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 ʃ (G — 2f)σ dS where s is a density 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 will be related to capacity and equilibrium distributions as developed in Section 4.4. The chapter will conclude with Wiener's necessary and sufficient condition for regularity of a boundary point for the Dirichlet problem.
Unable to display preview. Download preview PDF.