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Hardy’s Inequality on Domains

  • Alexander A. Balinsky
  • W. Desmond Evans
  • Roger T. Lewis
Part of the Universitext book series (UTX)

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 \(f \in W_{0}^{1,p}(\Omega ).\)

Keywords

Distance Function Convex Domain Lipschitz Domain Hardy Inequality Strong Barrier 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Alexander A. Balinsky
    • 1
  • W. Desmond Evans
    • 1
  • Roger T. Lewis
    • 2
  1. 1.School of MathematicsCardiff UniversityCardiffUK
  2. 2.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA

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