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 φ ∈ C ∞ 0(Ω),
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© 2004 Springer-Verlag Berlin Heidelberg
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Edmunds, D.E., Evans, W.D. (2004). Poincaré and Hardy inequalities. In: Hardy Operators, Function Spaces and Embeddings. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-07731-3_4
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DOI: https://doi.org/10.1007/978-3-662-07731-3_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-06027-4
Online ISBN: 978-3-662-07731-3
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