Local Minimizers of the Magnetic Ginzburg-Landau Functional with S¹-valued Order Parameter on the Boundary
It was shown in [L. Berlyand and V. Rybalko, Solution with Vortices of a Semi-Stiff Boundary Value Problem for the Ginzburg-Landau Equation, J. Eur. Math. Soc. 12 (2010), 1497{1531] that in doubly connected domains there exist local minimizers of the simplified Ginzburg-Landau functional with modulu...
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Rybalko, V. 2016-10-05T18:58:37Z 2016-10-05T18:58:37Z 2014 Local Minimizers of the Magnetic Ginzburg-Landau Functional with S¹-valued Order Parameter on the Boundary / V. Rybalko // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 1. — С. 134-151. — Бібліогр.: 23 назв. — англ. 1812-9471 DOI: 10.15407/mag10.01.134 MSC2000: 35A01, 35J20, 35Q56 https://nasplib.isofts.kiev.ua/handle/123456789/106788 It was shown in [L. Berlyand and V. Rybalko, Solution with Vortices of a Semi-Stiff Boundary Value Problem for the Ginzburg-Landau Equation, J. Eur. Math. Soc. 12 (2010), 1497{1531] that in doubly connected domains there exist local minimizers of the simplified Ginzburg-Landau functional with modulus one and prescribed degrees on the boundary, unlike global minimizers that typically do not exist. We generalize the results and techniques of the aforementioned paper to the case of the magnetic Ginzburg-Landau functional. В работе [L. Berlyand and V. Rybalko, Solution with Vortices of a Semi-Stiff Boundary Value Problem for the Ginzburg-Landau Equation, J. Eur. Math. Soc. 12 (2010), 1497-1531] было показано, что в двусвязных областях существуют локальные минимизанты упрощенного функционала Гинзбурга-Ландау, имеющие модуль один и заданные степени отображения на границе, в отличие от глобальных минимизантов, которые обычно не существуют. Результаты и методы упомянутой выше статьи обобщаются на случай "магнитного" функционала Гинзбурга-Ландау. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Local Minimizers of the Magnetic Ginzburg-Landau Functional with S¹-valued Order Parameter on the Boundary Article published earlier |
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Local Minimizers of the Magnetic Ginzburg-Landau Functional with S¹-valued Order Parameter on the Boundary |
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Local Minimizers of the Magnetic Ginzburg-Landau Functional with S¹-valued Order Parameter on the Boundary Rybalko, V. |
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Local Minimizers of the Magnetic Ginzburg-Landau Functional with S¹-valued Order Parameter on the Boundary |
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Local Minimizers of the Magnetic Ginzburg-Landau Functional with S¹-valued Order Parameter on the Boundary |
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Local Minimizers of the Magnetic Ginzburg-Landau Functional with S¹-valued Order Parameter on the Boundary |
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Local Minimizers of the Magnetic Ginzburg-Landau Functional with S¹-valued Order Parameter on the Boundary |
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local minimizers of the magnetic ginzburg-landau functional with s¹-valued order parameter on the boundary |
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Rybalko, V. |
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Rybalko, V. |
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2014 |
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Журнал математической физики, анализа, геометрии |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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It was shown in [L. Berlyand and V. Rybalko, Solution with Vortices of a Semi-Stiff Boundary Value Problem for the Ginzburg-Landau Equation, J. Eur. Math. Soc. 12 (2010), 1497{1531] that in doubly connected domains there exist local minimizers of the simplified Ginzburg-Landau functional with modulus one and prescribed degrees on the boundary, unlike global minimizers that typically do not exist. We generalize the results and techniques of the aforementioned paper to the case of the magnetic Ginzburg-Landau functional.
В работе [L. Berlyand and V. Rybalko, Solution with Vortices of a Semi-Stiff Boundary Value Problem for the Ginzburg-Landau Equation, J. Eur. Math. Soc. 12 (2010), 1497-1531] было показано, что в двусвязных областях существуют локальные минимизанты упрощенного функционала Гинзбурга-Ландау, имеющие модуль один и заданные степени отображения на границе, в отличие от глобальных минимизантов, которые обычно не существуют. Результаты и методы упомянутой выше статьи обобщаются на случай "магнитного" функционала Гинзбурга-Ландау.
|
| issn |
1812-9471 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/106788 |
| citation_txt |
Local Minimizers of the Magnetic Ginzburg-Landau Functional with S¹-valued Order Parameter on the Boundary / V. Rybalko // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 1. — С. 134-151. — Бібліогр.: 23 назв. — англ. |
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AT rybalkov localminimizersofthemagneticginzburglandaufunctionalwiths1valuedorderparameterontheboundary |
| first_indexed |
2025-11-25T06:28:37Z |
| last_indexed |
2025-11-25T06:28:37Z |
| _version_ |
1850509511078445056 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2014, vol. 10, No. 1, pp. 134–151
Local Minimizers of the Magnetic Ginzburg–Landau
Functional with S1-valued Order Parameter on the
Boundary
V. Rybalko
B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail: vrybalko@ilt.kharkov.ua
Received February 15, 2013, revised July 17, 2013
It was shown in [L. Berlyand and V. Rybalko, Solution with Vortices of
a Semi-Stiff Boundary Value Problem for the Ginzburg–Landau Equation,
J. Eur. Math. Soc. 12 (2010), 1497–1531] that in doubly connected domains
there exist local minimizers of the simplified Ginzburg–Landau functional
with modulus one and prescribed degrees on the boundary, unlike global mi-
nimizers that typically do not exist. We generalize the results and techniques
of the aforementioned paper to the case of the magnetic Ginzburg–Landau
functional.
Key words: superconductivity, Ginzburg–Landau functional, variational
problems with lack of compactness.
Mathematics Subject Classification 2010: 35A01, 35J20, 35Q56.
