  We prove \emph{uniform solvability estimates} for certain families of
elliptic problems posed in a bounded family of domains (for example, a sequence
that converges to another domain). We provide uniform estimates both in
weighted and in usual Sobolev spaces. When the limit domain is a \emph{polygon}
and the other domains are smooth, our results amount to rounding off'' the
corners of the limit domain. The technique of proof is based on a suitable
conformal modification of the metric, which makes the union of the domains a
manifold with boundary and relative bounded geometry.
