Distribution of Vorticity

We are now able to prove the uniform distribution of vorticity:

Proof of Theorem 1.1.

The proof follows very closely the proof of [CY, Theorem 3.3] and relies essentially on [SS1, Proposition 5.1].

The argument has to be slightly adapted depending on the value of the angular velocity: For any Ω ≤Ωε¯ ⁻¹, with ¯Ω<2/√

π, the proof of [CY, Theorem 3.3] applies with only one minor modification, since the cells in the lattice ˆLoccurring in the lower bound proof do not cover the whole ofB. However, since the region covered by cells tends to A^TF as ε →0 and the area of the excluded set close to the boundary is of order O(ε|logε|), i.e., much smaller than the cell area, such a difference in the lattice choice has no consequences for the final result.

We now discuss the modifications in the regimeε⁻¹ Ω ε⁻²|logε|⁻¹ which was not taken into account in [CY, Theorem 3.3]. The starting point is the localization of the energy bounds (3.15), (3.22) and (3.24), which can be rewritten as

X

In order to obtain a similar estimate inside one lattice cell, one first needs a suitable lower bound on the densityρ^TF and this can be obtained by restricting the analysis to the bulk of the condensate, i.e.,

Abulk:=n Moreover, the localization of the energy estimate requires that a certain number of bad cells be rejected:

As in [CY, Theorem 3.3] we first introduce a new small parameter :=

whereas the cell is bad if the inequality is reversed.

Now given any set S ⊂ A_bulk such that |S| |Q|, the upper bound (3.38), the definition of bad cells,ˆ (3.36) and the upper boundρ^TF≤ O(εΩ) imply that

NB ≤√

η γ⁻¹N=√

η|logη|N N, (3.41)

where NB and N stand for the number of bad cells and the total number of cells contained inside S respectively.

On the other hand by the definition (3.40) good cells satisfy the assumptions of [SS1, Proposition 5.1]

and therefore one can construct a finite collection of disjoint balls{Bi}:={B(~r_i, %_i)}such that|u|>1/2 on the boundary of each ball and %_i ≤ O(Ω^−1/2). Hence one can define the winding number d_i,ε ofuon

∂B_i, which coincides with the winding number of Ψ^GP and, using [SS1, Proposition 5.1]

2πX

di,ε= Ωˆ`²(1 +o(1)), 2πX

|di,ε|= Ωˆ`<sup>2</sup>(1 +o(1)). (3.42) The rest of the statement of Theorem 1.1 easily follows by noticing that one can always take ˆ` = Ω^−1/2|log(ε²Ω|logε|)|, which satisfies (3.21), obtaining the lower condition on the area of the setS.

4 The Giant Vortex Regime Ω ∼ ε

| log ε|

As a preparation for the proof of the main results contained in Theorems 1.3 and 1.4 we formulate and prove in Section 4.1 some important propositions about the properties of the giant vortex density profiles.

The proof of the absence of vortices in the bulk will follow the analysis of the ground state energy asymp-totics, which is achieved in several steps. The main ingredients are the energy decoupling (Section 4.2), the vortex ball construction and the jacobian estimate (Section 4.4). Each individual step is analogous to the corresponding one contained in [CRY] and we will often omit some details, only stressing the major differences with the analysis of [CRY] and referring to that paper for further details.

4.1 Giant Vortex Density Profiles

In this section we investigate the properties of the giant vortex profiles and the associated energy func-tional defined in (1.35). Actually for technical reasons which will be clearer later we consider a funcfunc-tional identical to (1.35) but on a different integration domain, i.e.,

A:={~r∈ B: r≥R<}, (4.1)

whereR<< Rhis suitably chosen in order to apply some estimates: All the conditions onR< occurring in the subsequent proofs are satisfied if we take

R_<:=R_h−ε^8/7. (4.2)

More precisely we define D˜^GP:=n

f ∈H¹(A) : f =f^∗, kfk_L2(A)= 1, f = 0 on∂Bo

(4.3) and set, for anyf ∈D˜^GP,

E˜_ω^gv[f] :=

Z

A

d~rn

|∇f|²+ ([Ω]−ω)²r⁻²f²−2([Ω]−ω)Ωf²+ε⁻²f⁴o

= Z

A

d~rn

|∇f|²+B_ω²f²−Ω²r²f²+ε⁻²f⁴o

. (4.4)

We recall that

B_ω= Ωr−([Ω]−ω)r⁻¹. The associated ground state energy is

E˜_ω^gv:= inf

f∈D˜^GP

E˜_ω^gv[f] (4.5)

and we denote g_ωany associated minimizer.