1. Introduction
The purpose of the paper is to generalize the results and methods of [6] (see
also [13]) to the magnetic Ginzburg–Landau model. We consider the functional
Fλ[u, A] =
1
2
∫
G
(
|∇u− iAu|2 +
λ
4
(|u|2 − 1)2
)
dx +
1
2
∫
Ω
(curlA)2 dx, (1.1)
where G ⊂ R2 is a bounded multiply connected domain and Ω is the smallest
simply connected domain containing G, u ∈ H1(G;C) is the order parameter,
A ∈ H1(Ω;R2) is the vector potential of the induced magnetic field, and λ > 0
is the coupling constant (
√
λ/2 is the Ginzburg–Landau parameter). We assume
c© V. Rybalko, 2014
Local Minimizers of the Magnetic Ginzburg–Landau Functional
that the order parameter u takes values in S1 on the boundary ∂G and study
critical points of the functional Fλ[u,A] in the space
(u,A) ∈ J = {u ∈ H1(G;C); |u| = 1 a.e. on ∂G} ×H1(Ω;R2). (1.2)
For the sake of simplicity, we will consider doubly connected domains G,
G = Ω \ ω, where ω, Ω are smooth simply connected domains (ω ⊂ Ω).
However, the results of the paper can be easily extended for general multiply
connected domains.
Since the external magnetic field is zero in (1.1), global minimizers of the
functional are trivial (up to a gauge transformation u ≡ const ∈ S1, A ≡ 0),
and we are interested in finding nontrivial local minimizers. A way to produce
these nontrivial local minimizers is to minimize the functional in the class of pairs
(u,A) with prescribed topological degree of u on the connected components of
the boundary. Recall that, given a simple closed curve γ, the topological degree
(winding number) of a map u ∈ H1/2(γ;S1) is an integer given by the classical
formula (cf., e.g., [8])
deg(u, γ) =
1
2π
∫
γ
u ∧ ∂u
∂τ
ds,
where the integral is understood via H1/2 − H−1/2 duality, and ∂
∂τ is the tan-
gential derivative with respect to the counterclockwise orientation of γ. The
functional deg( · , γ) : H1/2(γ; S1) → R is continuous with respect to the strong
H1/2-convergence. Therefore, for any prescribed p, q ∈ Z minimizers of Fλ[u,A]
over the set
Jpq = {(u, A) ∈ J ; deg(u, ∂ω) = p, deg(u, ∂Ω) = q}
are local minimizers of Fλ[u,A] in J . The problem however is that, in general,
global minimizers of Fλ[u,A] in Jpq do not exist ([4], see also [1],[3],[6]) because
of the lack of continuity of deg( · , γ) : H1/2(γ;S1) → R with respect to the weak
H1/2-convergence.
To construct local minimizers of Fλ[u, A] in J , we consider the constrained
minimization problem
mλ(p, q, d) := inf{Fλ[u,A]; (u,A) ∈ J (d)
pq }, (1.3)
where p, q and d are given integers,
J (d)
pq = {(u,A) ∈ Jpq; d− 1/2 ≤ Φ(u,A, V0) ≤ d + 1/2}, (1.4)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1 135
V. Rybalko
and Φ(u,A, V0) is given by
Φ(u,A, V0) =
1
2π
∫
G
(
u∧
(( ∂u
∂x2
− iA2u
)∂V0
∂x1
−( ∂u
∂x1
− iA1u
)∂V0
∂x2
)
+A ·∇⊥V0
)
dx,
(1.5)
with V0 being the unique solution of the boundary value problem
−∆V0 + V0 = 1 in G
V0 = 1 on ∂Ω
V0 = 0 on ∂ω.
(1.6)
The functional Φ(u,A, V ) (with harmonic V and A ≡ 0) was introduced in [6]
to find nontrivial local minimizers of the simplified Ginzburg–Landau functional
with prescribed degrees. Here we will make use of Φ(u,A, V ) for various smooth
functions V . Note that the functional Φ(u,A, V0) has the following properties
(cf. [6], Section 3):
(a) for every u ∈ H1(G; S1) and A ∈ H1(Ω;R2) we have
Φ(u,A, V0) = deg(u, ∂ω) = deg(u, ∂Ω);
(b) Φ( · , V0) : H1(G;C) × H1(Ω;R2) → R is continuous with respect to the
weak convergence.
Moreover, if we fix Λ > 0 and consider pairs (u,A) from the sublevel set
Fλ[u,A] ≤ Λ, then Φ(u,A, V0) is never half-integer for sufficiently large λ > 0.
More precisely, we prove (see Section )
Proposition 1. Fix Λ > 0. There exists λ0 = λ0(Λ) > 0 such that if λ ≥ λ0,
then for every integer d and every (u,A) ∈ H1(G;C) × H1(Ω;R2) satisfying
Fλ[u,A] ≤ Λ the closed constraint Φ(u,A, V0) ∈ [d− 1/2, d + 1/2] is equivalent to
an open one, that is,
d− 1/2 ≤ Φ(u, A, V0) ≤ d + 1/2 ⇐⇒ d− 1/2 < Φ(u, A, V0) < d + 1/2. (1.7)
Actually, it will be shown that Φ(u,A, V0) is close to integers uniformly in
(u,A) satisfying Fλ[u,A] ≤ Λ when λ is sufficiently large.
It follows from Proposition 1 that if
• the infimum mλ(p, q, d) in (1.3) is attained and (u,A) is a minimizer,
• mλ(p, q, d) < Λ and λ ≥ λ0, where λ0(= λ0(Λ)) is as in Proposition 1,
136 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
Local Minimizers of the Magnetic Ginzburg–Landau Functional
then (u,A) is a local minimizer of Fλ[u, A] in J . However the attainability of
mλ(p, q, d) is still a nontrivial question due to the lack of continuity of deg(u, ∂Ω)
and deg(u, ∂ω) with respect to the weak H1-convergence.
