Poincaré and Hardy inequalities

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


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),
$$ \int\limits_\Omega {\nabla u.\nabla \emptyset dx = - \int\limits_\Omega \upsilon } \emptyset dx. $$


Orlicz Space Hardy Inequality Continuous Norm Banach Function Space Young Function 
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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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