The TF-like functional obtained from (4.4) by dropping the kinetic term is denoted by ˜E_ω^TF (see (A.4)) and its minimization discussed in the Appendix.

Proposition 4.1 (Minimization ofE˜_ω^gv).

If Ω∝ε⁻²|logε|⁻¹ and|ω| ≤ O(ε^−5/4|logε|^−3/4)asε→0, then

E^TF≤E˜_ω^TF≤E˜_ω^gv≤E˜_ω^TF+O(ε^−5/2|logε|^−3/2)≤E^TF+O(ε^−5/2|logε|^−3/2). (4.6) There exists a minimizer gω that is unique up to a sign, radial and can be chosen to be positive away from the boundary ∂B. It solves inside Athe variational equation

−∆gω+B_ω²g_ω−Ω²r²g_ω+ 2ε⁻²g³_ω= ˜µ^gv_ω g_ω, (4.7) with boundary conditions gω(1) = 0 andg_ω⁰(R<) = 0 andµ˜^gv_ω = ˜E_ω^gv+ε⁻²kgωk⁴₄.

Moreover gω has a unique global maximum atR˜m withR<<R˜m<1.

Remark 4.1 (Composition of the energy)

Unlike the flat Neumann case, the remainder in the r.h.s. of (4.6) is of the same order even if the refined TF energy ˜E_ω^TF is extracted. The reason is that such a remainder is actually due to the radial kinetic energy of the giant vortex density profile and in particular to Dirichlet boundary conditions.

In order to give some heuristics to explain such a difference with the flat Neumann case, it is indeed sufficient to note that, by the pointwise estimate (2.21), the density gω goes from its maximum value

∼ε^1/2Ω^1/2∼ε^−1/2|logε|^−1/2to 0 in a region of width at mostO(ε^1/2Ω^−1/2|logε|^3/2) =O(ε^3/2|logε|²).

This yields an estimate for the kinetic energy ofgωin that region asO(ε^−5/2|logε|⁻³), i.e., approximately the same remainder as in (4.6), which is in any case much larger than the difference between the TF energies ˜E_ω^TF−E^TF (see (A.5)).

Proof of Proposition 4.1.

The proof of Proposition 2.1 applies to the functional ˜E_ω^gv as well by noticing that

|Bω(r)| ≤Ω sup

~r∈A

r⁻¹−r

+C|ω| ≤C ε⁻¹+|ω|

, (4.8)

which implies that the B_ω² term in the functional (see the second expression in (4.4)) is always smaller than the remainder in (2.2), provided |ω| ≤ O(ε^−5/4|logε|^−3/4). The Neumann condition at the inner boundary of Ais a direct consequence of the assumptionf ∈H¹(A).

Since the asymptotic behavior of the energy ˜E_ω^gv is the same as that of ˆE^GP (see (2.2)) for any

|ω| ≤ O(ε^−5/4|logε|^−3/4), most of the estimates proven for the density profile ghold true forgωas well, provided the phaseωsatisfies the estimate required in Proposition 4.1. We sum up such estimates in the following

Proposition 4.2 (Estimates for gω).

If Ω∼ε⁻²|logε|⁻¹ and|ω| ≤ O(ε^−5/4|logε|^−3/4)asε→0, g_ω²−ρ^TF

_L2(A)≤ O(ε^−1/4|logε|^−3/4), (4.9)

g_ω²( ˜R_m) =kg_ωk²_L∞(A)≤ ρ^TF

_L∞

(B)

1 +O(ε^1/4|logε|^−1/4)

. (4.10)

Moreover for any~r∈ Asuch that Rh+O(ε|logε|⁻¹)≤r≤max[ ˜Rm,1−ε^3/2|logε|²] g_ω²(r)−ρ^TF(r)

≤ O(ε^−3/4|logε|^−5/4)≤ O(ε^1/4|logε|^7/4)ρ^TF(r)ρ^TF(r), (4.11) and the maximum position R˜_m(ω)of g_ω satisfies the bounds

R˜m(ω)≥1− O(ε^9/8|logε|^7/8), kgωk²_L2(B\B( ˜Rm)) ≤ O(ε^1/8|logε|^1/8). (4.12) Finally for any~r such thatr≤Rh− O(ε^7/6),

g²_ω(r)≤Cε⁻¹|logε|⁻¹ expn

− c ε^1/6

o

. (4.13)

Proof. The results are proven exactly as the analogous statements contained in Propositions 2.3, 2.4, 2.5 and 2.6.