The main result of the paper is
Theorem 2. For any integers p, q and d > 0 (d < 0) with d ≥ max{p, q}
(d ≤ min{p, q}) there exists λ1 = λ1(p, q, d) > 0 such that the infimum in (1.3)
is always attained when λ ≥ λ1 and any minimizer of (1.3) is a local minimizer
of Fλ[u,A] in J .
Throughout the paper we will assume d > 0 (the case d < 0 follows by
taking (u,−A) instead of (u, A)). As already mentioned, the principal part of
Theorem 2 is to prove the attainability of the infimum in (1.3). To show this we
follow essentially the same scheme as in [6]; namely, we first prove the attainability
of the infimum in (1.3) for p = q = d and argue by induction, that is, pass to
p = d, q = d − 1 and p = d − 1, q = d, etc. The main technical result we use
to pass from the prescribed degrees p, q to p, q − 1 and p− 1, q is the following
strict inequalities:
mλ(p, q − 1, d) < mλ(p, q, d) + π and mλ(p− 1, q, d) < mλ(p, q, d) + π (1.8)
that hold provided that mλ(p, q, d) is attained and λ is sufficiently large. Bounds
(1.8) are proved by constructing testing pairs (u,A) ∈ J (d)
p(q−1) or (u,A) ∈ J (d)
(p−1)q
with Fλ[u,A] < mλ(p, q, d) + π, and this is the key point of the present paper.
The construction of testing pairs completely differs from that of [6] and makes use
of Bogomol’nyi’s representation of the functional [8] and the factorization idea of
C. Taubes [23].
There is a vast body of literature on 2D Ginzburg–Landau type problems.
Local minimizers of the simplified Ginzburg–Landau functional with the Dirichlet
condition on the boundary were studied in [11, 16], more general results were
obtained in [12]. Aforementioned works deal with large values of the Ginzburg–
Landau parameter and essentially rely on the results of the pioneering work [7];
more specifically, they use the reduction to the renormalized energy functional
introduced in [7].
Local minimizers with the Neumann boundary condition in multiply con-
nected domains are related to the phenomenon of permanent currents, see, e.g.,
[18, 14]. However, it is not known whether these local minimizers can have zeros
(vortices) in both simplified 2D model and magnetic Ginzburg–Landau 2D model
with zero external field. One can find some results on nonexistence of local min-
imizers with vortices in [15] and [21]. In the case of nonzero external magnetic
field, global minimizers and local ones can have vortices ([10, 19, 20, 22], see also
references therein).
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1 137
V. Rybalko
While the idea of local minimization is general, its implementation depends
strongly on the concrete problem one is dealing with. The distinguishing feature
of the problem studied in the present paper is its being a variational problem
with a possible lack of compactness. Also, the local minimizers obtained in
Theorem 2 have nonstandard behavior, their zeros are situated near the boundary
and approach it as λ → +∞ (this can be shown quite similarly to [6]).
In the paper, we use the following notation and conventions:
• Every closed curve is counterclockwise oriented. For such a curve, τ and ν
stand for the unit tangent and unit normal vectors, respectively, that agree
with the orientation ((ν, τ) is direct);
• The complex plane C is identified with R2 such that if x, y ∈ C, then
(x, y) = 1
2(xȳ + yx̄) and x ∧ y = i
2(xȳ − yx̄) are the scalar and the wedge
products, respectively;
• Given a fixed orthonormal frame (x1, x2) in R2,
∂
∂z
=
1
2
( ∂
∂x1
− i
∂
∂x2
)
and
∂
∂z̄
=
1
2
( ∂
∂x1
+ i
∂
∂x2
)
denote the classical Cauchy operators. For a
scalar (real-valued) function f , ∇⊥f is the vector field given by ∇⊥f =
(−∂f/∂x2, ∂f/∂x1).
2. Critical Points of Fλ[u,A] and Bogomol’nyi’s Representation
One of the main properties of the functional Fλ[u,A] is its invariance under
the gauge transformations u 7→ eiφu, A 7→ A +∇φ (where φ ∈ H2(Ω)). It is easy
to see that Φ(u,A, V0) also has the aforementioned property as well as deg(u, ∂Ω)
and deg(u, ∂ω). Thus, without loss of generality, we can assume that A is in the
Coulomb gauge, i.e., {
divA = 0 in Ω
A · ν = 0 on ∂Ω.
(2.1)
The critical points of Fλ[u,A] in J , in particular, the local minimizers, are
the solutions of the system of Euler–Lagrange equations
−(∇− iA)2u +
λ
2
u(|u|2 − 1) = 0 in G, (2.2)
−∇⊥h =
{
j in G
0 in ω,
(2.3)
where h = curl A is the magnetic field (scalar function), and
j = (iu,∇u− iAu)
138 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
Local Minimizers of the Magnetic Ginzburg–Landau Functional
is the current (note that h and j are gauge invariant). Additionally, h ∈ H1(Ω)
and the boundary conditions
|u| = 1, j · ν = 0 on ∂G, h = 0 on ∂Ω,
∂h
∂τ
= 0 on ∂ω (2.4)
are satisfied. We will assume that ∂G ∈ C∞, then every solution (u,A) of (2.2)–
(2.4) satisfies u ∈ C∞(Ḡ;C) and A ∈ C∞(Ḡ;R2)(see [5]). We also have the
pointwise inequality
|u| ≤ 1 in G
which is a consequence of the maximum principle applied to the equation for |u|2,
∆|u|2 = λ|u|2(|u|2 − 1) + 2|∇u− iAu|2 in G. (2.5)
Note that the equation −∇⊥h = j can be written either in the form
∂
∂z
(
h− 1
2
(|u|2 − 1)
)
= −u
(∂u
∂z
− A2 + iA1
2
u
)
or
∂
∂z̄
(
h +
1
2
(|u|2 − 1)
)
= u
(∂u
∂z̄
+
A2 − iA1
2
u
)
.
