J. Aust. Math. Soc.
75 (2003), 423440

On convexity and weak closeness for the set of
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
superharmonic functions in a bounded Lipschitz
domain in is considered. By using the fact of that
superharmonic functions are just the solutions
to an obstacle problem and establishing some
special properties of the obstacle problem, it is
shown that if satisfies
condition, then the set is not convex unless
or
. Nevertheless, it is found that the set is still
weakly closed in the corresponding OrliczSobolev
space.

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