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arXiv · 2104.03733

Riesz energy problems with external fields and related theory

Peter Dragnev·Ramon Orive·Edward B. Saff·Franck Wielonsky
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Abstract

In this paper, we investigate Riesz energy problems on unbounded conductors in $\R^d$ in the presence of general external fields $Q$, not necessarily satisfying the growth condition $Q(x)\to\infty$ as $x\to\infty$ assumed in several previous studies. We provide sufficient conditions on $Q$ for the existence of an equilibrium measure and the compactness of its support. Particular attention is paid to the case of the hyperplanar conductor $\R^{d}$, embedded in $\R^{d+1}$, when the external field is created by the potential of a signed measure $\nu$ outside of $\R^{d}$. Simple cases where $\nu$ is a discrete measure are analyzed in detail. New theoretic results for Riesz potentials, in particular an extension of a classical theorem by de La Vall\'ee-Poussin, are established. These results are of independent interest.

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Peter Dragnev, Ramon Orive, Edward B. Saff, Franck Wielonsky. 2021-04-08. Riesz energy problems with external fields and related theory. https://arxiv.org/abs/2104.03733

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