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A lower bound for electrostatic capacity in the plane

Published online by Cambridge University Press:  14 November 2011

L. E. Fraenkel
L. E. Fraenkel
Affiliation:
Mathematics Division, University of Sussex
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Synopsis

This note presents a lower bound, in terms of the diameter ratio of the inner and outer conductors, for the electrostatic capacity of certain two-dimensional condensers. We use double Steiner symmetrization to prove that the minimizing condenser consists of a line segment placed symmetrically within a circle; the capacity of this condenser is known explicitly.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 88 , Issue 3-4 , 1981 , pp. 267 - 273
Copyright
Copyright © Royal Society of Edinburgh 1981

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References

1Berger, M. S. and Fraenkel, L. E.. Nonlinear desingularization in certain free-boundary problems. Comm. Math. Phys. 77 (1980), 149–172.CrossRefGoogle Scholar
2Eggleston, H. G.. Convexity (Cambridge: University Press, 1969).Google Scholar
3Essen, M. R.. The cos πλ Theorem. Lecture Notes in Mathematics 467 (Berlin: Springer, 1975).Google Scholar
4Garnett, J.. Analytic capacity and measure. Lecture Notes in Mathematics 297 (Berlin: Springer, 1972).Google Scholar
5Hayman, W. K.. Multivalent functions (Cambridge: University Press, 1958).Google Scholar
6Landkof, N. S.. Foundations of modem potential theory (Berlin: Springer, 1972).CrossRefGoogle Scholar
7Lewy, H. and Stampacchia, G.. On the regularity of the solution of a variational inequality. Comm. Pure Appl. Math. 22 (1969), 53–188.CrossRefGoogle Scholar
8Maz'ja, V. G.. On (p,l)-capacity, imbedding theorems, and the spectrum of a selfadjoint elliptic operator. Math. USSR-Izv. 7 (1973), 357–387.CrossRefGoogle Scholar
9Payne, L. E.. Isoperimetric inequalities and their applications. SIAM Rev. 9 (1967), 453–488.CrossRefGoogle Scholar
10Pólya, G. and Szegö, G.. Inequalities for the capacity of a condenser. Amer. J. Math. 67 (1945), 1–32.CrossRefGoogle Scholar
11Pólya, G. and Szegö, G.. Isoperimetric inequalities in mathematical physics (Princeton: University Press, 1951).Google Scholar
12Sarvas, J.. Symmetrization of condensers in n-space. Ann. Acad. Sci. Fenn. Ser. AI 522 (1972).Google Scholar
13Stampacchia, G.. Variational inequalities. Proc. NATO Advanced Study Inst., Theory and applications of monotone operators (Gubbio: Ediz. Oderisi, 1969).Google Scholar
14Wendt, G.. Article in Encyclopedia of Physics (ed. Flügge, S.), vol. XVI, Electric fields and waves (Berlin: Springer, 1958).Google Scholar