  In this paper we concentrate on the analysis of the critical mass blowing-up
solutions for the cubic focusing Schr\"odinger equation with Dirichlet boundary
conditions, posed on a plane domain. We bound the blow-up rate from below, for
bounded and unbounded domains. If the blow-up occurs on the boundary, the
blow-up rate is proved to grow faster than $(T-t)^{-1}$, the expected one.
Moreover, we show that blow-up cannot occur on the boundary, under certain
geometric conditions on the domain.
