Rellich inequalities with weights

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

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