-dimen-sional convex domain (I), (II), J. London Math. Soc. 32 (1957), 295-302, 303-306. MR 19:848b;MR 19:848c
  • 11. D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Grundlehren Math. Wiss., vol. 224, Springer-Verlag, Berlin, 1983. MR 86c:35035
  • 12. W.K. Hayman and P.B. Kennedy, Subharmonic functions I, London Math. Soc. Monographs, vol. 9, Academic Press, London, 1976. MR 57:665
  • 13. R.A. Hunt and R.L. Wheeden, Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc. 147 (1970), 507-527. MR 43:547
  • 14. D.S. Jerison and C.E. Kenig, Boundary value problems on Lipschitz domains, Studies in partial differential equations (W. Littman, ed.), M.A.A. Studies in Math., vol. 23, 1982, pp. 1-68. MR 85f:35057
  • 15. O.D. Kellogg, Foundations of potential theory, Grundlehren Math. Wiss., vol. 31, Springer-Verlag, Berlin, 1929.
  • 16. C.E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Reg. Conf. Ser. in Math., vol. 83, Amer. Math. Soc., Providence, R.I, 1994. MR 96a:35040
  • 17. J. Korevaar, Asymptotically neutral distributions of electrons and polynomial approximation, Ann. of Math. 80 (1964), 403-410. MR 29:6031
  • 18. -, Equilibrium distributions of electrons on roundish plane conductors I, II, Indag. Math. 36 (1974), 423-437, 438-456. MR 50:13477a,b
  • 19. -, Problems of equilibrium points on the sphere and electrostatic fields, Report 76-03, Dept. of Mathematics, Univ. of Amsterdam, 1976.
  • 20. -, Chebyshev-type quadratures: use of complex analysis and potential theory, Complex Potential Theory (P.M. Gauthier and G. Sabidussi, eds.), Kluwer, Dordrecht, 1994, pp. 325-364. MR 96g:41029
  • 21. -, Fekete extreme points and related problems, Approximation theory and function series, Bolyai Soc. Math. Studies, vol. 5, Budapest, 1996, pp. 35-62. CMP 97:08
  • 22. J. Korevaar and T. Geveci, Fields due to electrons on an analytic curve, SIAM J. Math. Anal. 2 (1971), 445-453. MR 44:2939
  • 23. J. Korevaar and R.A. Kortram, Equilibrium distributions of electrons on smooth plane conductors, Indag. Math. 45 (1983), 203-219. MR 85j:30051.
  • 24. J. Korevaar and J.L.H. Meyers, Spherical Faraday cage for the case of equal point charges and Chebyshev-type quadrature on the sphere, Integral Trans. Spec. Funct. 1 (1993), 105-117. MR 97g:41046
  • 25. N.S. Landkof, Foundations of modern potential theory, Grundlehren Math. Wiss., vol. 180, Springer-Verlag, Berlin, 1972. MR 50:2520
  • 26. M.A. Liapounoff, Sur certaines questions qui se rattachent au problème de Dirichlet, J. Math. Pures Appl. (5) 4 (1898), 241-311.
  • 27. M.A. Monterie, Studies in potential theory, Ph.D. thesis, Free University, Amsterdam, 1995.
  • 28. R. Nevanlinna, Analytic functions, Grundlehren Math. Wiss., vol. 162, Springer-Verlag, Berlin, 1970, English transl. MR 43:5003
  • 29. G. Pólya and G. Szegö, Über den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen, J. Reine Angew. Math. 165 (1931), 4-49, Zbl 2-136.
  • 30. Ch. Pommerenke, Polynome und konforme Abbildung, Monatsh. Math. 69 (1965), 58-61. MR 30:4919
  • 31. -, Über die Verteilung der Fekete-Punkte, Math. Ann. 168 (1967), 111-127; II, Math. Ann. 179 (1969), 212-218. MR 34:6057; MR 40:377.
  • 32. P. Sjögren, On the regularity of the distribution of the Fekete points of a compact surface in $\mathbb{R}^{n}$, Ark. Mat. 11 (1973), 147-151. MR 49:5381
  • 33. J. Wermer, Potential theory, Lecture Notes in Math., vol. 408, Springer-Verlag, Berlin, 1974. MR 56:12284
  • 34. K.-O. Widman, On the boundary values of harmonic functions in $\mathbb{R}^{3}$, Ark. Mat. 5 (1964), 221-230. MR 34:1544
  • 35. -, Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand. 21 (1967), 17-37. MR 39:621

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): {31B15, 31B05, 31B10, 31B25}

Retrieve articles in all journals with MSC (1991): {31B15, 31B05, 31B10, 31B25}

Additional Information

J. Korevaar
Affiliation: Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, Netherlands

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

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

American Mathematical Society