Then, taking ∂/∂z̄ (or ∂/∂z) and using (2.5), we get
∆
(
h− 1
2
(|u|2− 1)
)− |u|2h = −4
∣∣∂u
∂z
− A2 + iA1
2
u
∣∣2− λ
2
|u|2(|u|2− 1) in G (2.6)
and
∆
(
h +
1
2
(|u|2 − 1)
)− |u|2h = 4
∣∣∂u
∂z̄
+
A2 − iA1
2
u
∣∣2 +
λ
2
|u|2(|u|2 − 1) in G. (2.7)
The representation valid for every (u, A) ∈ J ,
Fλ[u,A] = ±π(deg(u, ∂Ω)− deg(u, ∂ω)) + F±[u,A]
+
1
2
∫
ω
|curlA|2dx +
λ− 1
8
∫
G
(|u|2 − 1)2dx, (2.8)
where
F+[u,A] = 2
∫
G
∣∣∂u
∂z̄
+
A2 − iA1
2
u
∣∣2dx +
1
2
∫
G
∣∣curlA +
|u|2 − 1
2
∣∣2dx, (2.9)
F−[u, A] = 2
∫
G
∣∣∂u
∂z
− A2 + iA1
2
u
∣∣2dx +
1
2
∫
G
∣∣curlA− |u|2 − 1
2
∣∣2dx, (2.10)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1 139
V. Rybalko
plays an important role in the analysis of the functional Fλ[u,A]. This represen-
tation is due to a remarkable observation of E.B. Bogomol’nyi [8], for a detailed
derivation of (2.8) we refer to [9].
3. Properties of Functional Φ(u,A, V )
Let us rewrite Φ(u, A, V ) as the sum of two terms
Φ(u, A, V ) =
1
2π
∫
G
u ∧
( ∂u
∂x2
∂V
∂x1
− ∂u
∂x1
∂V
∂x2
)
dx +
1
2π
∫
G
A · ∇⊥V (1− |u|2) dx.
(3.1)
Then, using the results of [6] (Section 3) for the first term, we get that for every
fixed V ∈ C1(G) the functional Φ( · , V ) is continuous with respect to the weak
convergence in H1(G;C) × H1(Ω;R2). If, in addition, V is such that V = 0
on ∂ω and V = 1 on ∂Ω, then Φ(u,A, V ) = deg(u, ∂ω) = deg(u, ∂Ω) for every
u ∈ H1(G; S1).
Fix now Λ > 0. Let us show that
sup {dist(Φ(u,A, V0),Z); Fλ[u,A] ≤ Λ} → 0 as λ →∞. (3.2)
Since Φ(u, A, V0) is invariant under gauge transformations, we can always assume
(2.1). Then, given a sequence of pairs (uλ, Aλ), λ → ∞, satisfying the bound
Fλ[uλ, Aλ] ≤ Λ, we can extract a subsequence converging weakly to a limit (u,A).
Moreover, since ∫
G
(1− |uλ|2)2 dx ≤ 8Λ/λ → 0,
we have u ∈ H1(G;S1). Thus Φ(u, A, V0) ∈ Z, at the same time |Φ(u,A, V0) −
Φ(uλ, Aλ, V0)| → 0 as λ → ∞, thanks to the continuity of Φ( · , V0) with respect
to the weak convergence. So far we have proven (3.2), which in turn implies
Proposition 1.
4. Construction of Testing Pairs
As mentioned in Introduction, the main technical point in the proof of The-
orem 2 is to show strict inequalities (1.8). In this Section we provide a detailed
construction of testing pairs with energy control and prove (1.8) with their help.
Given a local minimizer (u,A) ∈ J (d)
pq of Fλ[u,A] in J , we construct (w(ξ), B(ξ))
∈ J (d)
p(q−1) in the form
w(ξ) = uaeφ/2, B(ξ) =
{
A + 1
2∇⊥φ, in G,
A +∇⊥θ +∇χ in ω,
(4.1)
140 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
Local Minimizers of the Magnetic Ginzburg–Landau Functional
where a is a conformal map from Ω onto the unit disk with zero at ξ ∈ G, and
φ ∈ H2(G), θ, χ ∈ H2(ω) are scalar functions. We assume that φ, θ, χ depend
on the parameter ξ omitted for brevity. In order to satisfy |w(ξ)| = 1 on ∂G, we
have the following conditions:
φ = 0 on ∂Ω, φ = −2 log |a| on ∂ω. (4.2)
Since the (exactly one) zero ξ of a lies in G, we have
deg(w(ξ), ∂ω) = p, deg(w(ξ), ∂Ω) = q − 1. (4.3)
We calculate Fλ[w(ξ), B(ξ)] using (2.8), (2.10),
Fλ[w(ξ), B(ξ)] =π + Fλ[u,A] + 2
∫
G
∣∣∂u
∂z
− A2 + iA1
2
u
∣∣2(|a|2eφ − 1) dx
+
1
2
∫
G
(
v
(
∆φ− |u|2(|a|2eφ − 1)
)
+
1
4
(
∆φ− |u|2(|a|2eφ − 1)
)2
)
dx
+
λ− 1
8
∫
G
(
(|u|2|a|2eφ − 1)2 − (|u|2 − 1)2
)
dx
+
1
2
∫
ω
(
(∆θ + curl A)2 − (curlA)2
)
dx,
(4.4)
where v = curl A− (|u|2 − 1)/2. Then we expand the integrands in the two last
terms of (4.4) and use the pointwise equality
4
∣∣∂u
∂z
− A2 + iA1
2
u
∣∣2(|a|2eφ − 1)
= (−∆v + |u|2v)(|a|2eφ − 1)− λ− 1
2
|u|2(|u|2 − 1)(|a|2eφ − 1)
(cf. (2.6)) to get
Fλ[w(ξ), B(ξ)] =π + Fλ[u,A] +
1
2
∫
G
(−∆v + |u|2v)(|a|2eφ − 1) dx
+
1
2
∫
G
(
v
(
∆φ− |u|2(|a|2eφ − 1)
)
+
1
4
(∆φ− |u|2(|a|2eφ − 1)
)2
)
dx
+
λ− 1
8
∫
G
|u|4(|a|2eφ − 1
)2 dx +
1
2
∫
ω
(
(∆θ)2 + 2∆θ curlA
)
dx.
