12 problems

Sharp zero-constant conjecture. Proposition should remain true with $C=0$; that is,

Let $K:\mathbb{R}^+\to\mathbb{R}$ be a $C^1$, strictly decreasing function satisfying … For a bounded planar domain $\Omega$, define its Riesz-type nonlocal energy by … Fix an inte…

Let $\varepsilon(m)$ be the stability separation curve, and let $m_2^c$ and $m_*$ be as above. Decreasing-branch conjecture. The stability separation curve $\varepsilon(m)$ is decr…

Let $\varepsilon(m)$ be the stability separation curve, let $m_2^c$ be the maximum point of the curve $\varepsilon_2$, and let $m_*$ be the supremum of the positive roots of the cu…

Periodic replication conjecture. Without requiring $(\widetilde U,\widetilde V)$ to be a global minimizer or compactly supported in $\Omega$, one has

Let $E\subset\mathbb{R}^n$ be a finite-perimeter set with mass $|E|=m$, surface energy … and nonlocal repulsion energy … The binding energy is…

One-dimensionality and periodicity conjecture. For $\varepsilon$ and $\tau$ sufficiently small, every minimizer is periodic and one-dimensional: there exist…

Let $d\in\mathbb{N}\setminus\{1\}$, and let $\mathcal{E}_h(w)$ denote the infimum of the energy among profiles with winding number $w$. Low-$h$ nonattainment conjecture. There exis…

Let $d\in\mathbb{N}$, and let $\mathcal{E}_h(w)$ denote the infimum of the energy among profiles with winding number $w$. Existence conjecture. There exists $H\in(0,1)$ such that…

Let $h\in[0,1)$, let $d\in\mathbb{N}=\{1,2,\dots\}$, and let $\mathcal{E}_h(w)$ denote the infimum of the energy among profiles with winding number $w$. Nonattainment conjecture. N…

Let $e_0(m)$ be the limiting energy for mass $m>0$, and let $m^\ast$ denote the threshold below which a single-ball minimizer exists. A configuration is a minimizer associated with…

Consider the limiting three-dimensional energy in which particles have masses $m^1,\ldots,m^n$ that play the role of electrical charges. Equal-mass conjecture. In three dimensions,…