4. Asymptotic analysis of the minimizers
4.1. Localizing the singularities
ε−1ρ bj
xj+reiθ if 0< ρ < ε bj
xj+reiθ ifε≤ρ < r.
Since vε ∈ Hg1(Ω, Rk), the minimality of uε entails Eε(uε) ≤ Eε(vε). Computing Eε(vε) will be enough to conclude (2.10). We find
Eε(vε;Ω\ m j=1
Bj) = 1 2
Ω\(m
j=1Bj
|∇v|2=C,
Eε(vε;Bj)≤
Bj
Cε−2≤C,
and, passing to polar coordinates,
Eε(vε;Bj\Bε(xj)) = 1 2
r ε
dρ ρ
S1dω bj(ω)2≤ 1
4πλ(ηj)2|logε|+C.
Combining these bounds, with the help of (4.2) we conclude.
4.1. Localizing the singularities
In this subsection, we will prove Proposition1.2. Namely, we will show that the image ofuεlies close to the vacuum manifold, except on the union of a finite number of small balls. Analogous results have been established for the Ginzburg−Landau model in [6], in case the domainΩ ⊆R2 is star-shaped. This technical assumption has been removed in [7,36] (see also [4] for more details).
We introduce a (small) parameter 0< α≤1, whose value is going to be adjusted later, and we set eε(uε) = 1
2|∇uε|2+ 1 ε2f(uε).
We claim the following
Proposition 4.5. Letδ∈(0, δ0)be fixed. For allε >0, there exists a finite setXε={x1, . . . , xk} ⊂Ω, whose cardinality is bounded independently ofε, such that
dist(uε(x), N)≤δ if dist(x, Xε)> λ0ε, (4.3) whereλ0>0is a constant independent of ε, and
ε4αeε(uε)(x)≤Cα if dist(x, Xε)> εα. (4.4)
Proposition 4.5 clearly implies Proposition1.2. As a first step in the proof, we show that eε(uε) solves an elliptic inequality, in the regions whereuε lies close to the vacuum manifold.
Lemma 4.6. Assumeω ⊂Ω is an open set, such that dist(uε(x),N )≤δ holds for all x∈ω and all ε >0.
Then, eε(uε)solves pointwise inω the inequality
−Δeε(uε)≤Ce2ε(uε).
Proof. Reminding thatuεis a solution of equation (2.2), we compute plainly
−1
2Δ|∇uε|2=−∇(Δuε)· ∇uε−∇2uε2=−1
ε2∇uε: D2f(uε)∇uε−∇2uε2, where∇2uε2=
i, j|∂i∂juε|2, and
−1
ε2Δf(uε) =−1
ε2∇(Df(uε))· ∇uε− 1
ε2Df(uε)·Δuε
=−1
ε2∇uε: D2f(uε)∇uε− 1
ε4|Df(uε)|2. Adding these contributions, we obtain
−Δeε(uε) +∇2uε2+ 1
ε4|Df(uε)|2=−2
ε2∇uε: D2f(uε)∇uε. (4.5) Hypothesis H2 provides
1
ε4|Df(uε)|2≥m0
ε4 dist2(uε,N ). (4.6)
Moreover, the imageuε(ω) lies close toN by assumption, so the right-hand side of (4.5) can be estimated by the local Lipschitz continuity of D2f:
−2
ε2∇uε: D2f(uε)∇uε≤ −2
ε2∇uε: D2f(π(uε))∇uε+ 2
ε2D2f(uε)−D2f(π(uε))|∇uε|2
≤ −2
ε2∇uε: D2f(π(uε))∇uε+ C
ε2dist(uε,N )|∇uε|2
≤ C
ε2dist(uε,N )|∇uε|2.
For the latter inequality, remind that every pointp∈N is a minimizer forf, so D2f(p)≥0. We infer
−2
ε2∇uε: D2f(uε)∇uε≤m0
ε4 dist2(uε,N ) +C|∇uε|4
and the first term can be reabsorbed in the left-hand side of (4.5), by means of (4.6). This concludes the
proof.
Our next ingredient is a Clearing Out lemma, which relies crucially on H2.