(4.5)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1 141
V. Rybalko
Set φ ∈ H2(G) to be the unique solution of the equation
−∆φ + |u|2(|a|2eφ − 1) = 0 in G (4.6)
subject to the boundary conditions (4.2) (this problem has the unique solution
φ ∈ H2(G), see, e.g., [9], Theorem 4.3), then (4.5) simplifies to
Fλ[w(ξ), B(ξ)] =π + Fλ[u,A] +
1
2
∫
G
(−∆v + |u|2v)(|a|2eφ − 1) dx
+
λ− 1
8
∫
G
|u|4(|a|2eφ − 1)2 dx +
1
2
∫
ω
(
(∆θ)2 + 2∆θ curlA
)
dx.
(4.7)
We set the requirement that
∂θ
∂ν
=
1
2
∂φ
∂ν
on ∂ω, (4.8)
which leads, after integrating by parts in (4.7), to
Fλ[w(ξ), B(ξ)] =π + Fλ[u,A] +
1
2
∫
G
v(−∆(|a|2eφ) + |u|2(|a|2eφ − 1)) dx
+
λ− 1
8
∫
G
|u|4(|a|2eφ − 1)2 dx +
1
2
∫
ω
(∆θ)2 dx,
(4.9)
where we have also used the facts that v = curl A on ∂ω, curl A = const in ω and
∫
∂ω
∂|a|2
∂ν
ds
|a|2 =
∫
ω
∆log |a|2 dx = 0
(a is a holomorphic function without zeros in ω).
Now set θ to be a solution of the equation
∆θ =
1
2|ω|
∫
∂ω
∂φ
∂ν
ds in ω
subject to the boundary condition (4.8). In order to have B(ξ) ∈ H1(Ω;R2), we
define χ ∈ H2(G) as a function satisfying the boundary conditions χ = 0 on ∂ω,
∂χ
∂ν = −1
2
∂φ
∂τ + ∂θ
∂τ on ∂ω (for definiteness, we can assume ∆2χ = 0 in ω). Thus,
for every ξ ∈ G we have (w(ξ), B(ξ)) ∈ Jp(q−1), and (4.9) yields
Fλ[w(ξ), B(ξ)] = π + Fλ[u,A] +
1
2
∫
G
v(−∆(|a|2eφ) + |u|2(|a|2eφ − 1)) dx
+
λ− 1
8
∫
G
|u|4(|a|2eφ − 1)2 dx +
1
8|ω|
(∫
∂ω
∂φ
∂ν
ds
)2
. (4.10)
142 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
Local Minimizers of the Magnetic Ginzburg–Landau Functional
Next we show that Fλ[w(ξ), B(ξ)] < π +Fλ[u,A] when ξ is sufficiently close to
∂Ω. To this end, we first study the asymptotic behavior of the conformal map a
and the solution of φ of problem (4.2), (4.6) as ξ → ∂Ω.
Lemma 3. Let ξ̃ be the nearest point projection of ξ ∈ G on ∂Ω and let
δ = |ξ − ξ̃| be sufficiently small. Then
(i) |∂a
∂z (x)| ≤ Cδ/(δ + |x− ξ̃|)2 and ‖1− |a|2‖L2(G) ≤ Cδ| log δ|1/2;
(ii) ‖φ‖W 2,r(G) ≤ C(r)δ for every 1 < r < 2 and ‖φ‖W 2,2(G) ≤ Cδ| log δ|1/2,
where a is the unique (up to a constant factor with modulus one) conformal map
from Ω onto the unit disk with prescribed zero at the point ξ, and φ is the unique
solution of problem (4.2), (4.6).
P r o o f. To see (i), we should note that a can be written as a = σ(F(x)−
F(ξ))/(1 − F(ξ)F(x)), where σ ∈ S1 is a constant and F is a fixed conformal
map from Ω onto the unit disk. Then the proof of the first bound in (i) is
straightforward. The second bound in (i) is shown in Section 8 of [4]. In the
proof of (ii) we also follow the lines of Section 8 from [4] (the main ingredient
here is the chain of pointwise inequalities 0 ≤ 1− |a|2eφ ≤ 1− |a|2 that follow by
the maximum principle; details are left to the reader).
It follows from Lemma 3 that
Fλ[w(ξ), B(ξ)] ≤ π+Fλ[u,A]+
1
2
∫
G
v(−∆(|a|2eφ)+|u|2(|a|2eφ−1)) dx+O(δ2| log δ|).
(4.11)
Observe that
∆(|a|2eφ)− |u|2(|a|2eφ − 1)
= |u|2(|a|2eφ − 1)2 + 4
∣∣∣∂a
∂z
∣∣∣
2
eφ + 4a
∂a
∂z
eφ ∂φ
∂z
+ 4a
∂a
∂z
eφ ∂φ
∂z
+ |a|2eφ|∇φ|2,
where we have used (4.6). Then we make use of Lemma 3 and the fact that v = 0
on ∂Ω to get, after routine calculations,
∫
G
v(∆(|a|2eφ)− |u|2(|a|2eφ − 1))dx = 4
∫
G
v
∣∣∣∂a
∂z
∣∣∣
2
dx + O(δ2| log δ|). (4.12)
Taking ṽ = ∂v
∂ν (ξ̃)ν(ξ̃)(x − ξ̃) in place of v in the right-hand side of (4.12) (note
that |v − ṽ| ≤ C|x− ξ̃|2), we obtain
4
∫
G
v
∣∣∣∂a
∂z
∣∣∣
2
dx = 4
∫
Ω
ṽ
∣∣∣∂a
∂z
∣∣∣
2
dx + O(δ2| log δ|). (4.13)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1 143
V. Rybalko
Finally, since ∆ log |a|2 = 4πδξ(x) in Ω (where δξ(x) stands for the Dirac delta
centered at ξ) and |a|2 = 1 on ∂Ω while ∆ṽ = 0 in Ω, we have
4
∫
Ω
ṽ
∣∣∣∂a
∂z
∣∣∣
2
dx =
∫
Ω
ṽ∆|a|2 dx =
∫
∂Ω
ṽ
∂|a|2
∂ν
ds =
∫
∂Ω
ṽ
∂ log |a|2
∂ν
ds = 4πṽ(ξ).
