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Hardy’s inequalities revisited. (English) Zbl 1011.46027

The paper contains various refinements and improvements of Hardy’s inequality. The authors study the quantity \[ J_{\lambda}= J_{\lambda}^{\Omega} = \inf \limits _{u\in H_0^1(\Omega)} \frac{\int _{\Omega} |\nabla u|^2 -\lambda \int _{\Omega}u^2}{\int _{\Omega}(u/\delta)^2}\quad , \quad \lambda \in \mathbb R , \] where \(\Omega\) is a smooth bounded domain. Note that \(J_0^{\Omega}=\mu (\Omega)\), \(\lambda \mapsto J_{\lambda}\) is concave and non-increasing, \(J_{\lambda}\to -\infty\) when \(\lambda\to \infty\) and \(J_{\lambda _1}=0\) where \(\lambda _1\) is the first eigenvalue of \(-\Delta\) on \(H_0^1(\Omega)\).
The main result of the paper is as follows. For every bounded domain \(\Omega\) of class \(C^2\) there exists a constant \(\lambda ^*=\lambda ^*(\Omega)\) such that \(J_{\lambda}=1/4\) for any \(\lambda \leq \lambda ^*\) and \(J_{\lambda}<1/4\) for any \(\lambda>\lambda^*\). The infimum of \(J_{\lambda}\) is achieved if and only if \(\lambda>\lambda^*\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators
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