Various Concepts of Riesz Energy of Measures and Application to Condensers with Touching Plates

Abstract

We develop further the concept of weak α-Riesz energy with α ∈ (0,2] of Radon measures μ on $\mathbb R^{n}$, $n\geqslant 3$, introduced in our preceding study and defined by $\int \limits (\kappa _{\alpha /2}\mu )^{2} dm$, m denoting the Lebesgue measure on $\mathbb R^{n}$. Here κ_α/2μ is the potential of μ relative to the α/2-Riesz kernel |x − y|^α/2−n. This concept extends that of standard α-Riesz energy, and for μ with κ_α/2μ ∈ L²(m) it coincides with that of Deny-Schwartz energy defined with the aid of the Fourier transform. We investigate minimum weak α-Riesz energy problems with external fields in both the unconstrained and constrained settings for generalized condensers (A₁,A₂) such that the closures of A₁ and A₂ in $\mathbb R^{n}$ are allowed to intersect one another. (Such problems with the standard α-Riesz energy in place of the weak one would be unsolvable, which justifies the need for the concept of weak energy when dealing with condenser problems.) We obtain sufficient and/or necessary conditions for the existence of minimizers, provide descriptions of their supports and potentials, and single out their characteristic properties. To this end we have discovered an intimate relation between minimum weak α-Riesz energy problems over signed measures associated with (A₁,A₂) and minimum α-Green energy problems over positive measures carried by A₁. Crucial for our analysis of the latter problems is the perfectness of the α-Green kernel, established in our recent paper. As an application of the results obtained, we describe the support of the α-Green equilibrium measure.