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Some Imbedding Theorems for Sobolev Spaces

Published online by Cambridge University Press:  20 November 2018

R. A. Adams  and
John Fournier
R. A. Adams
Affiliation:
University of British Columbia, Vancouver, British Columbia
John Fournier
Affiliation:
University of British Columbia, Vancouver, British Columbia
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We shall be concerned throughout this paper with the Sobolev space Wm,p(G) and the existence and compactness (or lack of it) of its imbeddings (i.e. continuous inclusions) into various LP spaces over G, where G is an open, not necessarily bounded subset of n-dimensional Euclidean space En. For each positive integer m and each real p ≧ 1 the space Wm,p(G) consists of all u in LP(G) whose distributional partial derivatives of all orders up to and including m are also in LP(G). With respect to the norm

1.1

Wm,p(G) is a Banach space. It has been shown by Meyers and Serrin [9] that the set of functions in Cm(G) which, together with their partial derivatives of orders up to and including m, are in LP(G) forms a dense subspace of Wm,p(G).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 517 - 530
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Adams, R. A., The Rellich-Kondrachov theorem for unbounded domains, Arch. Rational Mech. Anal. 29 (1968), 390–394.Google Scholar
2. Adams, R. A., Compact imbedding theorems for quasibounded domains, Trans. Amer. Math. Soc. 148 (1970), 445–459.Google Scholar
3. Adams, R. A., Capacity and compact imbeddings, J. Math. Mech. 19 (1970), 923–929.Google Scholar
4. Adams, R. A. and Fournier, J., A compact imbedding theorem for functions without compact support (to appear in Can. Math. Bull.).Google Scholar
5. Adams, R. D., Aronszajn, N., and Smith, K. T., Theory of Bessel potentials, Part II, Ann. Inst. Fourier (Grenoble), 17 (1967), 1–135.Google Scholar
6. Clark, C. W., An embedding theorem for function spaces, Pacific J. Math. 19 (1966), 243–251.Google Scholar
7. Gagliardo, E., Propriété di alcune classi difunzioni in piu variabili, Ricerche Mat. 7 (1958), 102–137.Google Scholar
8. Kondrachov, V. I., Certain properties of functions in the spaces D>, Dokl. Akad. NaukSSSR, 48 (1955), 563–566.,+Dokl.+Akad.+NaukSSSR,+48+(1955),+563–566.>Google Scholar
9. Meyers, N. G. and Serrin, J., H = W, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 1055–1056.Google Scholar
10. Rellich, F., Ein Satz über mittlere Konvergenz, Gö;ttingen Nachr. (1930), 30–35.Google Scholar