(4.14)
We combine (4.11)–(4.14) to get
Fλ[w(ξ), B(ξ)] ≤ π + Fλ[u,A] + 2π
∂v
∂ν
(ξ̃)δ + O(δ2| log δ|), (4.15)
where ξ̃ is the nearest point projection of ξ on ∂Ω. It is clear now that (4.15) yields
the required inequality Fλ[w(ξ), B(ξ)] < π + Fλ[u,A] when δ is sufficiently small,
provided that there is ξ̃ on ∂Ω where ∂v
∂ν < 0. The following result establishes
the existence of the point ξ̃ for large λ.
Lemma 4. Let (u(λ), A(λ)) be a local minimizer of Fλ[u,A] in J . Assume
that (u(λ), A(λ)) ∈ J (d) =
⋃
p,q∈Z J (d)
p,q (where d is a fixed positive integer number)
and Fλ[u(λ), A(λ)] ≤ Λ for some fixed Λ > 0. Then h(λ) = curlA(λ) satisfies
∂h(λ)
∂ν
(ξ(λ)) < 0 for some ξ(λ) ∈ ∂Ω
when λ is sufficiently large, λ ≥ λ2(Λ).
P r o o f. Assume by contradiction that ∂h(λ)
∂ν ≥ 0 on ∂Ω for a sequence
λ = λk, λk →∞.
Since divA(λ) = 0 in Ω and A(λ) · ν = 0 on ∂Ω, we have ‖A(λ)‖H1(Ω;R2) ≤
C‖h‖L2(Ω). It follows that ‖A(λ)‖H1(Ω;R2) ≤ C thanks to the bound Fλ[u(λ), A(λ)]
≤ Λ. These two bounds imply that ‖u(λ)‖H1(G;C) ≤ C, where C is independent
of λ. Therefore, up to extracting a subsequence, (u(λ), A(λ)) → (u,A) weakly in
H1(G;C)×H1(Ω;R2) as λ →∞, and |u| = 1 a.e. in G (since ‖|u(λ)|2−1‖2
L2(G) ≤
8Λ/λ → 0).
It follows from (2.3) that
−∆h(λ) + h(λ) = 2
∂u(λ)
∂x1
∧ ∂u(λ)
∂x2
+ curl
(
(1− |u(λ)|2)A(λ)
)
in G. (4.16)
We also have h(λ) = 0 on ∂Ω and h(λ) = const in ω. Let V ∈ C1(Ω) be the unique
solution of the equation
{
−∆V + V = 0 in G,
V = g on ∂Ω, V = 0 on ∂ω,
(4.17)
144 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
Local Minimizers of the Magnetic Ginzburg–Landau Functional
where g ∈ C1(∂Ω) is some nonnegative function. Multiply (4.16) by V and
integrate on G to get, after integrating by parts,
−
∫
∂Ω
∂h(λ)
∂ν
g ds =
∫
∂ω
∂V
∂ν
h(λ) ds +
∫
∂Ω
u(λ) ∧ ∂u(λ)
∂τ
g ds− 2πΦ(u(λ), A(λ), V ).
We know that u(λ) → u weakly in H1/2(∂Ω; S1). Due to the result from [17] (see
Lemma 3.2 therein) there is a subsequence such that for every g ∈ C1(∂Ω),
∫
∂Ω
u(λ) ∧ ∂u(λ)
∂τ
g ds →
∫
∂Ω
u ∧ ∂u
∂τ
g ds + 2π
∑
finite
Dkg(αk),
where the points αk ∈ ∂Ω and integers Dk are independent of g. Now choose a
nonnegative function g 6≡ 0 such that g(αk) = 0 for every αk. Then we have
∫
∂Ω
u(λ) ∧ ∂u(λ)
∂τ
g ds− 2πΦ(u(λ), A(λ), V ) →
∫
∂Ω
u ∧ ∂u
∂τ
g ds− 2πΦ(u,A, V )
= 2
∫
G
∂u
∂x1
∧ ∂u
∂x2
V dx = 0,
where we have used the continuity of Φ( · , V ) with respect to the weak conver-
gence in H1(G;C)×H1(Ω;R2) and the pointwise equality ∂u
∂x1
∧ ∂u
∂x2
= 0 a.e. in
G (which holds for every u ∈ H1(G;S1)). Thus,
0 ≤
∫
∂Ω
∂h(λ)
∂ν
g ds = −
∫
∂ω
∂V
∂ν
h(λ) ds + o(1) when λ →∞.
On the other hand, since g ≥ 0 and g 6≡ 0, by the strong maximum principle
and Hopf’s boundary lemma applied to (4.17) we have ∂V
∂ν > 0 on ∂ω. Therefore
we will have a contradiction, if we show that
∫
∂ω h(λ) ds ≥ c > 0 for sufficiently
large λ.
