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Poincaré and Friedrichs inequalities in abstract Sobolev spaces

Published online by Cambridge University Press:  24 October 2008

D. E. Edmunds ,
B. Opic  and
L. Pick
D. E. Edmunds
Affiliation:
Mathematics Division, University of Sussex
B. Opic
Affiliation:
Mathematical Institute, Czechoslovak Academy of Sciences, Prague, Czechoslovakia
L. Pick
Affiliation:
School of Mathematics, University of Wales, College of Cardiff
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Extract

Given any domain Ω in ℝN we consider spaces X = X(Ω),Y = Y(Ω) which are Banach function spaces in the sense of Luxemburg [12]. From these we form what we call the abstract Sobolev space W(X, Y), which is denned to be the linear space of all ƒ ∈ X such that for i = 1, …, N, the distributional derivative ∂ƒ/∂xi belongs to Y; equipped with the norm

W(X, Y) is a Banach space. The closure of (Ω) in W(X, Y) is denoted by W0(X, Y). Let w to bea weight function on Ω, that is, a measurable function which is positive and finite almost everywhere on Ω. We say that [w, X, Y] supports the (weighted) Poincaré inequality if there is a positive constant K such that for all u ∈ W(X, Y),

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

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 2 , March 1993 , pp. 355 - 379
Copyright
Copyright © Cambridge Philosophical Society 1993

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