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/ε2 and we are interested in the limit ε→0 (Thomas-Fermi limit) with the angular velocity Ω depending on ε. We derive rigorously the leading asymptotics of the ground state energy and the density profile when Ω tends to infinity as a power of 1/ε. If Ω(ε)=Ω0/ε a ''hole'' (i.e., a region where the density becomes exponentially small as 1/ε→∞) develops for Ω0 above a certain critical value. If Ω(ε)>>1/ε 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 vertical bar log ε vertical bar <Ω(ε) < or approx. const/ε