To this end, note that in view of (2.3),
2πΦ(u(λ), A(λ), V0) = −
∫
G
∇h(λ) · ∇V0 dx +
∫
G
A(λ) · ∇⊥V0 dx
=
∫
∂ω
∂V0
∂ν
h(λ) ds +
∫
G
h(λ)∆V0 dx +
∫
G
A(λ) · ∇⊥V0 dx.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1 145
V. Rybalko
Since V0 solves (1.6), we have
2πΦ(u(λ), A(λ), V0) =
∫
∂ω
∂V0
∂ν
h(λ) ds +
∫
G
h(λ)(V0 − 1) dx +
∫
G
A(λ) · ∇⊥V0 dx
=
∫
∂ω
∂V0
∂ν
h(λ) ds +
∫
G
curl
(
(V0 − 1)A(λ)
)
dx. (4.18)
Using Stokes’ theorem twice, we get
∫
G
curl
(
(V0 − 1)A(λ)
)
dx =
∫
∂ω
A(λ) · τ ds =
∫
ω
h(λ) dx =
|ω|
|∂ω|
∫
∂ω
h(λ) ds,
while
∫
∂ω
∂V0
∂ν
h(λ) ds =
∫
∂ω
∂V0
∂ν
ds
∫
∂ω
h(λ) ds
|∂ω| =
∫
G
(|∇V0|2 + (V0 − 1)2
)
dx
∫
∂ω
h(λ) ds
|∂ω| .
Thus, passing to the limit in (4.18) yields
lim
λ→∞
∫
∂ω
h(λ) ds = 2πd|∂ω|/
(
|ω|+
∫
G
(|∇V0|2 + (V0 − 1)2
)
dx
)
> 0,
where we have used the fact that Φ(u(λ), A(λ), V0) → d as λ →∞.
R e m a r k 5. In exactly the same way as in Lemma 4, one can show that
∂h(λ)
∂ν > 0 at a point on ∂ω for sufficiently large λ.
Corollary 6. If ξ(λ) is as in Lemma 4, then ∂v(λ)
∂ν (ξ(λ)) < 0, where v(λ) =
curlA(λ) − (|u(λ)|2 − 1)/2.
P r o o f. Since |u(λ)| = 1 on ∂G and |u(λ)| ≤ 1 in G, we have ∂|u(λ)|2
∂ν (ξ(λ)) ≥ 0.
Now, if we take ξ ∈ G sufficiently close to the point ξ(λ), where ∂v(λ)
∂ν (ξ(λ)) < 0,
by (4.15) and Corollary 6, we have Fλ[w(ξ), B(ξ)] < π + Fλ[u, A]. On the other
hand, (w(ξ), B(ξ)) ∈ Jp(q−1) and (w(ξ), B(ξ)) converges weakly to (γu, A) (γ =
const ∈ S1) as ξ → ∂Ω, up to a subsequence. Consequently, Φ(w(ξ), B(ξ), V0) →
Φ(u,A, V0). Thus, if d− 1/2 < Φ(u,A, V0) < d + 1/2, then (w(ξ), B(ξ)) ∈ J (d)
p(q−1),
when ξ is sufficiently close to ξ(λ). Quite similarly, we can show that there exists
a testing pair from J (d)
(p−1)q whose Ginzburg–Landau energy is strictly less than
π + Fλ[u,A]. These results are summarized in
146 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
Local Minimizers of the Magnetic Ginzburg–Landau Functional
Lemma 7. Given integers d > 0, p and q, if mλ(p, q, d) is attained and λ
is sufficiently large, λ ≥ λ3(p, q, d), then mλ(p − 1, q, d) < mλ(p, q, d) + π and
mλ(p, q − 1, d) < mλ(p, q, d) + π.
P r o o f. It is not hard to prove that the bound mλ(p, q, d) ≤ Λ(p, q, d)
holds for some Λ(p, q, d) independent of λ (see, e.g., [6]). Then the above results
in conjunction with Proposition 1 yield the statement of the lemma.
5. Existence of Minimizers
To begin with, let us quote the following result which is an important tool in
the proof of Theorem 2.
Lemma 8. ([2]) Let (u(n), A(n)) ∈ Jpq be a sequence converging weakly to
(u,A) in H1(G;C)×H1(Ω;R2). Then
lim inf
1
2
∫
G
|∇u(n) − iA(n)u(n)|2 dx ≥π(|p− deg(u, ∂ω)|+ |q − deg(u, ∂Ω)|)
+
1
2
∫
G
|∇u− iAu|2 dx.
Now consider the auxiliary minimization problem
Mλ(d) := inf{Fλ[u,A]; (u,A) ∈ J (d)}, (5.1)
where J (d) =
⋃
p,q∈Z J (d)
pq = {(u,A) ∈ J ; d− 1/2 ≤ Φ(u,A, V0) ≤ d + 1/2}. Note
that Mλ(d) is always attained. Indeed, Φ( · , V0) is continuous with respect to
the weak convergence (in H1(G;C) ×H1(Ω;R2)), and J (d) 6= ∅ (J (d) contains,
in particular, pairs (u, 0) with u ∈ H1(G; S1) and deg(u, ∂Ω) = d). Therefore
every minimizing sequence (u(n), A(n)) contains a subsequence converging weakly
in H1(G;C)×H1(Ω;R2) to a minimizer (u, A) ∈ J (d).
Lemma 9. For sufficiently large λ, λ ≥ λ4(d), Mλ(d) = mλ(d, d, d), and
minimizers of (1.3) (with p = q = d) and (5.1) coincide.
P r o o f. Clearly, Mλ(d) ≤ mλ(d, d, d). Now assume by contradiction that
for a sequence λ = λk, λk →∞, we have
Mλ(d) = Fλ[u(λ), A(λ)] and (u(λ), A(λ)) ∈ J (d) \ J (d)
dd . (5.2)
In other words, either deg(u(λ), ∂Ω) 6= d or deg(u(λ), ∂ω) 6= d. We assume that
divA(λ) = 0 in Ω and A(λ) ·ν = 0 on ∂Ω (Coulomb gauge). Thanks to the obvious
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1 147
V. Rybalko
bound
Mλ(d) ≤ M∞(d) := inf
{1
2
∫
G
|∇u− iAu|2 dx +
1
2
∫
Ω
(curlA)2 dx;
(u,A) ∈ J (d), u ∈ H1(G; S1)
}
< ∞ (5.3)
we can extract a subsequence such that u(λ) → u weakly in H1(G;C), A(λ) → A
weakly in H1(Ω;R2) as λ → ∞, and deg(u(λ), ∂Ω) = q, deg(u(λ), ∂ω) = p with
integers p, q independent of λ. Note that (u, A) ∈ J (d) and u ∈ H1(G;S1).
