Definition:Barrier

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Definition

A complex function $\varphi \in \C \left({\overline \Omega}\right)$ is a barrier for $\Omega$ at $z \in \partial \Omega$ if and only if:

$\varphi$ is subharmonic
$\varphi ( z ) = 0$
$\varphi < 0$ on $\partial \Omega \setminus \{ z \}$







We call the boundary point $z \in \partial \Omega$ regular if there is a barrier for $\Omega$ at $z \in \partial \Omega$.