Rapidly Rotating Bose-Einstein Condensates in Strongly Anharmonic Traps

We study a rotating Bose-Einstein condensate in a strongly anharmonic trap flat trap with a finite radius in the framework of two-dimensional Gross-Pitaevskii theory.We write the coupling constant for the interactions between the gas atoms as 1/\epsilon^2 and we are interested in the limit \epsilon→0 (Thomas-Fermi limit) with the angular velocity \Omega depending on \epsilon. We derive rigorously the leading asymptotics of the ground state energy and the density profile when \Omega tends to infinity as a power of 1/\epsilon. If \Omega(\epsilon) = \Omega_0/\epsilon a “hole” i.e., a region where the density becomes exponentially small as 1/\epsilon→\infty develops for \Omega_0 above a certain critical value. If \Omega(\epsilon) \gg 1/\epsilon the hole essentially exhausts the container and a “giant vortex” develops with the density concentrated in a thin layer at the boundary. While we do not analyze the detailed vortex structure we prove that rotational symmetry is broken in the ground state for const log\epsilon < \Omega(\epsilon) \lesssim const/\epsilon.