In the early development of potential theory, the singularities of superharmonic functions gave rise to the concept of a polar set. At the same time, the classic notion of capacity of a conductor in a grounded sphere became an object of interest to mathematicians. Both of these concepts are fully developed in this chapter and will culminate in the characterization of polar sets as sets of capacity zero. An essential part of this development is Choquet's theory of capacities which has important applications to stochastic processes as well as potential theory. These concepts are used to settle questions pertaining to equilibrium distribution of charges, the existence of Green function for regions in R n , and the boundary behavior of Green functions. It will be shown that an open subset of R 2 has a Green function if and only if the region supports a positive superharmonic function.
Unable to display preview. Download preview PDF.