  Let $\Omega\subset\mathbb{C}$ be an open set. We show that
$\overline{\partial}$ has closed range in $L^{2}(\Omega)$ if and only if the
Poincar\'e-Dirichlet inequality holds. Moreover, we give necessary and
sufficient potential-theoretic conditions for the
$\overline{\partial}$-operator to have closed range in $L^{2}(\Omega)$. We also
give a new necessary and sufficient potential-theoretic condition for the
Bergman space of $\Omega$ to be infinite dimensional.
