×

Electrostatic characterization of spheres. (English) Zbl 0948.31004

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.

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)
PDF BibTeX XML Cite
Full Text: DOI

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.