  We study the two-dimensional Gross-Pitaevskii theory of a rotating Bose gas
in a disc-shaped trap with Dirichlet boundary conditions, generalizing and
extending previous results that were obtained under Neumann boundary
conditions. The focus is on the energy asymptotics, vorticity and qualitative
properties of the minimizers in the parameter range $|\log\eps| \ll \Omega
\lesssim \eps^{-2}|\log\eps|^{-1}$ where $ \Omega $ is the rotational velocity
and the coupling parameter is written as $ \eps^{-2} $ with $ \eps \ll 1 $.
Three critical speeds can be identified. At $ \Omega = \Omega_{\mathrm{c_1}}
\sim|\log\eps| $ vortices start to appear and for $ |\log\eps| \ll \Omega <
\Omega_{\mathrm{c_2}} \sim \eps^{-1} $ the vorticity is uniformly distributed
over the disc. For $ \Omega \geq \Omega_{\mathrm{c_2}} $ the centrifugal forces
create a hole around the center with strongly depleted density. For $ \Omega
\ll \eps^{-2}|\log\eps|^{-1} $ vorticity is still uniformly distributed in an
annulus containing the bulk of the density, but at $ \Omega =
\Omega_{\mathrm{c_3}} \sim \eps^{-2}|\log\eps|^{-1} $ there is a transition to
a giant vortex state where the vorticity disappears from the bulk. The energy
is then well approximated by a trial function that is an eigenfunction of
angular momentum but one of our results is that the true minimizers break
rotational symmetry in the whole parameter range, including the giant vortex
phase.
