Rellich inequalities with weights

Abstract

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 $u\in C_c^2(\bar{\Omega }\setminus \{0\})$. We also estimate the best constant for the same inequality on $C_c^2(\Omega )$. Moreover we show improved Rellich inequalities with remainder terms involving logarithmic weights on cone-like domains.

Cited by in corpus (1)

• Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator: existence of extremal functions, breaking positivity and breaking symmetry