On approximation of Ginzburg-Landau minimizers by $\mathbb S^1$-valued maps in domains with vanishingly small holes

0.28 MB·

by  Leonid Berlyand

On approximation of Ginzburg-Landau minimizers by
S1-valued maps in domains with vanishingly small holes
7
Leonid Berlyand∗, Dmitry Golovaty†, Oleksandr Iaroshenko∗, Volodymyr Rybalko‡
1
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2 January 27, 2017

n
a
J
Abstract: We consider a two-dimensional Ginzburg-Landau problem on an arbitrary domain

5
withafinitenumberofvanishinglysmallcircularholes. Aspecialchoiceofscalingrelationbetween
2
the material and geometric parameters (Ginzburg-Landau parameter vs hole radius) is motivated

] by a recently discovered phenomenon of vortex phase separation in superconducting composites.
P We show that, for each hole, the degrees of minimizers of the Ginzburg-Landau problems in the
A classesofS1-valuedandC-valuedmaps,respectively,arethesame. Thepresenceoftwoparameters
. that are widely separated on a logarithmic scale constitutes the principal difficulty of the analysis
h
t that is based on energy decomposition techniques.
a
m
1 Introduction
[

2
Thepresentstudyismotivatedbythepinningphenomenonintype-IIsuperconductingcomposites.
v
4 Type-II superconductors are characterizedby vanishing resistivity and complete expulsion of mag-
3 netic fields from the bulk of the material at sufficiently low temperatures. When the magnitude
5 h of an external magnetic field h exceeds a certain threshold, the field begins to penetrate
ext ext
1
the superconductoralongisolatedvortexlinesthatmaymove,resultinginenergydissipation. This
0
motion and related energy losses can be inhibited by pinning the lines to impurities or holes in
.
1 a superconducting composite. Understanding the role of imperfections in a superconductor can
0
thus be used to design more efficient superconducting materials. In what follows, we will consider
7
a cylindrical superconducting sample containing rod-like inclusions or columnar defects elongated
1
: along the axis of the cylinder, so that the sample can be represented by its cross-section Ω R2.
v ⊂
Then the vortex lines penetrate each cross-sectionat isolated points, called vortices.
i
X Superconductivity is typically modeled within the framework of the Ginzburg-Landau theory
r [11] in terms of an order-parameter u C and the vector potential of the induced magnetic field
a A R2. The appearance and behavi∈or of vortices for the minimizers of the Ginzburg-Landau
∈
functional
1 1 1
GLε[u,A]= ( iA)u2dx+ (1 u2)2dx+ (curlA h )2dx (1)
2 | ∇− | 4ε2 −| | 2 − ext
ZΩ ZΩ ZΩ
∗DepartmentofMathematics, PennsylvaniaStateUniversity,[email protected], [email protected]
†DepartmentofMathematics, TheUniversityofAkron,[email protected]
‡MathematicalDivision,B.I.VerkinInstituteforLowTemperature,PhysicsandEngineering,Kharkiv,Ukraine,
[email protected]
1
have been studied, in particular, in [16, 18] where the existence of two critical magnetic fields, H
c1
and H , was established rigorously for simply-connected domain when ε > 0 is small. When the
c2
external magnetic field is weak (h <H ) it is completely expelled from the bulk semiconductor
ext c1
(Meissnereffect) andthere arenovortices. Whenthe fieldstrengthis rampedup fromH to H ,
c1 c2
the magneticfieldpenetratesthe superconductorthroughanincreasingnumberofisolatedvortices
while the superconductivity is destroyed everywhere, once the field exceeds H .
c2
The pinning phenomenon that we consider in this paper is observed in non-simply-connected
domains with holes that may or may not contain another material. If a hole ”pins” a vortex, the
order parameter u has a nonzero winding number on the boundary of the hole. We refer to this
object as a hole vortex. Note that degrees of the hole vortices increase along with the strength of
the external magnetic field. This situation is in contrast with the regular bulk vortices that have
degree 1 and increase in number as the field becomes stronger.
±
An alternative way to model the impurities is to consider a potential term (a(x) u2)2 where
−| |
a(x) varies throughout the sample. It was proven in [9] that the impurities corresponding to the
weakest superconductivity (where a(x) is minimal) pin the vortices first. This model was studied
further in [1] and [4] to demonstrate the existence of nontrivial pinning patterns and in [2] to
investigate the breakdown of pinning in an increasing external magnetic field, among other issues.
A composite consisting of two superconducting samples with different critical temperatures was
considered in [5, 14] where nucleation of vortices near the interface was shown to occur.
Inourmodel we considera superconductorwithholes, similarto the setupin [3]. Inthatwork,
the authors considered the asymptotic limits of minimizers of GLε as ε 0 and determined that
→
holes act as pinning sites gaining nonzero degree for moderate but bounded magnetic fields. For
magnetic fields below the threshold of order lnε the degree of the order parameter on the holes
| |
continues to grow without bound, however beyond the critical field strength, the pinning breaks
down and vortices appear in the interior of the superconductor. Since the contribution to the
energy from the hole vortices has a logarithmic dependence on the diameter of the holes, the hole
size can be used as an additional small parameter to enforce a finite degree of the hole vortex in
