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Poincaré and Hardy inequalities

  • David E. Edmunds
  • W. Desmond Evans
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

Spectral matters form a powerful motivation for the study of these inequalities. Thus let Ω be an open subset of Rn with volume |Ω| and let Δ D,Ω ,Δ N,Ω be respectively the Dirichlet Laplacian and the Neumann Laplacian on Ω. We recall that —Δ D,Ω is the Friedrichs extension of -Δ on C 0(Ω) and that its domain D(—Δ D,Ω )is contained in the Sobolev space W 1 2(Ω): v is the Dirichlet Laplacian of u ∈ D(—Δ D,Ω ), v = Δu, if it belongs to L 1, 1oc (Ω) and for all φC 0(Ω),
$$ \int\limits_\Omega {\nabla u.\nabla \emptyset dx = - \int\limits_\Omega \upsilon } \emptyset dx. $$

Keywords

Orlicz Space Hardy Inequality Continuous Norm Banach Function Space Young Function 
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-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • David E. Edmunds
    • 1
  • W. Desmond Evans
    • 2
  1. 1.Department of MathematicsSussex UniversityBrightonUK
  2. 2.School of MathematicsCardiff UniversityCardiffUK

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