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

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 |