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Shape optimisation for nonlocal anisotropic energies

Riccardo Cristoferi, Maria Giovanna Mora, Lucia Scardia

Abstract

In this paper we consider shape optimisation problems for sets of prescribed mass, where the driving energy functional is nonlocal and anisotropic. More precisely, we deal with the case of attractive/repulsive interactions in two and three dimensions, where the attraction is quadratic and the repulsion is given by an anisotropic variant of the Coulomb potential. Under the sole assumption of strict positivity of the Fourier transform of the interaction potential, we show the existence of a threshold value for the mass above which the minimiser is an ellipsoid, and below which the minimiser does not exist. If, instead, the Fourier transform of the interaction potential is only nonnegative, we show the emergence of a dichotomy: either there exists a threshold value for the mass as in the case above, or the minimiser is an ellipsoid for any positive value of the mass.

Shape optimisation for nonlocal anisotropic energies

Abstract

In this paper we consider shape optimisation problems for sets of prescribed mass, where the driving energy functional is nonlocal and anisotropic. More precisely, we deal with the case of attractive/repulsive interactions in two and three dimensions, where the attraction is quadratic and the repulsion is given by an anisotropic variant of the Coulomb potential. Under the sole assumption of strict positivity of the Fourier transform of the interaction potential, we show the existence of a threshold value for the mass above which the minimiser is an ellipsoid, and below which the minimiser does not exist. If, instead, the Fourier transform of the interaction potential is only nonnegative, we show the emergence of a dichotomy: either there exists a threshold value for the mass as in the case above, or the minimiser is an ellipsoid for any positive value of the mass.
Paper Structure (15 sections, 6 theorems, 118 equations)

This paper contains 15 sections, 6 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Motivation and comparison with the radially symmetric case
  3. Method of proof
  4. Outline of the paper
  5. Preliminary results
  6. Minimisation of $\mathcal{I}$ on $\mathcal{A}_{m,1}$ and on $\mathcal{A}_{m}$.
  7. The Fourier transform
  8. Ellipsoids
  9. The potential
  10. Proofs of our main results
  11. The non-degenerate case: Proof of Theorem \ref{['Mainthm']}
  12. Proof of Theorem \ref{['Mainthm']}: The critical and subcritical cases $m\leq m^\ast$.
  13. Proof of Theorem \ref{['Mainthm']}: The super-critical case $m> m^\ast$.
  14. The degenerate case: Proof of Theorem \ref{['Mainthm2']}
  15. Further results

Key Result

Theorem 1.1

Let $W$ be a potential defined as in potdef-dd and satisfying Assumption assumption-k. Assume that $\widehat{W}> 0$ on $\mathbb{S}^{d-1}$. Then there exists a critical value $m^\ast>0$ such that for $m<m^\ast$ the energy $\mathcal{I}$ in en:general-dd has no miminiser in $\mathcal{A}_m$, while for $

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Remark 4.1: Monotonicity of the minimiser with respect to $m$
  • Proposition 4.2: Minimality of the ball for finite mass $m$
  • proof
  • Remark 4.3: Bound on the critical mass
  • Lemma 4.4
  • ...and 1 more