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In 1828, Green used a physical argument to introduce a function, which he called a potential function, to calculate charge distributions on conducting bodies. The lack of mathematical rigor led Gauss in a 1840 paper to propose a procedure for finding equilibrium distributions based on the fact that such a distribution should have minimal potential energy. This led to the study of the functional \(\int (G - 2f)\sigma \,dS\) where \(\sigma \) is a density function, \(G\) is a Green function, \(f\) is a continuous function, and \(dS\) denotes integration with respect to surface area on the boundary of a conducting body. Gauss assumed the existence of a distribution minimizing the functional. Frostman proved the existence of such a minimizing distribution in a 1935 paper. After defining the energy of a measure, properties of energy are related to capacity and equilibrium distributions. The chapter concludes with Wiener’s necessary and sufficient condition for regularity of a boundary point for the Dirichlet problem.
KeywordsCalculated Charge Distribution Minimum Potential Energy Vague Convergence Norm-bounded Sequence Regular Boundary Point