Hardy’s Inequality on Domains

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


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
$$\displaystyle{ \int _{\Omega }\vert \nabla f(\mathbf{x})\vert ^{p}d\mathbf{x} \geq c(n,p,\Omega )\int _{ \Omega }\frac{\vert f(\mathbf{x})\vert ^{p}} {\delta (\mathbf{x})^{p}} d\mathbf{x},\ \ \ f \in C_{0}^{\infty }(\Omega ); }$$
equivalently, the inequality is to hold for all \(f \in W_{0}^{1,p}(\Omega ).\)


Distance Function Convex Domain Lipschitz Domain Hardy Inequality Strong Barrier 
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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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