J. Aust. Math. Soc.  75 (2003), 423-440
On convexity and weak closeness for the set of $\Phi$-superharmonic functions

Hongwei Lou
  Department of Mathematics
  Fudan University
  Shanghai 200433
  China
  hwlou@fudan.edu.cn


Abstract
Convexity and weak closeness of the set of $\Phi$-superharmonic functions in a bounded Lipschitz domain in $\mathbb{R}^n$ is considered. By using the fact of that $\Phi$-superharmonic functions are just the solutions to an obstacle problem and establishing some special properties of the obstacle problem, it is shown that if $\Phi$ satisfies $\Delta _2$-condition, then the set is not convex unless $\Phi (r)=Cr^2$ or $n=1$. Nevertheless, it is found that the set is still weakly closed in the corresponding Orlicz-Sobolev space.
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