Sharp interface limit for two components Bose-Einstein condensates

0.53 MB·

by  Michael Goldman

Sharp interface limit for two components Bose-Einstein
condensates
M. Goldman ∗ J. Royo-Letelier †
4
Abstract
1
0
We study a double Cahn-Hilliard type functional related to the Gross-Pitaevskii energy of
2
two-components Bose-Einstein condensates. In the case of large but same order intercomponent
p
e and intracomponent coupling strengths, we prove Γ-convergence to a perimeter minimisation
S
functional with an inhomogeneous surface tension. We study the asymptotic behavior of the

2 surface tension as the ratio between the intercomponent and intracomponent coupling strengths

becomes very small or very large and obtain good agreement with the physical literature. We
]
P
obtain as a consequence, symmetry breaking of the minimisers for the harmonic potential.
A
.
h
t 1 Introduction
a
m
For V a given trapping potential (see Hypothesis 3.1 below for more precise requirement) and a fixed
[

constant ε>0 let η be the (unique) positive minimiser of the Gross-Pitaevskii functional
5 ε
v
7 1(cid:90) 1 1
E (η):= |∇η|2+ V|η|2+ |η|4dx, (1.1)
2 ε 2 ε2 2ε2
7 Rn
1 under the constraint (cid:107)η(cid:107) =1, where (cid:107)η(cid:107) denotes the L2(Rn) norm of η. We then consider for β, α
2 2 1
.
1 and α positive constants, with α +α =1, the double Cahn-Hilliard type functional
2 1 2
0
4 1(cid:90) 1 1 1
1 F (v,ϕ):= η2|∇v|2+ η4(1−v2)2+ η2v2|∇ϕ|2+ βη4v4 sin2ϕdx, (1.2)
v: ε,β 2 Rn ε 2ε2 ε 4 ε 4ε2 ε
i under the mass constraints
X
(cid:90) (cid:90)
r
a ηε2v2dx=α1+α2 =1 and ηε2v2cosϕdx=α1−α2, (1.3)
Rn Rn
and study its behavior when the parameter ε tends to zero.
∗Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany, email: gold-
[e-mail],fundedbyaVonHumboldtPostDocfellowship
†Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria,
email:[e-mail]
1
This functional arises in the description of two-components Bose-Einstein condensates with equal in-
tracomponent coupling strengths (see Section 3). The parameter 1 represents the intracomponent
ε2
couplingstrengthwhereas1+β istheratiobetweentheintercomponentandintracomponentcoupling
strengths.
The Gross-Pitaevskii functional (1.1), which describes the energy of a single component condensate
with density |η |2, has been extensively studied in the literature [1, 2, 17, 18]. As ε goes to zero, η
ε ε
√
converges to the Thomas-Fermi profile ρ, given by
ρ(x):=(λ2−V(x)) (1.4)
+
(cid:90)
withλdeterminedbytheconstraint ρdx=1. ThesupportofρisadomaindenotedbyDandcor-
Rn
respondstotheregionwherethedensityofthesinglecomponentcondensatedoesnotvanishasε→0.
ThemainresultofthepaperistheΓ-convergence[12,11]ofεF toaperimeterminimisationproblem
ε,β
with an inhomogeneous surface tension σ , defined in D by σ (x):=ρ(x)3/2σ with
β β β
σ :=inf(cid:26)1(cid:90) +∞v(cid:48)2+ 1(cid:0)1−v2(cid:1)2+ 1v2ϕ(cid:48)2+ βv4sin2ϕdt : limϕ=0 and limϕ=π(cid:27) , (1.5)
β 2 2 4 4 −∞ +∞
−∞
where in the infimum, the function v (respectively ϕ) denotes a function from R to [0,1] (respectively
from R to [0,π]).
