The Transition to a Giant Vortex Phase in a Fast Rotating Bose-Einstein Condensate

We study the Gross-Pitaevskii (GP) energy functional for a fast rotating Bose-Einstein condensate on the unit disc in two dimensions. Writing the coupling parameter as $1 / \eps^2$ we consider the asymptotic regime $\eps \to 0$ with the angular velocity $\Omega$ proportional to $(\eps^2|\log\eps|)^{-1}$. We prove that if $\Omega = \Omega_0 (\eps^2|\log\eps|)^{-1}$ and $\Omega_0 > 2(3\pi)^{-1}$ then a minimizer of the GP energy functional has no zeros in an annulus at the boundary of the disc that contains the bulk of the mass. The vorticity resides in a complementary `hole' around the center where the density is vanishingly small. Moreover, we prove a lower bound to the ground state energy that matches, up to small errors, the upper bound obtained from an optimal giant vortex trial function, and also that the winding number of a GP minimizer around the disc is in accord with the phase of this trial function.