A rigorous framework for the Landau and Lifshitz approach to Thomson electrostatics. (English) Zbl 1359.35187

Grinfeld, Michael; Grinfeld, Pavel

J. Geom. Symmetry Phys. 41, 69-75 (2016).

Summary: L. D. Landau and E. M. Lifshitz [Electrodynamics of continuous media. Translated from the Russian by J. B. Sykes and J. S. Bell. New York-Paris: Pergamon Press (1960; Zbl 0122.45002)] proposed a novel formulation of the famous Thomson theorem, also known as the Thomson variational principle. In an attempt to explain, rather than postulate, the distribution of electrical charge exclusively on the surface of the conductor, Landau and Lifshitz allow the admissible variations in the electrical charge to penetrate the interior of the conductor. This is a valuable generalization of their predecessors’ work, as well as a step towards basing more of the analysis on first principles. Landau and Lifshitz’ approach has not received the attention it deserves because it was not formulated as a rigorous technique, but rather as a slight of hand to arrive at a known result. In this paper, we construct a rigorous mathematical framework based on the Landau and Lifshitz idea. In particular, we prove that surface distribution of charges corresponds to the absolute minimum of electrostatic energy.

MSC:

35Q60	PDEs in connection with optics and electromagnetic theory
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J50	Variational methods for elliptic systems
78A30	Electro- and magnetostatics

Keywords:

boundary value problems; electrostatics; mathematical physics; Poisson and Laplace equation; variational calculus

Citations:

Zbl 0122.45002