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On convexity and weak closeness for the set of Φ-superharmonic functions

Part of: Other generalizations

Published online by Cambridge University Press:  09 April 2009

Hongwei Lou
Hongwei Lou
Affiliation:
Department of Mathematics, Fudan University Shanghai, 200433, China, e-mail:hwlou@fudan.edu.cn
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Abstract

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Convexity and weak closeness of the set of Φ-superharmonic functions in a bounded Lipschitz domain in Rn is considered. By using the fact of that Φ-superharmonic functions are just the solutions to an obstacle problem and establishing some special properties of the obstacle problem, it is shown that if Φ satisfies Δ2-condition, then the set is not convex unless Φ(r) = Cr2 or n = 1. Nevertheless, it is found that the set is still weakly closed in the corresponding Orlicz-Sobolev space.

Keywords

convexityweak closenesssuperharmonic functions

MSC classification

Secondary: 31C45: Other generalizations (nonlinear potential theory, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 75 , Issue 3 , December 2003 , pp. 423 - 440
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Adams, R. A., Sobolev spaces (Academic Press, New York, 1975).Google Scholar
[2]Gelfand, I. M. and Shilov, G. E., Generalized functions, Vol. I (Academic Press, New York, 1964).Google Scholar
[3]Lions, J. L., Optimal control of systems governed by partial differential equations (Springer, New York, 1971).CrossRefGoogle Scholar
[4]Rao, M. M. and Ren, Z. D., Theory of Orlicz spaces (Marcel Dekker, New York, 1991).Google Scholar
[5]Schwartz, L., Théorie des distributions (Hermann, Paris, 1966).Google Scholar
[6]Simon, L., Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis ANU (Australian National University, Canberra, 1983).Google Scholar
[7]Yosida, K., Functional analysis, 6th edition (Springer, Berlin, 1980).Google Scholar
[8]Zygmund, A., Trigonometric series (Cambridge University Press, Cambridge, 1959).Google Scholar