## 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

### Keywords:

Hardy’s inequality; Sobolev space
Full Text:

### References:

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