  For certain annuli in $\mathbb{C}^n$, $n\geq 2$, with non-smooth holes, we
show that the $\bar{\partial}$-operator from $L^2$ functions to $L^2$
$(0,1)$-forms has closed range. The holes admitted include products of
pseudoconvex domains and certain intersections of smoothly bounded pseudoconvex
domains. As a consequence, we obtain estimates in the Sobolev space $W^1$ for
the $\bar{\partial}$-equation on the non-smooth domains which are the holes of
these annuli.
