5 problems

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

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

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…