We investigate a model corresponding to the experiments for a two-dimensional rotating Bose-Einstein condensate. It consists in minimizing a Gross-Pitaevskii functional defined in R 2 under the unit mass constraint. We estimate the critical rotational speed 1 for vortex existence in the bulk of the condensate and we give some fundamental energy estimates for velocities close to 1 .

We continue the analysis started in [14] on a model describing a two-dimensional rotating Bose-Einstein condensate. This model consists in minimizing under the unit mass constraint, a Gross-Pitaevskii energy defined in R 2 . In this contribution, we estimate the critical rotational speeds Ω d for having exactly d vortices in the bulk of the condensate and we determine their topological charge and their precise location. Our approach relies on asymptotic energy expansion techniques developed by Serfaty [20][21][22] for the Ginzburg-Landau energy of superconductivity in the high κ limit.

We study a class of symmetric critical points in a variational 2D Landau -de Gennes model where the state of nematic liquid crystals is described by symmetric traceless 3×3 matrices. These critical points play the role of topological point defects carrying a degree k 2 for a nonzero integer k. We prove existence and study the qualitative behavior of these symmetric solutions. Our main result is the instability of critical points when k = ±1, 0.