Therefore,
Fλ[u(λ), A(λ)] ≤ Mλ(d) ≤ Fλ[u,A] =
1
2
∫
G
(
|∇u− iAu|2 dx +
1
2
∫
Ω
(curlA)2 dx,
while
lim inf
λ→∞
Fλ[u(λ), A(λ)] ≥ π(|d−p|+ |d−q|)+
1
2
∫
G
|∇u− iAu|2 dx+
1
2
∫
Ω
(curlA)2 dx
by virtue of Lemma 8. Thus p = q = d for sufficiently large λ, i.e., u(λ) ∈ J (d)
dd ,
which contradicts (5.2).
Thus, Lemma 9 shows that mλ(d, d, d) is always attained for λ ≥ λ4(d).
Moreover, thanks to Proposition 1 every minimizer of (1.3) with p = q = d is a
local minimizer of Fλ[u,A] in J × H1(Ω;R2) when λ ≥ λ0(M∞(d) + 1), where
M∞(d) is defined in (5.3) (note that mλ(d, d, d) < M∞(d) + 1).
We next prove the attainability of the infimum (1.3) in
Proposition 10. For every K = 0, 1, 2, . . . there is λ5 = λ5(K) > 0 such
that for all λ ≥ λ5 and all integers p and q satisfying p ≤ d, q ≤ d, and
|q − d|+ |p− d| ≤ K:
(i) the infimum mλ(p, q, d) is attained,
(ii) if p ≤ p′ ≤ d, q ≤ q′ ≤ d and either p 6= p′ or q 6= q′, then
mλ(p, q, d) < mλ(p′, q′, d) + π(|p− p′|+ |q − q′|).
P r o o f. By Lemma 9, Proposition 10 holds for K = 0 (induction base). Now
assume that (i) and (ii) hold for given K ≥ 0. Then by Lemma 7, (ii) holds for
K +1 in place of K when λ ≥ max
{
λ5(K), max{λ3(p, q, d); |q−d|+ |p−d| = K,
p ≤ d, q ≤ d}}.
148 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
Local Minimizers of the Magnetic Ginzburg–Landau Functional
To show ( i), we consider a minimizing sequence (u(n), A(n)) for problem (1.3).
This minimizing sequence exists since mλ(p, q, d) < mλ(d, d, d) + π(|p− d|+ |q −
d|) ≤ M∞(d)+π(K +1). Thanks to this bound, up to extracting a subsequence,
u(n) → u ∈ J weakly in H1(G;C), and A(n) → A weakly in H1(Ω;R2). By virtue
of Lemma 8, we have
mλ(p, q, d) = lim
n→∞Fλ[u(n), A(n)] ≥ Fλ[u,A]+π(|q−deg(u, ∂Ω)|+|p−deg(u, ∂ω)|).
(5.4)
Let us show that deg(u, ∂Ω) = q and deg(u, ∂ω) = p. To this end, we need the
following.
Lemma 11. For every Λ > 0 there is λ6 = λ6(Λ) such that mλ(p, q, d) ≥
M∞(d)+π(|p−d|+|q−d|−1/2) when λ ≥ λ6 and M∞(d)+π(|p−d|+|q−d|) ≤ Λ.
This lemma and (5.4) imply that if λ ≥ λ6, then
mλ(p, q, d) ≥ mλ(deg(u, ∂ω),deg(u, ∂Ω), d)+π(|q−deg(u, ∂Ω)|+|p−deg(u, ∂ω)|)
≥ M∞(d) + π(|p− deg(u, ∂ω)|) + |deg(u, ∂ω)− d|)
+ π(|q − deg(u, ∂Ω)|+ |deg(u, ∂Ω)− d|))− π/2.
On the other hand, (ii) guaranties that mλ(p, q, d) < M∞(d)+π(|p−d|+ |q−d|),
therefore p ≤ deg(u, ∂ω) ≤ d and q ≤ deg(u, ∂Ω) ≤ d. Furthermore, if we
assume that either deg(u, ∂Ω) 6= q or deg(u, ∂ω) 6= p, then (5.4) implies that
mλ(p, q, d) ≥ mλ(deg(u, ∂ω),deg(u, ∂Ω), d)+π(|q−deg(u, ∂Ω)|+|p−deg(u, ∂ω)|),
but this contradicts (ii). Thus mλ(p, q, d) is always attained for sufficiently large
λ when p ≤ d, q ≤ d, and |q − d|+ |p− d| ≤ K + 1. Proposition 10 is proved.
P r o o f of Lemma 11. Similarly to Lemma 9, we argue by contradiction.
Namely, assume that mλ(p, q, d) < M∞(d) + π(|p − d| + |q − d| − 1/2) for some
integers p, q, d and a sequence λ = λk, λk → ∞. In other words, there are
(u(λ), A(λ)) ∈ J (d)
pq such that Fλ[u(λ), A(λ)] < M∞(d) + π(|p− d|+ |q − d| − 1/2)
for λ = λk. Since Fλ[u(λ), A(λ)] is bounded, up to extracting a subsequence,
u(λ) → u weakly in H1(G;C), A(λ) → A weakly in H1(Ω;R2). Besides, u ∈
J (d) ∩H1(G; S1). Therefore,
1
2
∫
G
|∇u− iAu|2 dx +
1
2
∫
Ω
(curlA)2 dx ≥ M∞(d).
On the other hand, by virtue of Lemma 8, we have
lim inf
λ→∞
Fλ[u(λ), A(λ)] ≥ 1
2
∫
G
|∇u− iAu|2 dx+
1
2
∫
Ω
(curlA)2 dx+π(|p−d|+ |q−d|),
and thus we arrive at a contradiction with the bound Fλ[u(λ), A(λ)] < M∞(d) +
π(|p− d|+ |q − d| − 1/2).
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1 149
V. Rybalko
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