10 problems

Explicit-form conjecture. An extremal for the general case $p\neq 2$ has a similar explicit form.

Weighted-minimizer profile conjecture. The minimizers are conjectured to have the form

Let $C_{\text{\tiny{BE}}}$ denote the best Bianchi–Egnell constant for the one-dimensional fractional Sobolev inequality, and let $C_{\text{\tiny{BE}}}^{\text{\tiny{loc}}}$ denote…

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

Brasco's classification conjecture. All minimizers for $S_{s,p}$ are of the form

Let $p\neq 2$, and consider extremal functions for the $p$-fractional Sobolev inequality, with $C$ a constant. Up to translation and dilation, the extremal-function conjecture. Suc…

Classification conjecture. Up to a multiplicative constant, every minimizer of $S$ is of the form $U((x-x_0)/\varepsilon)$ for some $x_0\in\mathbb{R}^N$ and $\varepsilon>0$.

Ball minimizer conjecture. The infimum of $S(n,\sigma,\Omega)$ over all bounded open sets $\Omega$ is achieved by a ball. This question concerns the domain dependence of the best c…

Let $s\in(0,1)$ and $p\in(1,n/s)$, and consider minimizers of the fractional Sobolev quotient … The critical exponent is $p^\ast=np/(n-sp)$. Explicit-form conjecture. Every minimiz…

Let $\Omega\subset {\mathbb{R}}^n$, with $n\geq 2$, be an $s$-John domain. Let $\delta\in(0,1)$, $1\leq p<n/\delta$, and $s<\frac{n}{n-p\delta}$. The fractional $(q,p)$-Sobolev-Poi…