Theorem 1.1. (Γ-convergence) Let β > 0 be fixed. Under the Hypothesis 3.1, the Γ-limit in
L1 (D)×L1 (D) as ε→0 of εF with mass constraint (1.3) is given by the functional F defined
loc loc ε,β β
as
 (cid:90) σ
 πβ|Dϕ| if v =1 a.e. in D and ϕ∈BVloc(D;{0,π})
D
F (v,ϕ):= (1.6)
β
 +∞ otherwise,
with mass constraint
(cid:90)
ρcosϕdx=α −α . (1.7)
1 2
Rn
Since F is finite only for v =1, we will denote by F (ϕ):=F (1,ϕ). It is worth noticing that since
β β β
σ (cid:90)
F (ϕ)= β ρ3/2|Dϕ|,theminimizersofF donotdependonβ. Thisfact,whichisquitepeculiar
β π β
D
to BEC interfaces, was already well known in the physics literature (see [32]). The functional F
ε,β
shares at the same time some features with the celebrated Ambrosio-Tortorelli functional which is
approximating the Mumford-Shah functional (see [5, 4]), and some other with functionals appearing
inthestudyofphasetransitionssuchastheModica-Mortolaenergy[26](alsoknownasCahn-Hilliard
or Allen-Cahn functional) or more general weighted functionals [10] (see also [11, 12]). Indeed, F
ε,β
consists of the sum of two singularly perturbed, weighted double-well potentials which are coupled
2
together. As in [5, 11, 3, 10] our proof is based on the slicing method described in Section 2.2.
In experiments realised with two-components Bose-Einstein condensates [25, 16, 29], the segregation
of the components is observed for large values of the intercomponent coupling strengths. This has
also been supported by numerical simulations in respectively, one ([20]), two ([19, 23]) and three
([28]) space dimensions. In our setting, at the level of F this means that for large values of β, ϕ
ε,β
takes approximately only values 0 and π while v is almost everywhere close to one. Moreover, for
the harmonic potential V =|x|2 in dimension n=2 [25, 23], one also observes a symmetry breaking
in the sense that while V is radially symmetric, the support of each component (which correspond
respectively to A := {ϕ = π} and D\A = {ϕ = 0}) are not. The numerical simulations also show
that near ∂A, the function v is close to a small positive constant. For β <0 the two components do
not segregate and their densities are both proportional to ρ.
Wementionthatsegregationoftwo-componentscondensateshasbeenwidelystudiedforboundedin-
tracomponentcouplingstrengthsandlargeintercomponentcouplingstrength. In[30]segregationand
symmetry breaking is proven in R2 for small intracomponent coupling strengths. In [33], working on
aboundeddomainofR2 andtakingthetrappingpotentialV tobezero,theauthorsshowsegregation
and local uniform convergence of the two components. In [15, 27] the regularity of ∂A is studied for
the same model. The profile of the components near ∂A is analysed in [8, 9].
In[3]thefunctionalF isstudiedforn=2whenβ goesto+∞asεtendstozero. Theauthorsalso
ε,β
prove Γ-convergence to a perimeter minimisation problem with an inhomogeneous surface tension.
The main difference with our setting is that for β →+∞, the limiting energy is given by the first two
terms of εF while the last two terms go to zero as ε → 0. This leads to some decoupling of the
ε,β
energywhichallowstocomputeexplicitlythelimitingsurfacetension. Inourcase,allthetermsinthe
energyεF areofthesameordersothatthesurfacetensionisgivenbytheonedimensionaloptimal
ε,β
transition problem (1.5). Thus, we need to precisely analyse the behavior of σ and of the associated
β
optimal profile. We prove existence and qualitative properties of minimisers of σ , an equipartition
β
of the energy and compare our results with the physical literature [32, 7, 6, 31, 24]. In particular,
we prove that minimisers (v,ϕ) of σ satisfy infv = m(β) > 0, as was expected from numerical
β
simulations. We remark that we are unable to prove uniqueness of the optimal profile. We study the
asymptotic behavior of σ when β tends to zero or infinity. On the one hand, we prove that when
β
β → +∞, we recover the functional derived in [3]. We show that in this regime, σ (cid:39) β−1/4 as pre-
β
dicted by formal asymptotic expansions [32]. This estimate follows from the fact that m(β)∼β−1/4
(see Proposition 4.3). This fact is related to some open questions raised in [8] (see also the discussion
√
in[3]). Ontheotherhand,weshowthatasexpectedfrom[6,31,24,7],σ (cid:39) β whenβ goestozero.
