A minimal interface problem arising from a two component Bose Einstein condensate via $\G$-convergence

We consider the energy modeling a two component Bose-Einstein condensate in the limit of strong coupling and strong segregation. We prove the $Γ$-convergence to a perimeter minimization problem, with a weight given by the density of the condensate. In the case of equal mass for the two components, this leads to symmetry breaking for the ground state. The proof relies on a new formulation of the problem in terms of the total density and spin functions, which turns the energy into the sum of two weighted Cahn-Hilliard energies. Then, we use techniques coming from geometric measure theory to construct upper and lower bounds. In particular, we make use of the slicing technique introduced in Ambrosio-Tororelli (CPAM, 1990).

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