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Hardy’s Inequality on Domains

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The Analysis and Geometry of Hardy's Inequality

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Abstract

Let \(\Omega \) be a domain (an open, connected set) in \(\mathbb{R}^{n}\) with non-empty boundary, \(1 < p < \infty \), and denote by δ(x) the distance from a point \(\mathbf{x} \in \Omega \) to the boundary \(\partial \Omega \) of \(\Omega,\) i.e.,

$$\displaystyle{\delta (\mathbf{x}):=\inf \{ \vert \mathbf{x} -\mathbf{y}\vert: \mathbf{y} \in \mathbb{R}^{n}\setminus \Omega \}.}$$

The basic inequality to be considered in this chapter is

$$\displaystyle{ \int _{\Omega }\vert \nabla f(\mathbf{x})\vert ^{p}d\mathbf{x} \geq c(n,p,\Omega )\int _{ \Omega }\frac{\vert f(\mathbf{x})\vert ^{p}} {\delta (\mathbf{x})^{p}} d\mathbf{x},\ \ \ f \in C_{0}^{\infty }(\Omega ); }$$
(3.1.1)

equivalently, the inequality is to hold for all \(f \in W_{0}^{1,p}(\Omega ).\)

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Authors and Affiliations

  1. School of Mathematics, Cardiff University, Cardiff, UK

    Alexander A. Balinsky & W. Desmond Evans

  2. Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama, USA

    Roger T. Lewis

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  2. W. Desmond Evans
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Balinsky, A.A., Evans, W.D., Lewis, R.T. (2015). Hardy’s Inequality on Domains. In: The Analysis and Geometry of Hardy's Inequality. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-22870-9_3

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