β
The fact that σ vanishes in this limit, reflects the non segregation of the two components. Finally,
β
in Proposition 6.6, we extend the symmetry breaking result for minimizers of F (for the harmonic
∞
potential V = |x|2) obtained in [3] to space dimensions n = 1 and n = 3. We notice that since the
3
minimizers of F coincide with the minimizers of F , this symmetry breaking result extends to any
β ∞
β >0 and by Γ−convergence to minimizers of the original functional F for ε small enough.
ε,β
Thepaperisorganisedasfollows: inSection2werecallthedefinitionandmainpropertiesoffunctions
of bounded variation and the slicing method. In Section 3, we explain how the functional F arises
ε,β
from the coupled Gross-Pitaevskii energy of a two-components Bose-Einstein condensate. In Section
4 we study the variational problem (1.6) and β >0, and prove existence and qualitative properties of
minimisers. In Section 5, we prove our main Γ-convergence theorem. Finally, in Section 6 we analyse
the asymptotic behavior of σ when β tends to zero or infinity and prove as a consequence symmetry
β
breaking of the minimisers.
2 Notation
Forx∈Rn andr >0,wedenotebyB (x)theballofradiusr centeredatxandsimplywriteB when
r r
x=0. We let Sn−1 be the unit sphere in Rn and for k ∈[0;n], we denote by Hk the k−dimensional
Hausdorff measure. Given a set E ⊂ Rn, we let 1 be the characteristic function of the set E. The
E
letters, c,C denote universal constants which can vary from line to line. We also make use of the
usual o and O notation. For a and b real numbers we let a∧b := min(a,b) and a∨b := max(a,b).
Throughout the paper, with a small abuse of language, we call sequence a family (u ) of functions
ε
labeled by a continuous parameter ε ∈ (0,1]. A subsequence of (u ) is any sequence (u ) such that
ε εk
ε →0 as k →+∞. We mention that ρ will denote a positive constants in Sections 4 and 6, while in
k
the rest of the paper it will be the function given in (1.4).
2.1 BV(Ω) functions
For Ω an open set of Rn, let BV(Ω) be the space of functions u ∈ L1(Ω) having as distributional
derivativeDuameasurewithfinitetotalvariation. Foru∈BV(Ω),wedenotebyS thecomplement
u
1 (cid:90)
of the Lebesgue set of u. That is, x∈/ S if and only if lim |u(y)−z| dy =0 for some
u r→0+ |B |
r Br(x)
z ∈R. We say that x is an approximate jump point of u if there exist ν ∈Sn−1 and distinct a,b∈R
such that
1 (cid:90) 1 (cid:90)
lim |u(y)−a| dy =0 and lim |u(y)−b| dy =0,
r→0|Br+(x,ν)| Br+(x,ν) r→0|Br−(x,ν)| Br−(x,ξ)
where B±(x,ν) := {y ∈ B (x) : ±(cid:104)y −x,ν(cid:105) > 0}. Up to a permutation of a and b and a change
r r
of sign of ν, this characterizes the triplet (a,b,ν) which is then denoted by (u+,u−,ν ). The set of
u
approximated jump points is denoted by J . The following theorem holds [4].
u
Theorem 2.1. The set S is countably Hn−1-rectifiable and Hn−1(S \J )=0. Moreover Du J =
u u u u
(u+−u−)ν Hn−1 J .
u u
4
We indicate by Du = ∇u dx + Dsu the Radon-Nikodym decomposition of Du. Setting Dcu :=
Dsu (Ω\S ) we get the decomposition
u
Du=∇u dx + (u+−u−)ν Hn−1 J + Dcu,
u u
where denotestherestriction. Inparticular,ifu=π1 ∈BV(Ω,{0,π})thenDu=πνEHn−1 ∂∗E
E
where ∂∗E is the reduced boundary of E defined by
(cid:26) (cid:27)
D1 (B (x))
∂∗E := x∈Spt(|D1 |) : νE(x):=−lim E r exists and |νE(x)|=1
E r↓0 |D1E|(Br(x))
and νE is the outward measure theoretic normal to the set E which is countably Hn−1-rectifiable.
