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

Aftalion, Amandine; Royo-Letelier, Jimena

doi:10.1007/s00526-014-0708-y

Cited by 15 publications

(46 citation statements)

References 30 publications

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“…In the present paper, we are interested in the segregation case. The Γ limit of (1.26)-(1.27) with Ω = 0 in the case where g 2 > g 1 g 2 , g 1 = g 2 and g = g ε → ∞ has been studied in [2]. A change of functions is used, namely (v, ϕ), where v 2 = u 2 1 + u 2 2 and cos ϕ =…”

Section: Physical Motivation and Known Resultsmentioning

confidence: 99%

“…The segregation behaviour in two-component condensates has been widely studied in the mathematics literature: regularity of the wave function [13,14,30,34,35,39,40], regularity of the limiting interface [11,37,41], asymptotic behaviour near the interface [8,9], Γ-convergence in the case of a trapped problem [2,19,20]. This paper deals with the case of segregation, and more precisely, the study of the behaviour of the wave functions near the interface.…”

Section: Introductionmentioning

confidence: 99%

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Interface layer of a two-component Bose–Einstein condensate

Aftalion

Sourdis

2016

Commun. Contemp. Math.

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This paper deals with the study of the behaviour of the wave functions of a two-component Bose-Einstein condensate near the interface, in the case of strong segregation. This yields a system of two coupled ODE's for which we want to have estimates on the asymptotic behaviour, as the strength of the coupling tends to infinity. As in phase separation models, the leading order profile is a hyperbolic tangent. We construct an approximate solution and use the properties of the associated linearized operator to perturb it into a genuine solution for which we have an asymptotic expansion. We prove that the constructed heteroclinic solutions are linearly nondegenerate, in the natural sense, and that there is a spectral gap, independent of the large interaction parameter, between the zero eigenvalue (due to translations) at the bottom of the spectrum and the rest of the spectrum. Moreover, we prove a uniqueness result which implies that, in fact, the constructed heteroclinic is the unique minimizer (modulo translations) of the associated energy, for which we provide an expansion. 4

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Section: Physical Motivation and Known Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Interface layer of a two-component Bose–Einstein condensate

Aftalion

Sourdis

2016

Commun. Contemp. Math.

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“…There are results about the regularity and connectedness, and the fact that the interface goes from one part of the boundary to another [7,33,43]. There are also results about the Γ limit [4,21,20] which rely on similar techniques to those used for the Mumford Shah functional [8,9]. The order of magnitude of δ has a strong impact on m α,ε,δ and the boundary layer between the two components.…”

Section: Introductionmentioning

confidence: 99%

Vortex patterns and sheets in segregated two component Bose–Einstein condensates

Aftalion

Sandier²

2019

Calc. Var.

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We study minimizers of a Gross-Pitaevskii energy describing a two-component Bose-Einstein condensate set into rotation. We consider the case of segregation of the components in the Thomas-Fermi regime, where a small parameter ε conveys a singular perturbation. We estimate the energy as a term due to a perimeter minimization and a term due to rotation. In particular, we prove a new estimate concerning the error of a Modica Mortola type energy away from the interface. For large rotations, we show that the interface between the components gets long, which is a first indication towards vortex sheets.

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“…On the other hand, the case K > √ g, where it is expected both experimentally [33,38], numerically [32,37] and theoretically [28,10,41,11,42] that segregation occurs, is maybe not yet so well understood. In the symmetric case g = 1, it has been proved in [4,25] that, as expected from the physics literature, the Thomas-Fermi limit i.e. the limit of εF ε only imposes segregation but does not give any information about the actual shape of the minimizers.…”

Section: Introductionmentioning

confidence: 84%

Phase Segregation for Binary Mixtures of Bose--Einstein Condensates

Goldman¹,

Merlet²

2017

SIAM J. Math. Anal.

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

Cited by 15 publications

References 30 publications

Interface layer of a two-component Bose–Einstein condensate

Interface layer of a two-component Bose–Einstein condensate

Vortex patterns and sheets in segregated two component Bose–Einstein condensates

Phase Segregation for Binary Mixtures of Bose--Einstein Condensates

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