A family of interaction energy minimizers supported on two intervals

In this paper, we consider the one-dimensional interaction energy $\frac{1}{2}\int_{\mathbb{R}}(W*\rho)(x)d\rho(x) + \int_{\mathbb{R}}U(x)d\rho(x)$ 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 $\rho$ 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.