When n=1 we use the symbol u(cid:48) in place of ∇u, and u(x±) to indicate the right and left limits at x.
2.2 Slicing method
In this section we recall the slicing method for functions with bounded variation [11, Ch. 4] which
willbeusedintheproofofthelowerΓ-limit. ConsideranopensetA⊂Rn andletν ∈Sn−1. Wecall
Π the hyperplane orthogonal to ν and A the projection of A on Π . We define the one dimensional
ν ν ν
slices of A, indexed by x∈A , as
ν
A :={t∈R; x+tν ∈A}.
νx
For every function f in Rn, we note f the restriction of f to the slice A , defined by f (t) :=
νx νx νx
f(x+tν) . Functions in BV(Ω) can be characterised by one-dimensional slices (see [11]).
Theorem 2.2. Let u∈BV(A). Then for all ν ∈Sn−1 we have
u ∈BV(A ) for Hn−1−a.e. x∈A .
νx νx ν
Moreover, for such points x, we have
u(cid:48) (t)=(cid:104)∇u(x+tν),ν(cid:105) for a.e. t∈A , (2.1)
νx νx
J ={t∈R:x+tν ∈J }, (2.2)
uνx u
and
u (t±)=u±(x+tν) or u (t±)=u∓(x+tν), (2.3)
νx νx
according to whether (cid:104)ν ,ν(cid:105)>0 or (cid:104)ν ,ν(cid:105)<0. Finally, for every Borel function g :A→R,
u u
(cid:90) (cid:90)
(cid:88)
g (t) dHn−1(x)= g |(cid:104)ν ,ν(cid:105)| dHn−1. (2.4)
νx u
Aν t∈Juνx Ju
Conversely if u∈L1(A) and if for all ν ∈{e ,...,e }, where (e ,...,e ) is a basis of Rn, and almost
1 n 1 n
every x∈A we have u ∈BV(A ) and
ν νx νx
(cid:90)
|Du |(A ) dHn−1(x)<+∞,
νx νx
Aν
then u∈BV(A).
5
3 Derivation of the energy F from the coupled Gross-Pitaevskii
ε,β
functional
A two-components condensate is described by two functions u and u , where |u |2 and |u |2 respec-
1 2 1 2
tively represent the densities of the first and second component. The energy of the two-components
condensate is given by a coupled Gross-Pitaevskii functional. When the intracomponent coupling
strength of each component is equal to 1/ε2, and when the intercomponent coupling strength is equal
to (1+β)/ε2, the functional is given by
1+β (cid:90)
E (u ,u ):=E (u )+E (u )+ |u |2|u |2dx,
ε 1 2 ε 1 ε 2 2ε2 1 2
Rn
where E is defined in (1.1). Assuming that the mass of each component is preserved, the functional
ε
E is minimised under the restrictions
ε
(cid:90) (cid:90)
|u |2dx=α and |u |2dx=α (3.1)
1 1 2 2
Rn Rn
with α ,α >0 and α +α =(cid:107)η(cid:107) =1.
1 2 1 2 2
Standard arguments used in the study of a single component condensate yield that the minimisers of
E under the constraint (3.1) are smooth positive functions, up the multiplication by constant terms
ε
ofmodulus1,withL∞ normuniformlyboundedwithrespecttoε(see[1,3,17,18]). Noticealsothat
foraradialpotentialV,if(u ,u )isaminimiser,thenforanyrotationRofthespace,(u ◦R,u ◦R)
1 2 1 2
is also a minimiser. In the single component case, the Euler-Lagrange equations imply uniqueness of
the minimiser from which one can infer its radial symmetry. For two components condensates, this is
not the case anymore.
