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Article Dans Une Revue Analysis and Mathematical Physics Année : 2023

Threshold condensation to singular support for a Riesz equilibrium problem

Résumé

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.

Dates et versions

hal-03850097 , version 1 (12-11-2022)

Identifiants

Citer

Djalil Chafaï, Edward Saff, Robert Womersley. Threshold condensation to singular support for a Riesz equilibrium problem. Analysis and Mathematical Physics, 2023, 13 (1), pp.19. ⟨10.1007/s13324-023-00779-w⟩. ⟨hal-03850097⟩
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