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Explicit minimisers for anisotropic Riesz energies.

R L Frank1,2,3, J Mateu4,5, M G Mora6

  • 1Mathematisches Institut, Ludwig-Maximilians Universität München, München, Germany.

Calculus of Variations and Partial Differential Equations
|December 16, 2025
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Summary
This summary is machine-generated.

This study characterizes energy minimizers for nonlocal interactions with quadratic attraction and anisotropic Riesz repulsion. Findings reveal that under specific conditions, the minimizer adopts a Barenblatt-type profile on an ellipsoid.

Keywords:
Primary 31A15Secondary 49K20

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Area of Science:

  • Mathematical Physics
  • Analysis

Background:

  • Nonlocal interaction energies are crucial in various scientific fields.
  • Characterizing energy minimizers is fundamental for understanding system stability and behavior.
  • Previous work established formulas for Coulomb interactions.

Purpose of the Study:

  • To characterize energy minimizers for a specific class of nonlocal interaction energies.
  • To analyze systems with quadratic attraction and anisotropic Riesz-like repulsion.
  • To extend existing mathematical frameworks to more complex interaction potentials.

Main Methods:

  • Utilizing Fourier analysis to represent potentials of measures.
  • Extending previously established formulas for Coulombic potentials.
  • Analyzing the properties of energy minimizers under specific conditions.

Main Results:

  • Demonstrated that energy minimizers are supported on fully-dimensional ellipsoids.
  • Showed that the density profile of the minimizer follows a Barenblatt-type distribution.
  • Established these results when the Fourier transform of the repulsive potential is positive.

Conclusions:

  • The study provides a detailed characterization of energy minimizers for a novel class of nonlocal interactions.
  • The findings contribute to the mathematical understanding of systems with complex interaction potentials.
  • The developed Fourier representation technique offers a powerful tool for future research in this area.