Capacity and its applications

Capacity appears in electrostatics as a characteristic of a body made of conducting material: it is the maximal charge that can be put on this body so that the electric potential of the field created by this charge is bounded by 1. In 1924 Norbert Wiener introduced capacity in mathematics as a positive function of compact sets. It is not additive as a measure but subadditive. Starting with Wiener’s famous work, the theory was fast developed and applied in many problems of analysis, partial differential equations, mathematical physics and geometry. This paper presents a short summary of my lectures given in Summer 2004 for graduate students in Northeastern University (Boston) and in Conference-School on Analysis and Geometry (Novosibirsk). It starts with a definition and simplest properties of the Wiener capacity. Then we describe some classical applications of capacity in partial differential equations: removable singularities of bounded harmonic functions and regularity of boundary points for the Dirichlet boundary value problem (Wiener). In the last section we formulate a recent result by V.Maz’ya and M.Shubin on two-sided estimates for the bottom of the spectrum of the Laplacian with the Dirichlet boundary conditions in open subsets of R . We comment on corollaries of this result and formulate unsolved problems.