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Flow approach on Riesz type nonlocal energies

Jiaxin He, Qinfeng Li, Juncheng Wei, Hang Yang

TL;DR

This work develops a flow-based framework to study the Riesz-type nonlocal energy $D( abla\Omega)=\iint_{\Omega}\iint_{\Omega} K(|x-y|) \,dx\,dy$ under area-preserving deformations. By deriving an evolution equation for $D$ along smooth flows and establishing key boundary comparisons for the boundary potential $V_{\Omega}$, the authors prove new monotonicity results for triangles and quadrilaterals, including height/leg-based deformations and flows that avoid Steiner symmetrization. Notably, they show monotonic increases toward isosceles/equilateral configurations for triangles and toward squares for quadrilaterals, with alternative proofs of the extremal properties of equilateral and square shapes. The results extend the toolbox for shape optimization of nonlocal energies, offering a cohesive, non-symmetrization-based route to classical extremizers and new monotonicity phenomena across polygon classes.

Abstract

Via continuous deformations based on natural flow evolutions, we prove several novel monotonicity results for Riesz-type nonlocal energies on triangles and quadrilaterals. Some of these results imply new and simpler proofs for known theorems without relying on any symmetrization arguments.

Flow approach on Riesz type nonlocal energies

TL;DR

This work develops a flow-based framework to study the Riesz-type nonlocal energy under area-preserving deformations. By deriving an evolution equation for along smooth flows and establishing key boundary comparisons for the boundary potential , the authors prove new monotonicity results for triangles and quadrilaterals, including height/leg-based deformations and flows that avoid Steiner symmetrization. Notably, they show monotonic increases toward isosceles/equilateral configurations for triangles and toward squares for quadrilaterals, with alternative proofs of the extremal properties of equilateral and square shapes. The results extend the toolbox for shape optimization of nonlocal energies, offering a cohesive, non-symmetrization-based route to classical extremizers and new monotonicity phenomena across polygon classes.

Abstract

Via continuous deformations based on natural flow evolutions, we prove several novel monotonicity results for Riesz-type nonlocal energies on triangles and quadrilaterals. Some of these results imply new and simpler proofs for known theorems without relying on any symmetrization arguments.

Paper Structure

This paper contains 6 sections, 18 theorems, 83 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Boundary comparison of $V_\Omega$ and evolution equation of $D(\cdot)$ along flows
  3. Monotonicity of $D(\cdot)$ on triangles via height stretching and compressing flows
  4. Monotonicity of $D(\cdot)$ on triangles via leg stretching and parallel movement flow
  5. Monotonicity results of $D(\cdot)$ on rhombuses and rectangles via vertical stretching flows
  6. Continuous monotonic deformation of quadrilaterals

Key Result

Theorem 1.2

Let $\Omega=\triangle_{ABC}$ be a triangle, with $AB$ being the longest side and lying on the $x$-axis, and with $C$ lying on the positive $y$-axis. Suppose that $|AB|>|BC|\ge |AC|$, and we let $C_t=(1+t)C$, $\Omega_t=\sqrt{\tfrac{|OC|}{|OC_t|}}\triangle_{ABC_t}=\triangle_{A_tB_t\tilde{C}_t}$, where

Figures (16)

  • Figure 1.1: An illustration of Theorem \ref{['yangsheng2']} for the case when $t\in (0,t_1]$. Stretching the shortest height $OC$ above while scaling to keep the area fixed. Then, $|\triangle_{A_tB_t\tilde{C}_t}|=|\triangle_{ABC}|$ and $D(\triangle_{A_tB_t\tilde{C}_t})$ is strictly increasing until $t=t_1$, at which time the triangle becomes isosceles.
  • Figure 1.2: An illustration of Theorem \ref{["yangsheng2'"]} for the case when $0<t\le t_2$. Compressing below the tallest height $OC$ while scaling to keep the area fixed. Then, $|\triangle_{A_tB_t\tilde{C}_t}|=|\triangle_{ABC}|$, and $D(\triangle_{A_tB_t\tilde{C}_t})$ is strictly increasing until $t=t_2$, at which time the triangle becomes isosceles.
  • Figure 2.1: If $|BC|>|AC|$, then $V_\Omega(P)<V_\Omega(P')$.
  • Figure 2.2: If $|AC|<|AB|$, then $V_\Omega(P)<V_\Omega(P')$.
  • Figure 2.3: Picture illustration of the proof of Proposition \ref{['not-equidistribution-thmintro']}
  • ...and 11 more figures

Theorems & Definitions (33)

  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Proposition 2.1
  • proof
  • ...and 23 more