4 problems

Extremal-profile conjecture. For general $p\neq 2$, up to translation and scaling, an extremal function is conjectured to have the form

Let $N$, $calpha$, $p$, and $q$ be admissible parameters, and let $\mathscr{C}_{N,\alpha,p,q}$ denote the optimal constant in the variational quotient considered in the paper. For…

Brezis's conjecture. There exist constants $C>0$ and $\delta_0>0$ such that the displayed inequality holds for every continuous $f\in C^0({\mathbb S}^1;{\mathbb S}^1)$ and every…

Let $c$ be a sufficiently regular immersed closed curve, let $\tilde H^\alpha$ denote the first-variation expression for O'Hara's energy after the relevant decomposition, and let…