Proposition 4.7 (Clearing Out). There exist some positive constants λ0 and μ0 with the following property:
for allx0∈Ω and alll∈[λ0ε,1], if the minimizer uε satisfies 1
ε2
B(x0,2l)∩Ω
f(uε)≤μ0 (4.7)
then
dist(uε(x), N)≤δ for allx∈Ω∩B(x0, l). (4.8)
Proof. Set
f0:= min{f(v) : dist(v,N )≥δ,|v| ≤1},
and remark thatf0>0, because it is the minimum of a strictly positive function on a compact set. We define λ0:= δ
2C, μ0:= π 2λ20min
f0, 1
8m0δ2
, (4.9)
whereCis a constant such that|∇uε| ≤Cε−1(such a constant exists, by Lem.4.1). We are going to check that this choice ofλ0, μ0works. To do so, we proceed by contradiction and assume there is some pointx∈B(x0, l) such that dist(uε(x), N )> δ. Firstly, we remark that this assumption implies dist(x, ∂Ω)> λ0ε. Indeed, if it were dist(x, ∂Ω)≤λ0εthen, in view of H4, we would have
dist(uε(x), N)≤ ∇uεL∞dist(x, ∂Ω)≤Cλ0=δ 2.
It follows that the ballB(x, λ0ε) is entirely contained inΩ∩B(x0,2l). In addition, for ally ∈B(x, λ0ε) we have
dist(uε(y), N)≥dist(uε(x),N )− |uε(x)−uε(y)|> δ−λ0ε∇uεL∞ ≥ δ 2. Due to Remark2.5and Lemma4.1, this implies
1 ε2
Ω∩B(x0,2l)f(uε)≥ 1 ε2
B(x, λ0ε)f(uε)≥πλ20min
f0, 1 8m0δ2
= 2μ0,
which contradicts the hypothesis (4.7).
The two following results can be found in Section IV.5 of [4]. The proofs carry over to our setting, without any change.
Lemma 4.8. Let x0∈Ω. There exists a constantCα, depending only on α,g andΩ, such that 1
ε2
B(x0, εα)∩Ω
f(uε)≤Cα.
Proposition 4.9. There exists a constant ηα > 0, independent of ε, with the following property: if a point
x0∈Ωverifies
B(x0,2εα)∩Ω
|∇uε|2≤ηα|logε|+C (4.10)
thendist(uε(x), N)≤δ for allx∈B(x0, εα).
Proposition4.9provides a concentration result for the energy, which will be crucial in our argument. Reducing, if necessary, the value ofηα, we are able to show another estimate for minimizers satisfying (4.10). This will be the final ingredient in our proof of Proposition4.5.
Proposition 4.10. There exist constantsηα, Cα>0 (withCαdepnding only onα, ηα) such that, ifuεverifies the condition (4.10)for somex0∈Ω, then
ε4αeε(uε)(x0)≤Cα.
Proof. We suppose, at first, thatB(x0, εα)⊂Ω. In view of Proposition4.9, we can assume that dist(uε,N )≤δ onB(x0, εα). Furthermore, (4.10) and Lemma4.8provide
Eε(uε, B(x0, εα)∩Ω)≤ηα|logε|+Cα. (4.11) We claim that there exists a radiusr∈(ε2α, εα) such that
∂B(x0, r)∩Ω
|∇uε|2+ 1 ε2f(uε)
≤2ηα
αr . (4.12)
Indeed, if (4.12) were false, integrating over (ε2α, εα) we would obtain Eε(uε, B(x0, εα)∩Ω)≥2ηα
α εα
ε2α
dr
r = 2ηα|logε|, which contradicts (4.11), forε1. Thus, the claim is established.
Set
:=ε/r and v(x) :=uε
x r +x0
forx∈B :=B(0,1).
As a consequence of the scaling, we deducee(v) =r2eε(uε), hence E(v, B) =Eε(uε, B(x0, r))
and v minimizes the energy E among the maps w ∈ H1(B,Rd), with w|∂B = v|∂B. Moreover, (4.12)
transforms into
∂B
e(v)≤2ηα
α · (4.13)
We will take advantage of the following property, whose proof is postponed.
Lemma 4.11. The energy ofv is controlled by ηα, that is,
B
e(v)≤Cαηα.
Recall also that, due to Lemma4.6,e(v) solves an elliptic inequality. Thus, we are in position to invoke a result by Chen and Struwe ([9] – the reader is also referred to [32], Thm. 2.2): provided thatηαis small enough, Lemma 4.11implies the estimate
r2eε(uε)(x0) =e(v)(x)≤
B
e(v)≤Cαηα. This concludes the proof, in caseB(x0, εα) does not intersect the boundary.
We still have to cover the case B(x0, εα)Ω, but this entail no significant change in the proof (nor in the proof of Lem.4.11). As we deal with a local result, we can straighten the boundary and assume thatΩcoincides locally with the set Rn+. In place of the Chen–Struwe result we can exploit ([32], Thm. 2.6), which deals with
the Dirichlet boundary condition.
Proof of Lemma 4.11. We split the proof in steps, for clarity.
