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Hardy, Sobolev, Maz’ya (HSM) Inequalities

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

Abstract

From the Hardy and Sobolev inequalities
$$\displaystyle{\|\nabla u\|_{p,\Omega }^{p} \geq C_{ H}\|u/\delta \|_{p,\Omega }^{p},\ \ \ \|\nabla u\|_{ p,\Omega }^{p} \geq C_{ S}\|u\|_{p^{{\ast}},\Omega }^{p},\ \ \ u \in D_{ 0}^{1,p}(\Omega ),}$$
where \(\delta (\mathbf{x}) =\mathrm{ dist}(\mathbf{x},\partial \Omega ),C_{H},C_{S}\) are the optimal constants and p = np∕(np), it follows that for 0 < α ≤ C H , 
$$\displaystyle\begin{array}{rcl} \|\nabla u\|_{p,\Omega }^{p} -\alpha \| u/\delta \|_{ p,\Omega }^{p}& \geq & \left (1 -\alpha /C_{ H}\right )\|\nabla u\|_{p,\Omega }^{p} \\ & \geq & \left (1 -\alpha /C_{H}\right )C_{S}\|u\|_{p^{{\ast}},\Omega }^{p}.{}\end{array}$$
(4.1.1)

Keywords

Quadratic Form Distance Function Sobolev Inequality Convex Domain Exterior Domain 
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.

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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