Choksi–Peletier liquid drop minimizer conjecture

Let $\Omega\subset\mathbb{R}^3$ be measurable, and define the liquid drop energy

$$\mathcal E(\Omega)=P(\Omega)+D(\Omega),$$

where

$$D(\Omega)=\frac12\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|}$$

and $P(\Omega)$ is the perimeter. For a prescribed volume $V$, write

$$E(V)=\inf_{|\Omega|=V}\mathcal E(\Omega).$$

Let $B_1\subset\mathbb{R}^3$ be the unit ball and set

$$V_* = \frac{2-2^{\frac23}}{2^{\frac23}-1}\frac{|B_1|P(\partial B_1)}{\frac12\iint_{B_1\times B_1}|x-y|^{-1}\,dx\,dy}=5\frac{2-2^{\frac23}}{2^{\frac23}-1}\approx 3.51.$$

Choksi–Peletier conjecture. For $V\leq V_*$, the round ball of volume $V$ uniquely minimizes $\mathcal E(\cdot)$ among measurable sets $\Omega\subset\mathbb{R}^3$ with $|\Omega|=V$; for $V>V_*$, no minimizer exists.

The conjecture describes the expected transition from a single spherical droplet to nonexistence caused by splitting into droplets at infinite separation. The paper's abstract states that the first assertion is proved for volumes at most $1$, while the full conjecture and the nonexistence assertion are not resolved here.