TherelationbetweenE andF wasestablishedin[3]. Usingthenonlinearsigmamodelrepresenta-
ε ε,β
tion[19,23]andtheLassoued-Mironescutricktodecomposetheenergyofarotatingsinglecondensate
[21], the authors introduced the change of variables
(cid:112) (cid:32) (cid:33)
|u |2+|u |2 ϕ |u |+i|u |
v := 1 2 and :=Arg 1 2 (3.2)
(cid:112)
ηε 2 |u1|2+|u2|2
for any pair (u ,u ) such that E (u ,u )<∞ and |u |2+|u |2 >0. The equality
1 2 ε 1 2 1 2
E (u ,u )=F (v,ϕ)+E (η ) (3.3)
ε 1 2 ε,β ε ε
then holds, and the mass constraints in (3.1) rewrite as in (1.3). Let us point out that in [3], only
the case n = 2 is considered but the proof carries over verbatim to any space dimension. As seen
from (3.3) and the expression (1.2) of F , there are two main advantages of the formulation of the
ε,β
problem in terms of the functions (v,ϕ). On the one hand, it naturally identifies the leading order
termE (η ). Ontheotherhand,itclearlyshowsthatthesecondordercontributionF isasingular
ε ε ε,β
6
perturbation type functional.
Notice that since the minimisers (u ,u ) are uniformly bounded and η does not vanish, for every
1 2 ε
compact set K of D, there exists a constant C(K) such that 0 < v ≤ C(K) in K. Moreover, it is
readily seen from the definition that ϕ ∈ [0,π]. We are thus naturally led to minimize F in the
ε,β
class
Y(D):={(v,ϕ) : for every compact set K ⊂D, 0<v ≤C(K) in K and ϕ∈[0,π]}
under the mass constraints (1.3). For a subset A of D, we introduce the localised version of F :
ε,β
1(cid:90) 1 1 1
F (v,ϕ;A):= η2|∇v|2+ η4(1−v2)2+ η2v2|∇ϕ|2+ βη4v4 sin2ϕdx,
ε,β 2 ε 2ε2 ε 4 ε 4ε2 ε
A
and
Y(A):={(v,ϕ) : for every compact set K ⊂D, 0<v ≤C(K) in K∩A and ϕ∈[0,π]}.
Notice that, for any (v,ϕ)∈Y(D), defining
u :=η vcos(ϕ/2) and u :=η vsin(ϕ/2) (3.4)
1 ε 2 ε
relation (3.3) holds and we have |u |2+|u |2 >0.
1 2
In the following we are going to make the following assumptions on V:
Hypothesis 3.1. V is such that V(x)→+∞ when |x|→+∞ and there exist C,a,b,c>0 such that
if ρ is the Thomas-Fermi profile defined in (1.4),
(cid:107)η (cid:107) < C (3.5)
ε ∞
(cid:107)ηε(cid:107)L2(Rn\D) ≤ Cεa (3.6)
√
|η (x)− ρ(x)| ≤ Cεc if dist(x,∂D)>Cεb (3.7)
ε
WeremarkthatfortheharmonicpotentialV(x)=|x|2,itwasprovenin[17]thattheseconditionshold
true in dimension n=2. Moreover, it can be checked that their proof carries over almost verbatim to
any space dimension. Recently, Karali and Sourdis [18], obtained that if n=2, Hypothesis 3.1 holds
if V satisfies:
(i) V is nonnegative and C1,
(ii) there exist C >1, p≥2 such that 1(1+|x|p)≤V(x)≤C(1+|x|p),
C
(iii) D isasimplyconnectedboundeddomaincontainingtheoriginwithsmoothboundaryandsuch
that ∂V >0 on ∂D.
∂ν
√
Notice that in their paper, Karali and Sourdis prove that (cid:107)ηε − ρ(cid:107)L∞(R2) ≤ Cε1/3 [18, Rem. 4.4]
which is stronger than (3.7). They also claim that their proof should extend to any space dimension
(see [18, Rem. 3.12]) and that the fact that D is simply connected is superfluous (see [18, Rem. 1.1]).
7
4 The surface tension at finite β > 0
In this section, for β >0 fixed, we study the following variational problem:
(cid:26) (cid:27)
σ :=inf G (v,ϕ) : v ≥0, 0≤ϕ≤π, limϕ=0 and limϕ=π , (4.1)
β β
−∞ +∞
where
1(cid:90) +∞ 1 β
G (v,ϕ):= v(cid:48)2+W(v)+ v2ϕ(cid:48)2+ v4sin2ϕdt, (4.2)
β 2 4 4
−∞
with W(v):= 1(cid:0)1−v2(cid:1)2.
