Energy on spheres and discreteness of minimizing measures

Dmitriy Bilyk; Alexey Glazyrin; Ryan Matzke; Josiah Park; Oleksandr Vlasiuk

doi:10.1016/j.jfa.2021.108995

Energy on spheres and discreteness of minimizing measures

Dmitriy Bilyk, Alexey Glazyrin, Ryan Matzke, Josiah Park, Oleksandr Vlasiuk

Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the p-frame energy has empty interior whenever p is not an even integer. A similar effect is also demonstrated for energies with analytic potentials which are not positive definite. In addition, we establish the existence of discrete minimizers for a large class of energies, which includes energies with polynomial potentials.

Original language	English (US)
Article number	108995
Journal	Journal of Functional Analysis
Volume	280
Issue number	11
DOIs	https://doi.org/10.1016/j.jfa.2021.108995
State	Published - Jun 1 2021

Bibliographical note

Funding Information:
We express our gratitude to the following organizations that hosted subsets of the authors during the work on this paper: AIM, ICERM, INI, CIEM, Georgia Tech. The first author was supported by the grant DMS-1665007 , the third author was supported by the Graduate Fellowship 00039202 , and the fourth author was supported in part by the grant DMS-1600693 , all from the US National Science Foundation . This paper is based upon work supported by the National Science Foundation grant DMS-1439786 while D.B., A.G., and O.V. were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the “Point Configurations in Geometry, Physics and Computer Science” program. This work is also supported in part by EPSRC grant No. EP/K032208/1 .

Publisher Copyright:
© 2021 Elsevier Inc.

Keywords

Potential energy minimization
Spherical codes
Spherical designs

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abstract = "In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the p-frame energy has empty interior whenever p is not an even integer. A similar effect is also demonstrated for energies with analytic potentials which are not positive definite. In addition, we establish the existence of discrete minimizers for a large class of energies, which includes energies with polynomial potentials.",

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note = "Funding Information: We express our gratitude to the following organizations that hosted subsets of the authors during the work on this paper: AIM, ICERM, INI, CIEM, Georgia Tech. The first author was supported by the grant DMS-1665007 , the third author was supported by the Graduate Fellowship 00039202 , and the fourth author was supported in part by the grant DMS-1600693 , all from the US National Science Foundation . This paper is based upon work supported by the National Science Foundation grant DMS-1439786 while D.B., A.G., and O.V. were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the “Point Configurations in Geometry, Physics and Computer Science” program. This work is also supported in part by EPSRC grant No. EP/K032208/1 . Publisher Copyright: {\textcopyright} 2021 Elsevier Inc.",

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AU - Matzke, Ryan

AU - Park, Josiah

AU - Vlasiuk, Oleksandr

N1 - Funding Information: We express our gratitude to the following organizations that hosted subsets of the authors during the work on this paper: AIM, ICERM, INI, CIEM, Georgia Tech. The first author was supported by the grant DMS-1665007 , the third author was supported by the Graduate Fellowship 00039202 , and the fourth author was supported in part by the grant DMS-1600693 , all from the US National Science Foundation . This paper is based upon work supported by the National Science Foundation grant DMS-1439786 while D.B., A.G., and O.V. were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the “Point Configurations in Geometry, Physics and Computer Science” program. This work is also supported in part by EPSRC grant No. EP/K032208/1 . Publisher Copyright: © 2021 Elsevier Inc.

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N2 - In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the p-frame energy has empty interior whenever p is not an even integer. A similar effect is also demonstrated for energies with analytic potentials which are not positive definite. In addition, we establish the existence of discrete minimizers for a large class of energies, which includes energies with polynomial potentials.

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KW - Potential energy minimization

KW - Spherical codes

KW - Spherical designs

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Energy on spheres and discreteness of minimizing measures

Abstract

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Keywords

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