Rellich inequalities with weights

Let $\Omega$ be a cone in $\mathbb{R}^{n}$ with $n\ge 2$. For every fixed $\alpha\in\mathbb{R}$ we find the best constant in the Rellich inequality $\int_{\Omega}|x|^{\alpha}|\Delta u|^{2}dx\ge C\int_{\Omega}|x|^{\alpha-4}|u|^{2}dx$ for $u\in C^{2}_{c}(\bar\Omega\setminus\{0\})$. We also estimate the best constant for the same inequality on $C^{2}_{c}(\Omega)$. Moreover we show improved Rellich inequalities with remainder terms involving logarithmic weights on cone-like domains

Rellich inequalities with weights

Let $\Omega$ be a cone in $\mathbb{R}^{n}$ with $n\ge 2$. For every fixed $\alpha\in\mathbb{R}$ we find the best constant in the Rellich inequality $\int_{\Omega}|x|^{\alpha}|\Delta u|^{2}dx\ge C\int_{\Omega}|x|^{\alpha-4}|u|^{2}dx$ for $u\in C^{2}_{c}(\bar\Omega\setminus\{0\})$. We also estimate the best constant for the same inequality on $C^{2}_{c}(\Omega)$. Moreover we show improved Rellich inequalities with remainder terms involving logarithmic weights on cone-like domains

Sharp nonexistence results for a linear elliptic inequality involving Hardy and Leray potentials

In this paper we deal with nonnegative distributional supersolutions for a class of linear elliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results

A note on truncations in fractional Sobolev spaces

We study the Nemytskii operators $u\mapsto |u|$ and $u\mapsto u^{\pm}$ in fractional Sobolev spaces $H^s(\mathbb R^n)$, $s>1$.Comment: 9 page