2
Let us point out that if G (v,ϕ) is finite then lim v(x)=1.
β x→±∞
We start by evaluating the energy necessary to connect v from a given value m>0 to 1.
Lemma 4.1. Let m∈[0,1] then
(cid:26)(cid:90) +∞ (cid:27) √ (cid:18)2 m3(cid:19)
inf v(cid:48)2+W(v)dt : v(0)=m = 2 −m+ ,
3 3
0
(cid:32)(cid:114) (cid:33)
1
and the optimal profile is given by v :=tanh t+c where c :=tanh−1(m).
m 2 m m
Proof. As in the usual Modica-Mortola problem,
(cid:90) +∞ √ (cid:90) 1 √ (cid:18)2 m3(cid:19)
inf v(cid:48)2+W(v)dt= 2 (1−t2)dt= 2 −m+ .
v(0)=m 0 m 3 3
We now prove that we can restrict ourselves to functions v which stay away from zero.
Proposition 4.2. For every β >0, there exists m∗ =m∗(β)>0 such that
(cid:26) (cid:27)
σ =inf G (v,ϕ) : v ∈[m∗,1], limϕ=0 and limϕ=π .
β β
−∞ +∞
Proof. First, let us notice that by truncation, we can reduce ourselves to minimise among functions
v ∈[0,1]. UptotranslationwecanalsoassumethatinfRv =v(0). Letm≥0,thenforeveryfunction
v such that infRv =v(0)=m and every admissible ϕ,
1(cid:20) (cid:90) 0 (cid:21) 1(cid:20) (cid:90) +∞ (cid:21)
G (v,ϕ)≥ inf v(cid:48)2+W(v)dt + inf v(cid:48)2+W(v)dt
β 2 v(0)=m −∞ 2 v(0)=m 0
1(cid:90) 1 β
+ v2ϕ(cid:48)2+ v4sin2ϕdt
2 4 4
R
(cid:20) (cid:90) +∞ (cid:21) 1(cid:90)
≥ inf v(cid:48)2+W(v)dt + β1/2v3|sinϕ||ϕ(cid:48)|dt
v(0)=m 0 4 R
√ (cid:18)2 m3(cid:19) β1/2m3 (cid:90)
≥ 2 −m+ + |sinϕ||ϕ(cid:48)|dt
3 3 4
R
√ (cid:18)2 m3(cid:19) β1/2m3 (cid:90) π
= 2 −m+ + |sinx|dx
3 3 4
0
√ (cid:18)2 (cid:18)1 β1/2(cid:19)(cid:19)
= 2 −m+m3 + √ .
3 3 2 2
8
Now, for m≥0 and T >0, consider the test functions defined by
 
vm(−t−T) t<−T 0 t<−T
vm,T := m t∈[−T,T] and ϕT := π (t+T) t∈[−T,T]
2T
v (t−T) t≥T π t≥T,
m
then
√ (cid:18)2 m3(cid:19) T m2π2 1 (cid:90) T (cid:16) π (cid:17)
G (v ,ϕ )= 2 −m+ + (1−m2)2+ + βm4 sin2 (t+T) dt
β m,T T 3 3 2 16T 4 2T
0
√ (cid:18)2 m3(cid:19) T m2π2 β
= 2 −m+ + (1−m2)2+ + m4T. (4.3)
3 3 2 16T 8
Optimizing in T we find Tm := 2√2((1−mm2)π2+βm4)1/2 and
4
√
√ (cid:18)2 m3(cid:19) 2 (cid:18) β (cid:19)1/2
G (v ,ϕ )= 2 −m+ + mπ (1−m2)2+ m4 . (4.4)
β m,Tm Tm 3 3 4 4
Let now (see Figure 1)
(cid:18)m3 (cid:19) 1 (cid:18) β (cid:19)1/2
Ψ(m):= −m + mπ (1−m2)2+ m4
3 4 4
√
so that G (v ,ϕ )= 2(Ψ(m)+ 2) and let
β m,Tm Tm 3
m:=argminΨ(m).
