Optimizing improved Hardy inequalities
Let \(N\geq 3\) and \(\Omega\) be a bounded domain in \({\mathbb R}^N\) such that \(0\in \Omega\). The goal of the authors is to study a general improved Hardy inequality: For all \(u\in H^1_0(\Omega)\), \[ \int_{\Omega} \left|\nabla u\right|^2\geq \left(\frac{N-2}2\right)^2\int_{\Omega} \frac{\left|u\right|^2}{\left|x\right|^2}dx+b\int_{\Omega} Vu^2dx \] for a suitable chosen potential \(V\). First, they find necessary conditions on the potential \(V\), under which the previous inequality can or cannot be further improved. They show that the best constant \(b\) is never achieved in \(H^1_0(\Omega)\). Moreover, a ``series expansion'' for the Hardy inequality is obtained. In establishing these results, they derive various sharp improved Hardy-Sobolev inequalities.
scientific article; zbMATH DE number 1797597