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

Transactions of the American Mathematical Society

Volumn 350, Issue 6, 1998, Pages 2329-2348

  Korevaar, J ^a   Monterie, M A ^a  

^a UNIVERSITY OF AMSTERDAM   (Netherlands)

Author keywords

Capacity; Capacity regular sets; Electrostatic fields; Energies; Equilibrium distributions; Fekete points

Indexed keywords

EID: 22044446913     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-98-02187-4     Document Type: Article

References (35)

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 N 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 àla 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 19:848
10. ____, Further results concerning the level surfaces of the Green's function for a 3-dimensional convex domain (I), (II), J. London Math. Soc. 32 (1957), 295-302, 303-306. MR 19:848.
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, éd.), 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, SpringerVerlag, 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. 325364. 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 transi. MR 43:5003
29. G. Pölya 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; //, 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 Rn, 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 R3, 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