A family of interaction energy minimizers supported on two intervals

TL;DR

The paper analyzes the one-dimensional interaction energy with kernel $W(x)=-|x|^b/b$ and quartic confinement $U(x)=|x|^4/4$, proving that for $1<b<2$ the unique minimizer is supported on two symmetric intervals and exhibits a positive density on its support. It introduces a novel variant of the iterated balayage algorithm (IBA) to construct and understand the minimizer via a carefully engineered one-parameter family $\mu_\lambda$ supported on $K_{\lambda,1}$ and a rescaling to $K_{R_1,R_2}$, culminating in a zero condition $F(\lambda_*)=0$. Key steps include an explicit seed measure $\mu_0$, balayage representations, and continuity/limiting analysis that drive the existence of a critical $\lambda_*$ yielding the two-interval minimizer, with precise regularity results for the density. The methodology provides a robust framework for extending to higher dimensions where annulus-like supports are possible, connecting equilibrium theory, balayage, and gradient-flow perspectives in a cohesive variational approach.

Abstract

In this paper, we consider the one-dimensional interaction energy $\frac{1}{2}\int_{\mathbb{R}}(W*ρ)(x)dρ(x) + \int_{\mathbb{R}}U(x)dρ(x)$ where the interaction potential $W(x)= -\frac{|x|^b}{b},\,1\le b \le 2$ and the external potential $U(x)=\frac{|x|^4}{4}$, and $ρ$ is a compactly supported probability measure on the real line. Our main result shows that the minimizer is supported on two intervals when $1<b<2$, showing in particular how the support of the minimizer transits from an interval (when $b=1$) to two points (when $b=2$) as $b$ increases. As a crucial part of the proof, we develop a new version of the iterated balayage algorithm, the original version of which was designed by Benko, Damelin, Dragnev and Kuijlaars for logarithmic potentials in one dimension. We expect the methodology in this paper can be generalized to study minimizers of interaction energies in $\mathbb{R}^d$ whose support is possibly an annulus.

Figures (2)

• Figure 1: The minimizer $\rho$ of $\mathcal{E}$ with $b=1.3$, and its generated potential $\mathcal{W}*\rho+\mathcal{U}$. We have $R_1\approx 0.6532$ and $R_2\approx 0.9411$. We only sketch the graphs on $x>0$ due to symmetry. We sketch the adjusted version $50(\mathcal{W}*\rho+\mathcal{U}+0.6)$ of the generated potential for clearer illustration.
• Figure 2: The signed measure $\mu_\lambda$ and the corresponding $V[\mu_\lambda]$. Upper row: $s=0.7$ (corresponding to the case in Figure \ref{['fig1']}), with $\lambda=0.2$, $\lambda=\lambda_\infty\approx 0.4440$ and $\lambda=\lambda_*\approx 0.6941$ (see Sections \ref{['sec_iba']} and \ref{['sec_main']} for the definition of $\lambda_\infty$ and $\lambda_*$). Lower row: $s=0.3$, with $\lambda=0.2$, $\lambda=\lambda_\infty\approx 0.7880$ and $\lambda=\lambda_*\approx 0.9876$. For this row, the graph of $\frac{1}{10}\mu_\lambda$ is sketched for better illustration. For $\lambda=\lambda_\infty$, the number $y$ appeared in the proof of Item 4 of Proposition \ref{['prop_iba']} is marked, and the graphs near $x=\lambda$ is zoomed in to show the regularity near this point. For $\lambda=\lambda_*$, the numbers $y$ and $z$ appeared in Section \ref{['sec_main']} are marked.

Theorems & Definitions (36)

• Proposition 1.1
• proof
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• proof
• Proposition 3.1
• Proposition 3.2