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On a class of weighted Sobolev spaces and the minimization of quadratic forms

Published online by Cambridge University Press:  14 November 2011

Frans Penning  and
Niko Sauer
Frans Penning
Affiliation:
Department of Mathematics
Niko Sauer
Affiliation:
Department of Applied Mathematics, University of Pretoria, South Africa
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Synopsis

In this paper a class of weighted Sobolev spaces defined in terms of square integrability of the gradient multiplied by a weight function, is studied. The domain of integration is either the space Rn or a half-space of Rn. Conditions on the weight functions that will ensure density of classes of smooth functions or functions with compact support, and compact embedding theorems, are derived. Finally the results are applied to a class of isoperimetrical problems in the calculus of variations in which the domain of integration is unbounded.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 81 , Issue 1-2 , 1978 , pp. 119 - 130
Copyright
Copyright © Royal Society of Edinburgh 1978

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References

1Adams, R. A.Sobolev Spaces (London: Academic Press, 1975).Google Scholar
2Adams, R. A.Compact imbeddings of weighted Sobolev spaces on unbounded domains. J. Differential Equations 9 (1971), 325–334.CrossRefGoogle Scholar
3Adams, R. A. and Fournier, J.Some imbedding theorems for Sobolev Spaces. Canad. J. Math. 23 (1971), 517–530.CrossRefGoogle Scholar
4Amos, R. J. and Everitt, W. N.On integral inequalities and compact embeddings associated with ordinary differential expressions, submitted for publication.CrossRefGoogle Scholar
5Amos, R. J. and Everitt, W. N.On integral inequalities associated with ordinary regular differential expressions, submitted for publication.CrossRefGoogle Scholar
6Bradley, J. S. and Everitt, W. N.Inequalities associated with regular and singular problems in the calculus of variations. Trans. Amer. Math. Soc. 182 (1973), 303–321.CrossRefGoogle Scholar
7Clark, C. W.The Hilbert-Schmidt property for embedding maps between Sobolev spaces. Canad. J. Math. 18 (1966), 1079–1084.CrossRefGoogle Scholar
8Courant, R. and Hilbert, D.Methods of Mathematical Physics I (New York: Interscience, 1953).Google Scholar
9Dunford, N. and Schwartz, J. T.Linear Operators I (New York: Interscience, 1957).Google Scholar
10Friedman, A.Partial Differential Equations (New York: Holt, 1969).Google Scholar
11Goldberg, S.Unbounded linear operators (New York: McGraw-Hill, 1966).Google Scholar
12Hildebrandt, S.Rand-und-Eigenwertaufgaben bei stark elliptischen Systemen linearer Differentialgleichungen. Math. Ann. 148 (1962), 411–429.CrossRefGoogle Scholar
13Meyer, R. D.Some embedding theorems for generalized Sobolev spaces and applications to degenerate elliptic differential operators. J. Math. Mech. 16 (1967), 739–760.Google Scholar
14Mizohata, S.The theory of partial differential equations (Cambridge: University Press, 1973).Google Scholar
15Nikol'skii, S. M.On embedding, continuation and approximation theorems for differentiable functions of several variables. Russian Math. Surveys 16 (1961), 55–104.CrossRefGoogle Scholar
16Penning, F. D. and Sauer, N. Note on the minimization of Pretoria Univ. Dept. Appl. Math. Res. Report UPTW2 (1976).Google Scholar
17Poulsen, E. T.Boundary value properties connected with some improper Dirichlet integrals. Math. Scand. 8 (1960), 5–14.CrossRefGoogle Scholar
18Putnam, C. R.An application of spectral theory to a singular calculus of variations problem. Amer. J. Math. 70 (1948), 780–803.CrossRefGoogle Scholar
19Uspenskii, S. V.Imbedding theorems for weighted classes. Amer. Math. Soc. Transl. 87 (1970), 121–145.Google Scholar