Electrostatic characterization of spheres. (English) Zbl 0948.31004

Mendez, Osvaldo; Reichel, Wolfgang

Forum Math. 12, No. 2, 223-245 (2000).

The classical results on the distribution of electrostatic charge in a three-dimensional Euclidean space go back to Gauss. A property of a closed sphere is that the charge is uniformly distributed on its surface. The authors prove that a sphere in \(\mathbb{R}^{n}, n\geq 2\), is the only surface boundary of convex bodies for which electric charges distribute uniformly (with a constant density) in the absence of exterior fields.

Reviewer: R.P.Srivastav (Stony Brook)

MSC:

31B15	Potentials and capacities, extremal length and related notions in higher dimensions
78A30	Electro- and magnetostatics
52A20	Convex sets in \(n\) dimensions (including convex hypersurfaces)
32J25	Transcendental methods of algebraic geometry (complex-analytic aspects)

Keywords:

single-layer potential; charge distribution; convex bodies

References:

[1]	Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Ranton 1992 · Zbl 0804.28001
[2]	Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften 153. Springer-Verlag, New York 1969
[3]	Henrot, A., Philippin, G.A., PreAbet, H.: Overdetermined problems on ring shaped domains. Preprint (1998)
[4]	Leichtweiu, K.: Konvexe Mengen. Hochschultext. Springer-Verlag, Berlin-New York 1980
[5]	Leichtweiu, K.: Convexity and di erential geometry. In: Handbook of convex geometry B (ed. by P. Gruber, J. Wills), pp. 1045-1080. North-Holland, Amsterdam 1993
[6]	Martensen E., Math. Mech. 72 pp T596– (1992)
[7]	Martensen E., Arch. Math. 26 pp 620– (1975)
[8]	Payne L., SIAM J. Math. Anal. 10 pp 96– (1979)
[9]	Philippin G., Math. Meth. Appl. Sci. 12 pp 387– (1990)
[10]	Pohozaev S.I., Soviet Math. Dokl. 6 pp 1408– (1965)
[11]	Pucci P., Indiana Univ. Math. J. 35 pp 681– (1986)
[12]	Reichel W., Arch. Rational Mech. Anal. 137 pp 381– (1997)
[13]	Reichel W., Z. Anal. Anwendungen 15 pp 619– (1996)
[14]	Rellich F., Math. Z. 46 pp 635– (1940)
[15]	Schneider, R.: Convex surfaces, curvature and surface area measures. In: Handbook of Convex Geometry A (ed. by P. Gruber, J. Wills), pp. 273-300. North-Holland, Amsterdam 1993 · Zbl 0817.52003
[16]	Verchota G., J. Func. Anal. 59 pp 572– (1984)
[17]	Wermer, J.: Potential Theory. Lecture Notes in Mathematics 408. Springer-Verlag, Berlin-Heidelberg-New York 1974 · Zbl 0297.31001