A note on approximation of plurisubharmonic functions

We extend a recent result of Avelin, Hed, and Persson about approximation of functions $u$ that are plurisubharmonic on a domain $Ω$ and continuous on $\barΩ$, with functions that are plurisubharmonic on (shrinking) neighborhoods of $\barΩ$. We show that such approximation is possible if the boundary of $Ω$ is $C^0$ outside a countable exceptional set $E\subset\partial Ω$. In particular, approximation is possible on the Hartogs triangle. For Hölder continuous $u$, approximation is possible under less restrictive conditions on $E$. We next give examples of domains where this kind of approximation is not possible, even when approximation in the Hölder continuous case is possible.

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