Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Explicit minimisers for anisotropic Riesz energies

Rupert L. Frank, Joan Mateu, Maria Giovanna Mora, Luca Rondi, Lucia Scardia, Joan Verdera

TL;DR

This work characterises explicit minimisers for a class of anisotropic nonlocal energies with quadratic confinement. It develops a Fourier-analytic framework that reduces the problem to constructing an ellipsoidally deformed Barenblatt profile and verifying Euler–Lagrange conditions via a deformation argument across anisotropy, with a detailed treatment of Bessel and hypergeometric representations. The main result establishes a unique minimiser supported on an ellipsoid for d≥3 and s in [d-3,d)∩(0,5], extending prior 2D/3D symmetry-restricted cases and handling the full range up to s<5 through analytic continuation. The paper also discusses degenerate, non-strictly positive anisotropy cases where minimisers may lose dimensionality, highlighting the role of the Fourier-transform positivity of the kernel. Altogether, the work provides explicit, high-resolution descriptions of how anisotropy deforms isotropic minimisers in nonlocal interaction energies.

Abstract

In this paper we describe explicitly the energy minimisers of a class of nonlocal interaction energies where the attraction is quadratic, and the repulsion is Riesz-like and anisotropic.

Explicit minimisers for anisotropic Riesz energies

TL;DR

This work characterises explicit minimisers for a class of anisotropic nonlocal energies with quadratic confinement. It develops a Fourier-analytic framework that reduces the problem to constructing an ellipsoidally deformed Barenblatt profile and verifying Euler–Lagrange conditions via a deformation argument across anisotropy, with a detailed treatment of Bessel and hypergeometric representations. The main result establishes a unique minimiser supported on an ellipsoid for d≥3 and s in [d-3,d)∩(0,5], extending prior 2D/3D symmetry-restricted cases and handling the full range up to s<5 through analytic continuation. The paper also discusses degenerate, non-strictly positive anisotropy cases where minimisers may lose dimensionality, highlighting the role of the Fourier-transform positivity of the kernel. Altogether, the work provides explicit, high-resolution descriptions of how anisotropy deforms isotropic minimisers in nonlocal interaction energies.

Abstract

In this paper we describe explicitly the energy minimisers of a class of nonlocal interaction energies where the attraction is quadratic, and the repulsion is Riesz-like and anisotropic.

Paper Structure

This paper contains 19 sections, 3 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence and uniqueness of a minimiser
  4. Properties of the Gamma function
  5. The Fourier transform.
  6. The Bessel function of first kind
  7. The hypergeometric function
  8. A formula relating Bessel and hypergeometric functions
  9. Ellipses and ellipsoids
  10. The Fourier transform of the candidate minimiser
  11. The Fourier representation of $W\ast \mu_E$ for $s\in (d-4,d)\cap(0,5]$.
  12. The case of $s\in (d-4,d)\cap(0,3)$
  13. The case of $s\in (d-4,d)\cap[3,5]$
  14. Proof of the main result
  15. The first Euler-Lagrange condition
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $s\in [d-3,d)\cap (0,5]$, and let $W$ be as in potdef with $\Psi$ even. Assume that $W$ and $\widehat{W}$ are strictly positive and continuous on $\mathbb{S}^{d-1}$. Then there exists a unique minimiser $\mu_0$ of $I$ over $\mathcal{P}(\mathbb{R}^d)$. It is given by the push-forward of the measu where $E$ is the ellipsoid $R D(a) \overline{B}$, and $c_d$ and $r_d$ are the constants from intro:

Theorems & Definitions (6)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 3.1