Hardy, Sobolev, Maz’ya (HSM) Inequalities

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


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 , 
$$\displaystyle\begin{array}{rcl} \|\nabla u\|_{p,\Omega }^{p} -\alpha \| u/\delta \|_{ p,\Omega }^{p}& \geq & \left (1 -\alpha /C_{ H}\right )\|\nabla u\|_{p,\Omega }^{p} \\ & \geq & \left (1 -\alpha /C_{H}\right )C_{S}\|u\|_{p^{{\ast}},\Omega }^{p}.{}\end{array}$$


Quadratic Form Distance Function Sobolev Inequality Convex Domain Exterior Domain 
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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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