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arXiv · 2206.04956

Threshold condensation to singular support for a Riesz equilibrium problem

Djalil Chafaï·Edward B. Saff·Robert S. Womersley
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Abstract

We compute the equilibrium measure in dimension d=s+4 associated to a Riesz s-kernel interaction with an external field given by a power of the Euclidean norm. Our study reveals that the equilibrium measure can be a mixture of a continuous part and a singular part. Depending on the value of the power, a threshold phenomenon occurs and consists of a dimension reduction or condensation on the singular part. In particular, in the logarithmic case s=0 (d=4), there is condensation on a sphere of special radius when the power of the external field becomes quadratic. This contrasts with the case d=s+3 studied previously, which showed that the equilibrium measure is fully dimensional and supported on a ball. Our approach makes use, among other tools, of the Frostman or Euler-Lagrange variational characterization, the Funk-Hecke formula, the Gegenbauer orthogonal polynomials, and hypergeometric special functions.

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Djalil Chafaï, Edward B. Saff, Robert S. Womersley. 2023-01-24. Threshold condensation to singular support for a Riesz equilibrium problem. https://doi.org/10.1007/s13324-023-00779-w

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