Phase transition problems in Two-Components Bose-Einstein Condensates

When the parameter describing the interaction between two trapped Bose-Einstein condensates is very large, the total system undergoes a phase transition. The condensates then segregate, break the symmetry of the trap and their supports became half-ball shaped. In this talk, we study this phenomenon using two phase transition models : optimal partition problems and a Cahn-Hilliard energy functional. We will first study the Gross-Pitaevskii functional describing the behaviour of a condensate and the main properties of its minimizers. Then, we will explain the analogous functional when two condensates are presents. We finally study the associated phase transition problems in two main regimes and explain the results obtained.