m∈[0,1]
Let us first notice that since Ψ(0) = 0 and Ψ(cid:48)(0) = π −1 < 0, the minimum of Ψ is negative for
4
every β >0. The function m3 −m is decreasing in [0,1] and Ψ(m)>−2 hence there exists a unique
3 3
m∗(β)∈(0,1) such that m∗(β)3 −m∗(β)=Ψ(m). We claim that
3
(cid:26) (cid:27)
σ =inf G (v,ϕ) : infv ≥m∗(β), limϕ=0 and limϕ=π .
β β
−∞ +∞
Indeed, if v is such that infv ≤ m∗(β) and if ϕ is any admissible function, then letting m := infv,
there holds
√ (cid:16) 2(cid:17) √ (cid:16)m3 2(cid:17)
G (v ,ϕ )= 2 Ψ(m)+ < 2 −m+
β m,Tm Tm 3 3 3
√ (cid:18)2 (cid:18)1 β1/2(cid:19)(cid:19)
≤ 2 −m+m3 + √ ≤G (v,ϕ)
3 3 2 2 β
so that we can construct a competitor with smaller energy than (v,ϕ).
Intheregimeβ →+∞,wecanproveamorepreciseboundoninfv. Noticethatinthecaseβ =+∞,
[3] proved that infv =0.
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Figure 1: The function Ψ
Proposition 4.3. There exist constants B,C >0 such that if β ≥B,
(cid:26) (cid:27)
1
σ =inf G (v,ϕ) : β−1/4 ≤infv ≤Cβ−1/4, limϕ=0 and limϕ=π .
β β C −∞ +∞
(cid:16) (cid:17)
Proof. Let M:={m∈[0,1]:m3 1 + β√1/2 −m>Ψ(m)} then arguing as in the previous proof, we
3 2 2
obtain
(cid:26) (cid:27)
σ =inf G (v,ϕ) : infv ∈/ M, limϕ=0 and limϕ=π .
β β
−∞ +∞
The claim is thus proven provided we can show that for β large enough, and for m∈[0,1] such that
m ≤ 1β−1/4 or m ≥ Cβ−1/4 then m ∈ M. We notice first that if β is large then if m ≥ Cβ−1/4,
C
(cid:16) (cid:17) (cid:16) (cid:104) (cid:105)(cid:17)1/4
m3 1 + β√1/2 −m > 0 > Ψ(m) hence m ∈ M. Taking m = m˜β−1/4 with 0 < m˜ < 4 16−π2
3 2 2 π2
(cid:16) (cid:17)1/2
so that 1π (1−m2)2+ βm4 <1, we obtain Ψ(m)≤−1β−1/4 and therefore, for m≤ 1β−1/4,
4 4 C C
(cid:16) (cid:17)
we have m3 1 + β√1/2 −m>Ψ(m), that is m∈M.
3 2 2
We can now prove the existence of an optimal profile.
Proposition4.4. Foreveryβ >0thereexistsaminimiserofσ . Moreover,itissmoothandsatisfies
β
the Euler-Lagrange equations
1 β
−v(cid:48)(cid:48)−(1−v2)v+ vϕ(cid:48)2+ v3sin2ϕ = 0 (4.5)
4 2
−(v2ϕ(cid:48))(cid:48)+βv4sinϕcosϕ = 0. (4.6)
Proof. Let(v ,ϕ )beaminimisingsequence. Uptotranslation,wecanassumethatϕ(0)= π. Letus
n n 2
notice that up to truncating v , we can also assume that v ∈[0,1]. Therefore, since v(cid:48) is uniformly
n n n
bounded in L2(R), up to extraction, the sequence v converges locally uniformly to some continuous
n
function v. Moreover, by lower semicontinuity,
(cid:90) (cid:90)
v(cid:48)2+W(v)dt≤ lim v(cid:48)2+W(v )dt.
n n
R n→+∞ R
10