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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

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