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A remark on Hardy type inequalities with remainder terms

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  • In this paper we focus our attention to some Hardy type inequalities with a remainder term. In particular we find the best value of the constant $h$ for the inequalities

    $\int_{\Omega}|\nabla u|^2 dx \geq c \int_{\Omega}\frac{u^2}{|x|^2} dx+ h\int_{\Omega}\frac{u^2}{|x|}dx, \forall u\in H_0^1( \Omega) $

    $ \int_{\Omega}|\nabla u|^2dx\geq c\int_{\Omega} \frac{u^2}{|x|^2}dx+ h(\int_{\Omega}|\nabla u| dx)^2, \forall u\in H_0^1 (\Omega)$

    where $c\geq 0$ is smaller than the optimal Hardy constant $(N-2)^2/4$.

    Mathematics Subject Classification: Primary: 35J20, 26D10; Secondary: 46E35.


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