Various Concepts of Riesz Energy of Measures and Application to Condensers with Touching Plates
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Various Concepts of Riesz Energy of Measures and Application to Condensers with Touching Plates Bent Fuglede1 · Natalia Zorii2 Dedicated to Professor Stephen J. Gardiner on the occasion of his 60th birthday Received: 27 September 2018 / Accepted: 22 August 2019 / © Springer Nature B.V. 2019
Abstract We develop further the concept of weak α-Riesz energy with α ∈ (0, 2] of Radon measures μ on Rn , n 3, introduced in our preceding study and defined by (κα/2 μ)2 dm, m denoting the Lebesgue measure on Rn . Here κα/2 μ 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 κα/2 μ ∈ 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 Rn 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 αGreen 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. Keywords Standard and weak Riesz energy · Deny-Schwartz energy · Minimum energy problems · Condensers with touching plates · External fields · Constraints
Natalia Zorii
[email protected] Bent Fuglede [email protected] 1
Department of Mathematical Sciences, University of Copenhagen, 2100 Copenhagen, Denmark
2
Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska 3, 01601, Kyiv-4, Ukraine
B. Fuglede, N. Zorii
Mathematics Subject Classification (2010) 31C15
1 Introduction Throughout the paper we fix a natural n 3 and a real α ∈ (0, 2]. Let M(Rn ) stand for the linear space of all real-valued Radon measures μ on Rn , equipped with the vague topology, i.e. the topology of pointwise convergence on the class C0 (Rn ) of all (real-valued finite) continuous functions on Rn with compact support. The standard concept of energy of a (signed) Radon measure μ ∈ M(Rn ) relative to the α-Riesz kernel κα (x, y) := |x − y|α−n on Rn , |x − y| being the Euclidean distance between x, y ∈ Rn , is introduced by Eα (μ) := Eκα (μ) :=
κα (x, y) d(μ ⊗ μ)(x, y)
(1.1)
provided that Eα (μ+ ) + Eα (μ− ) or Eα (μ+ , μ− ) is finite, and finiteness of Eα (μ) means that κα is (|μ| ⊗ |μ|)-integrable
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