Rellich inequalities with weights

Details Rellich inequalities with weights Rellich inequalities with weights

2011-03-31

arxiv.org/abs/1103.6184v1

Mathematics

Functional Analysis

Scientific paper

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.

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