5 problems

Higher-order Rellich inequality conjecture. One has

Let $B_R$ be the ball of radius $R$ in $\mathbb{R}^N$, let $N>k\geq 2$, and let $u\in C_c^k(B_R)$. Critical logarithmic Hardy conjecture. The inequality … holds. Moreover, the cons…

Let $B_R$ be the ball of radius $R$ in $\mathbb{R}^N$, let $1<p<\infty$, let $k\geq 2$, and let $u\in C_c^k(B_R)$. Write $\operatorname{dist}(x,\partial B_R)$ for the distance from…

Let $B_R$ be the ball of radius $R$ centred at the origin in $\mathbb{R}^N$, let $k\geq 2$, let $1<p<N/k$, let $0<\gamma\leq (N-kp)/(kp-1)$, and let $u\in C_c^k(B_R)$. Weighted hig…

Let $N>k\geq 2$, $m\in\mathbb{N}$, and $p=\frac{N}{k}$. Let $R_{k,\gamma}$ denote the optimal constant in the critical Rellich inequality and let $R^{\rm rad}_{k,\gamma}$ denote it…