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Poincaré and Friedrichs inequalities in abstract Sobolev spaces II

Published online by Cambridge University Press:  24 October 2008

D. E. Edmunds ,
B. Opic  and
J. Rákosník
D. E. Edmunds
Affiliation:
Mathematics Division, University of Sussex
B. Opic
Affiliation:
Mathematical Institute, Czechoslovak Academy of Sciences, Prague
J. Rákosník
Affiliation:
Mathematical Institute, Czechoslovak Academy of Sciences, Prague
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Extract

This paper is a continuation of [4]; its aim is to extend the results of that paper to include abstract Sobolev spaces of higher order and even anisotropic spaces. Let Ω be a domain in ℝN, let X = X(Ω) and Y = Y(Ω) be Banach function spaces in the sense of Luxemburg (see [4] for details), and let W(X, Y) be the abstract Sobolev space consisting of all f ∈ X such that for each i ∈ {l, …, N} the distributional derivative belongs to Y; equipped with the norm

W(X, Y) is a Banach space. Given any weight function w on Ω, the triple [w, X, Y] is said to support the Poincaré inequality on Ω. if there is a positive constant K such that for all u ∈ W(X, Y)

the pair [X, Y] is said to support the Friedrichs inequality if there is a positive constant K such that for all u ∈ W0(X, Y) (the closure of in W(X, Y))

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 1 , January 1994 , pp. 159 - 173
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

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