12 problems

Consider the stochastic differential equation and its equidistant Euler approximation $X^{\mathrm{E}}_{n,1}$ from the paper, with bounded drift coefficient $\mu\in W^{s,p}$, where…

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain, and let the fractional Korn inequalities under consideration be the fractional analogues of Korn's first and second i…

The paper considers the Besov-type borderline space $b^{2/3}_{3,\infty}$, which is larger than the fractional Sobolev space $W^{2/3,3}$. Borderline-space conjecture. The results of…

Let $d4a4^s(d6a9)$ denote the fractional Sobolev space on the domain $d6a9$, and let $d700_s(q)$ and $d4c0_s(q)$ be the two extremal functions defined in the source for $q>2$. Naza…

Let $\Omega$ be a finite domain, let $W^{\alpha,p}(\Omega)$ and $\widetilde{W}^{\alpha,p}(\Omega)$ be the fractional Sobolev spaces used in the paper, and suppose $\alpha p<1$. The…

Let ${^{\pm}}{W}^{\alpha,p}(\mathbb{R})$, $\widehat{W}^{\alpha,p}(\mathbb{R})$, and $\widetilde{W}^{\alpha,p}(\mathbb{R})$ denote the fractional Sobolev spaces defined using, respe…

Let $\Omega$ be a finite domain, let $0<\alpha<1$, and let $p>0$ satisfy $\alpha p<1$. Denote by $W^{\alpha,p}(\Omega)$ and ${\widetilde{W}}^{\alpha,p}(\Omega)$ the two fractional…

Let $\mathbb{R}$ denote the real line, let $0<\alpha<1$, and let $p\neq2$. Denote by ${^{\pm}}{W}{^{\alpha,p}}(\mathbb{R})$, $\widehat{W}^{\alpha,p}(\mathbb{R})$, and…

Let $\Omega\subset\mathds{R}^N$, let $w:\Omega\to\mathds{R}^M$, let $p$ and $s$ be as in the preceding construction, and let $\rho_\varepsilon$ be the corresponding regularizing ke…

Extremal-profile conjecture. The extremals for the embedding ${\mathcal {D}}^{s,p}({\mathbb {R}}^n)\subset L^{p^*_s}({\mathbb {R}}^n)$ are of the form $V_{\varepsilon,x_0}$.

Let $s>0$ be nonintegral, let $u\in H^s(\mathbb{R}^n)$ change sign, and assume that $u^+$ and $u^-$ belong to $H^s(\mathbb{R}^n)$. Write $\lfloor s\rfloor$ for the integer part of…

Ignat's kinetic formulation conjecture. Although this equation was proved for $p\in[1,2]$, it was conjectured that it continues to hold for every $p>2$. The surrounding proposition…