Riesz spherical potentials with external elds and minimal energy points separation

In this paper we consider the minimal energy problem on the sphere S for Riesz potentials with external …elds. Fundamental existence, uniqueness, and characterization results are derived about the associated equilibrium measure. The discrete problem and the corresponding weighted Fekete points are investigated. As an application we obtain the separation of the minimal senergy points for d 2 < s < d. The explicit form of the separation constant is new even for the classical case of s = d 1. Mathematics Subject Classi…cations (2005): 31B05, 31B15, 78A30 Key words: Minimal energy problems with external …elds, Riesz spherical potentials, Minimal s-energy points separation, Balayage, -superharmonic functions. The work of this author was initiated while visiting Vanderbilt University. yResearch supported, in part, by a National Science Foundation Research grant DMS 0532154. 1 1 Introduction and main results In this article we shall further develop and apply the theory of minimal s-energy problems for Riesz spherical potentials with external …eld, where the potential varies inversely with respect to the s-power of the Euclidean distance between points. The restriction to spherical potentials is mainly motivated by the applications to minimal energy points on the sphere, but the analysis may be carried out on more general manifolds, as well as with other kernels. This we intend to address in a subsequent work. For more on the general theory of equilibrium potentials with external …elds we refer to the recent works of Zorii [22], [23], and [24]. As the main application of our results we derive optimal order separation of the minimal senergy points on the sphere Sd Rd+1 for the range of the parameter d 2 < s < d. The explicit form of our separation constant is new even for the classical case s = d 1 considered by Dahlberg [2] in 1978, and improves upon the (mainly) implicit constants obtained in [15] by Kuijlaars, Sa¤, and Sun for the cases d 1 < s < d. In addition, for the important particular case of S2, our results with what was previously known settle the question of well-separation of minimal s-energy points for all s 0, except for the critical value s = 2. 1.1 Energy problems on the sphere with external …elds. Let Sd := fx 2 Rd+1 : jxj = 1g be the unit sphere in Rd+1, where j j denotes the Euclidean norm. Given a compact set E Sd, consider the classM(E) of unit positive Borel measures supported on E. For 0 < s < d the Riesz s-potential and Riesz s-energy of a measure 2 M(E) are given respectively by U s (x) := Z ks(x; y) d (y); Is( ) := Z Z ks(x; y) d (x)d (y);