Equal-size equal-distance minimizer conjecture for the generalized Ohta–Kawasaki model

Let $\omega\ll 1$ and $\gamma\gg 1$, and let $\mathcal{A}_N^{\omega}$ be the class of periodic step functions considered in the generalized Ohta–Kawasaki model. Let $U_N^{\omega}$ denote the $N$-bubble function whose bubbles have equal size and equal distance. Equal-size equal-distance minimizer conjecture. The function $u^*(x)$ is the unique minimizer of $E_0^{\emph{NOK}}$ in $\mathcal{A}_N^{\omega}$ if and only if $u^*(x)$ has equal-size and equal-distance bubbles. Equivalently,

$$U_N^{\omega}=\underset{u\in\mathcal{A}_N^{\omega}}{\operatorname{argmin}} E_0^{\emph{NOK}}[u].$$

The conjecture is motivated by numerical gradient-flow experiments for several nonlocal operators, while an analytical characterization of minimizers is unavailable in the generalized setting; its resolution therefore remains open.