On Some Basic Problems of Riesz Kernels in Potential Theory

Abstract

This dissertation contains discussions of three open problem areas in Potential Theory involving the Riesz kernels. In the first discussion, we consider the k-nearest neighbor logarithmic energies. We will establish first- and second-order asymptotics of the energies as the number of particles goes to infinity on any Jordan measurable set. The second discussion involves continuous energies with external fields. We will investigate a particular case of dimension reduction phenomena where the support of the equilibrium measure becomes a single sphere. For the third discussion, we study polarization problems or Chebyshev problems under the effect of external fields for Riesz-like kernels. In particular, we try to find a connection between discrete and continuous polarizations with external fields by establishing a minimum principle.