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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

REFERENCES

[1]Besov, O. V., Il'in, V. P. and Nikol'skii, S. M.. Integral Representations of functions and imbedding theorems. Vol. II. (Winston, Washington, 1979).Google Scholar
[2]Edmunds, D. E. and Evans, W. D.. Spectral theory and differential operators (Clarendon Press, 1987).Google Scholar
[3]Edmunds, D. E. and Opic, B.. Weighted Poincaré and Friedrichs inequalities. J. London Math. Soc. (2) 47 (1993), 79–93.Google Scholar
[4]Edmunds, D. E., Opic, B. and Pick, L.. Poincaré and Friedrichs inequalities in abstract Sobolev spaces. Math. Proc. Cambridge Phil. Soc. 113 (1993), 355–379.Google Scholar
[5]Goldberg, S.. Unbounded linear operators (McGraw-Hill, 1966).Google Scholar
[6]Kufner, A., John, O. and Fučík, S.. Function spaces (Academia, Prague and Noordhoff International Publishing, 1977).Google Scholar
[7]Nečas, J.. Les méthodes directes en théorie des équations elliptiques (Academia, Prague and Masson, 1967).Google Scholar
[8]Opic, B. and Rákosník, J.. Estimates for mixed derivatives of functions from anisotropic Sobolev-Slobodeckij spaces with weights. Quart. J. Math. Oxford (2) 42 (1991), 347–363.Google Scholar
[9]Ziemer, W.. Weakly differentiable functions (Springer-Verlag, 1989).CrossRefGoogle Scholar