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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.00120 |
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| _version_ | 1866911139178217472 |
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| author | Chafaï, Djalil Matzke, Ryan W. Saff, Edward B. Vu, Minh Quan H. Womersley, Robert S. |
| author_facet | Chafaï, Djalil Matzke, Ryan W. Saff, Edward B. Vu, Minh Quan H. Womersley, Robert S. |
| contents | We consider Riesz energy problems with radial external fields. We study the question of whether or not the equilibrium is the uniform distribution on a sphere. We develop general necessary as well as general sufficient conditions on the external field that apply to powers of the Euclidean norm as well as certain Lennard--Jones type fields. Additionally, in the former case, we completely characterize the values of the power for which dimension reduction occurs in the sense that the support of the equilibrium measure becomes a sphere. We also briefly discuss the relation between these problems and certain constrained optimization problems. Our approach involves the Frostman characterization, the Funk--Hecke formula, and the calculus of hypergeometric functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00120 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Riesz Energy with a Radial External Field: When is the Equilibrium Support a Sphere? Chafaï, Djalil Matzke, Ryan W. Saff, Edward B. Vu, Minh Quan H. Womersley, Robert S. Classical Analysis and ODEs 31B10, 31A10, 44A20, 33C20 We consider Riesz energy problems with radial external fields. We study the question of whether or not the equilibrium is the uniform distribution on a sphere. We develop general necessary as well as general sufficient conditions on the external field that apply to powers of the Euclidean norm as well as certain Lennard--Jones type fields. Additionally, in the former case, we completely characterize the values of the power for which dimension reduction occurs in the sense that the support of the equilibrium measure becomes a sphere. We also briefly discuss the relation between these problems and certain constrained optimization problems. Our approach involves the Frostman characterization, the Funk--Hecke formula, and the calculus of hypergeometric functions. |
| title | Riesz Energy with a Radial External Field: When is the Equilibrium Support a Sphere? |
| topic | Classical Analysis and ODEs 31B10, 31A10, 44A20, 33C20 |
| url | https://arxiv.org/abs/2405.00120 |