Distribution of point charges with small discrete energy

Abstract

We study the asymptotic equidistribution of points near arbitrary compact sets of positive capacity in Rd, d ≥ 2. Our main tools are the energy estimates for Riesz potentials. We also consider the quantitative aspects of this equidistribution in the classical Newtonian case. In particular, we quantify the weak convergence of discrete measures to the equilibrium measure, and give the estimates of convergence rates for discrete potentials to the equilibrium potential. 1. Asymptotic equidistribution of discrete sets Let E be a compact set in R, d ≥ 2. Denote the Euclidean distance between x ∈ R and y ∈ R by |x− y|. We consider potential theory associated with Riesz kernels kα(x) := |x|α−d, x ∈ R, 0 < α < d. For a Borel measure μ with compact support, define its energy by Iα[μ] := ∫∫ kα(x− y) dμ(x)dμ(y). A central theme in potential theory is the study of the minimum energy problem Wα(E) := inf μ∈M(E) Iα[μ], where M(E) is the space of all positive unit Borel measures supported on E. If Robin’s constant Wα(E) is finite, then the above infimum is attained by the equilibrium measure μE ∈ M(E) [14, p. 131–133], which is a unique probability measure expressing the steady state distribution of charge on the conductor E. The capacity of E is defined by Cα(E) := 1 Wα(E) , where we set Cα(E) = 0 when Wα(E) is infinite. For a more detailed exposition of Riesz potential theory, we refer the reader to the book of Landkof [14]. 2010 Mathematics Subject Classification. Primary 31C20; Secondary 31C15.