Step 1(Construction of the harmonic extension). The compositionπ(v) is well defined, since the image ofv
lies close to the vacuum manifold. Setσ:= dist(v,N ) =|v−π(v)|, and denote byωan harmonic extension
of π(v)|∂B on B. The existence of such an extension is a classical result by Morrey (see, for instance, [26]).
Lemma 4.2 in [33] and (4.11) imply that of the boundary condition on∂D.
Step 2(An auxiliary map). It will be useful to introduce an auxiliary mapϕ. Using polar coordinates onD, we defineϕ by the formula
and check it by a straightforward computation. Indeed, the existence of an orthonormal frame of normal vectors, defined on some neighborhood of ω(x0). Then, the construction ofν is completed by a partition of the unity argument.
Step 4(Construction of a comparison map). Setω:=ω+ν. It follows from the previous steps thatωenjoys these properties:
ω|∂B= v|∂B,
π(ω(x)) =v(x) and dist(ω(x),N ) =ϕ(x) for allx∈B.
In particular,ω is an admissible comparison map forv. By this information and Lemma2.6, we infer a bound for the gradient ofω:
|∇ω|2≤(1 +Cϕ)|∇ω|2+C
|∇ϕ|2+ϕ2
Since|ϕ| ≤δby construction, integrating this inequality overD and exploiting (4.13) we obtain
∇ω2L2(D)≤(1 +δ)∇ω2L2(D)+C. (4.16) The potential energy ofω is estimated by means of Remark2.5:
1
Combining this inequality with (4.15) and (4.16), we deduce
With this estimate and (4.14), we complete the proof.
Having established all these preliminary results, Proposition4.5follows easily from a covering argument as, for instance, the one in [4] (see also [6], Chap. IV).
Proof of Proposition4.5. By Vitali covering lemma, we can find a finite family of points{yi}i∈I such that Ω⊆
i∈I
B(yi,3εα) and
B(yi, εα)∩B(yj, εα) =∅ ifi=j.
Letηα=ηα(α, δ) be given by Proposition4.10. DefineJεas the subset of indexesi∈Ifor which the inequality
B(yi,3εα)|∇uε|2≥ηα(|logε|+ 1) holds; then, by Lemma4.4, we have
ηα(|logε|+ 1) card(Jε)≤
j∈Jε
B(yi,3εα)|∇uε|2≤C(|logε|+ 1) (4.18) (to prove the last inequality, recall that there is a universal constantCsuch that each point ofΩis covered by at mostCballs of radius 3εα). It follows that card(Jε) is bounded independently ofε. Moreover, Propositions4.9 and4.10imply that
dist(uε(x), N)≤δ ifx∈B(yi,3εα) andi∈I\Jε
ε4αeε(uε)(x)≤Cα ifx∈B(yi, εα) andi∈I\Jε.
Now, let us fix an index i∈ Jε, and let us focus on B(yi,3εα). Being λ0 =λ0(δ) and μ0=μ0(δ) given by Proposition4.7, we consider a finite covering{B(xim, 3λ0ε) :m∈Λε, i}of B(yi,3εα), such that
B(xim, λ0ε)∩B(xin, λ0ε) =∅ ifm=n, and we define the set Lε, i of indexesm∈Λε, i such that
1 ε2
B(xim,3λ0ε)f(uε)> μ0. Sinceε−2
B(xi,3εα)f(uε) is controlled by Lemma4.8, we can bound the cardinality ofLε, i, independently ofε, exactly as in (4.18). By Proposition4.7, we have that dist(uε(x),N )≤δ ifx∈B(xim,3λ0ε) andm /∈Lε, i.
Combining all these facts, we conclude easily.
Denote by xε1, xε2, . . . , xεkε the elements of Xε. For any given sequence εn 0 we can extract a renamed subsequence, such thatkεn is independent ofn(say,kεn=N) and
xεin→Li for i∈ {1,2, . . . , N},
for some point Li ∈ Ω Some of the points Li might coincide; therefore, we relabel them as a1, a2, . . . , aN, withN ≤N, in such a way thatai=aj ifi=j.
For the time being, we cannot exclude the possibility that ai ∈ ∂Ω, for some index i. To deal with this difficulty, we enlarge a little the domain Ω and consider a smooth, bounded domainΩ ⊇Ω, with the same homotopy type as Ω – for instance, we can defineΩ as ar-neighborhood ofΩ, forr small enough. Also, we fix a smooth functiong:Ω\Ω→N , such thatg=g on∂Ω and∇gL2(Ω\Ω)≤CgH1(∂Ω). From now on, we extend systematically any functionv:Ω→N withv=gon∂Ω to a mapv:Ω →N , by settingv=gon Ω\Ω.