Phase Segregation for Binary Mixtures of Bose--Einstein Condensates

Abstract

We study the strong segregation limit for mixtures of Bose-Einstein condensates modeled by a Gross-Pitaievskii functional. Our first main result is that in presence of a trapping potential, for different intracomponent strengths, the Thomas-Fermi limit is sufficient to determine the shape of the minimizers. Our second main result is that for asymptotically equal intracomponent strengths, one needs to go to the next order. The relevant limit is a weighted isoperimetric problem. We then study the minimizers of this limit problem, proving radial symmetry or symmetry breaking for different values of the parameters. We finally show that in the absence of a confining potential, even for non-equal intracomponent strengths, one needs to study a related isoperimetric problem to gain information about the shape of the minimizers. Proposition 1.5. For every α ∈ (0, α), there exists C = C(α) > 0 such that for every measurable set E ⊂ R 2 with E ρ = α, we have,where r is such that Br ρ = α.The second is an estimate on the potential instability of the ball for the weighted isoperimetric problem.Proposition 1.6. For every α ∈ (0, α), there exists c = c(α) > 0 such that for every set E ⊂ R 2 with locally finite perimeter and with E ρ = α, there holds ∂E ρ 3/2 − ∂Br ρ 3/2 ≥ −c E∆Br ρ 2 ,where r is such that Br ρ = α.

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