Rellich inequalities with weights

Let $\mathrm{\Omega }$ 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 $C_c^2(\mathrm{\Omega })$. Moreover we show improved Rellich inequalities with remainder terms involving logarithmic weights on cone-like domains.