A concept of weak Riesz energy with application to condensers with touching plates
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A concept of weak Riesz energy with application to condensers with touching plates Natalia Zorii1 Received: 9 January 2020 / Revised: 9 January 2020 / Accepted: 11 August 2020 © Springer Nature Switzerland AG 2020
Abstract We proceed further with the study of minimum weak Riesz energy problems for condensers with touching plates, initiated jointly with Fuglede (Potential Anal 51:197– 217, 2019). Having now added to the analysis constraint and external source of energy, we obtain a Gauss type problem, but with weak energy involved. We establish sufficient and/or necessary conditions for the existence of solutions to the problem and describe their potentials. Treating the solution as a function of the condenser and the constraint, we prove its continuity relative to the vague topology and the topologies determined by the weak and standard energy norms. We show that the criteria for the solvability thus obtained fail in general once the problem is reformulated in the setting of standard energy, thereby justifying an advantage of weak energy when dealing with condensers with touching plates. Keywords Standard and weak Riesz energies of Radon measures · Condensers with touching plates · Minimum energy problems · Constraints · External fields Mathematics Subject Classification Primary 31C15
1 Standard and weak Riesz energies of measures In potential theory on Rn , n ≥ 3, relative to the Riesz kernel κα (x, y) := |x − y|α−n of order α ∈ (0, 2], we proceed further with the study of minimum weak energy problems, initiated jointly with Bent Fuglede [17]. We are motivated by the observation that the standard concept of energy is too restrictive when dealing with condensers
Dedicated to Professor Edward B. Saff on the occasion of his 75th birthday
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Natalia Zorii [email protected] Institute of Mathematics Academy of Sciences of Ukraine, Tereshchenkivska 3, 01601 Kyiv, Ukraine 0123456789().: V,-vol
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with touching plates, while application of weak energy allows the treatment of much broader condenser problems. Let M(Rn ) stand for the linear space of all real-valued Radon measures μ on n R 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, and let M+ (Rn ) be the cone of all positive μ ∈ M(Rn ). The standard concept of α-Riesz energy of μ ∈ M(Rn ) is introduced by E α (μ) := E κα (μ) :=
κα (x, y) d(μ ⊗ μ)(x, y)
(1.1)
provided that E α (μ+ ) + E α (μ− ) or E α (μ+ , μ− ) is finite, and the finiteness of E α (μ) means that κα is (|μ| ⊗ |μ|)-integrable, i.e. E α (|μ|) < ∞. Here μ+ and μ− denote the positive and negative parts in the Hahn–Jordan decomposition of μ, E α (μ+ , μ− ) :=
κα (x, y) d(μ+ ⊗ μ− )(x, y)
is the (standard) α-Riesz mutual energy of μ+ and μ− , and |μ| := μ+ + μ− . The Riesz kernel is strictly positive definite in the sense that E α (μ) ≥ 0 for any μ ∈ M(Rn ) (whenever defined), and E α (μ) = 0 only for μ = 0. This implies that all μ ∈ M(R
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