thelimitofsmallε. Thedomainwithfinitelymanyshrinking(pinning)subdomainswithweakened
superconductivity was considered in [10] in the case of the simplified Ginzburg-Landau functional.
The model with a potential term (a(x) u2)2 with piecewise constant a(x) was used to enforce
−| |
pinning anditwasobservedthatthe vorticesarelocalizedwithin pinning domainsandconvergeto
their centers.
The problemconsideredinthis workwas inspiredbythe resultin[6]where a periodic lattice of
vanishingly small holes was considered. The main interest was in the regime when the radii of the
holes were exponentially small compared to the period a of the lattice; both of these parameters
were assumed to converge to zero along with ε. Using homogenization-type arguments, it was
shown in [6] that in the limit of ε 0 and when the external magnetic field of order O(a−2), the
→
minimizers can be characterized by nested subdomains of constant vorticity. The physical nature
of this result was discussed in [12]. The analysis in [6] relies on a conjecture that for small ε, the
degreesofthe hole vorticesare the same for both C- andS1-valuedmaps. The principalaimof the
present paper is to establish the validity of this conjecture in the case of finitely many vanishingly
small holes.
Our approach builds on that of [3], combined with the appropriately chosen lower bounds on
the energy and the ball construction method [7], [13], [15]-[19].
The paper is organized as follows. Section 2 contains the formulation of the problem, as well
as the main result described in Theorem 1. In Section 3, we prove that the minimizers in the class
2
of S1-valued maps are characterized by the unique set of integer degrees on the holes. In Section
4, we use the approach, similar to that in [3], to express the energy of a C-valued minimizer as
the sum of the the energy of the S1-valued minimizer and the remainder terms. Compared to [3],
additional complications arise in the analysis due to the fact that the radius of the holes is not
fixed in the present work. In particular, because of the presence of another small parameter, we
use a different ball construction method that incorporates both the Ginzburg-Landau parameter ε
andthe holes radius δ. In Section5 we showthat the minimizes cannothavevorticeswith nonzero
degrees outside of the holes. This section also provides sharp energy estimates that allow us to
prove the main theorem. Finally, in Section 6, the equality of degrees is established based on the
estimates obtained in the previous section.
2 Main results
Let B(x ,R) R2 denote a disk of radius R centered at x . Let Ω be an arbitrary smooth,
0 0
bounded, simp⊂ly connected domain and suppose that ωj =B(aj,δ) Ω, j =1...N represent the
δ ⊂
holes in Ω, where aj is the center of the hole j = 1,...,N and δ 1 is its radius. We introduce
≪
the perforated domain
N
Ω =Ω ωj (2)
δ \ δ
j=1
[
and consider the Ginzburg-Landau functional
1 1 1
GLε[u,A]= ( iA)u2dx+ (1 u2)2dx+ (curlA h )2dx. (3)
δ 2 | ∇− | 4ε2 −| | 2 − ext
Z Z Z
Ωδ Ωδ Ω
The domain Ω represents a cross-section of a superconducting sample. Here u : Ω C is an
δ δ
order parameter, A : Ω R2 is a vector potential of the induced magnetic field, and→h is the
ext
→
magnitude of the external magnetic field. By ε we denote the inverse of the Ginzburg-Landau
parameterthatdeterminesthe radiusofatypicalvortexcore. Inwhatfollows,wewillassumethat
the cores radii are much smaller than the radius of the holes ωj.
δ
The functional GLε[u,A] is gauge-invariant, i.e., for any ϕ H2(Ω,R) and any admissible
δ ∈
pair (u,A), the equality GLε[u,A] = GLε ueiϕ,A+ ϕ always holds. This degeneracy can be
δ δ ∇
eliminated by imposing the Coulomb gauge, that is requiring that
(cid:2) (cid:3)
A H(Ω,R2):= a H1(Ω,R2) diva=0 in Ω, a ν =0 on ∂Ω , (4)
∈ ∈ | ·
where ν is an outward unit norm(cid:8)al vector to ∂Ω. We will fix the Coulomb gaug(cid:9)e throughout the
rest of this work.
We consider the minimizers of the two variational problems
(uε,Aε):=arg min GLε[u,A] u H1(Ω ;C),A H(Ω;R2) , (5)
δ δ δ | ∈ δ ∈
and (cid:8) (cid:9)
(u ,A ):=arg min GLε[u,A] u H1(Ω ;S1),A H(Ω;R2) . (6)
δ δ δ | ∈ δ ∈
Note that, trivially, (cid:8) (cid:9)
(u ,A ):=arg min GL [u,A] u H1(Ω ;S1),A H(Ω;R2) , (7)
δ δ δ δ
| ∈ ∈
(cid:8) (cid:9)
3
where
1 1
GL [u,A]= u iAu2dx+ (curlA h )2dx. (8)
δ ext
2 |∇ − | 2 −
Z Z
Ωδ Ω
For any hole center aj, j = 1,...,N and R > 0, let γj = ∂B(aj,R) be a circle of radius R
R
centered at aj. In what follows we make a frequent use of the following
Definition 1. Given a u H1(Ω ,C) and aj, j =1,...,N, suppose there exists an R=δ+o(δ)
δ
such that the winding num∈ber d=deg(u/u,γj)=0. Then u is said to have a hole vortex of the
| | R 6
degree d inside ωj.
δ
The existence of γj is established in the Theorem 1 and they are specified using the results
R
of Theorem 3. Hole vortices may exist inside ωj for the minimizers of both (5) and (7) and our
δ
principal goal is to prove that the respective degrees of the hole vortices arising in both problems
coincide for the same external magnetic field as long as the parameter δ is sufficiently small. This
resultimpliesthatthenon-linearpotentialtermcanbeeffectivelyreplacedbytheconstraint u =1
| |
when one is interestedin studying the distribution of degreesof the hole vortices for the minimizer
of the problem (5).
The main result of this work is the following theorem.
Theorem 1. Assume that the parameters ε and δ satisfy
logε logδ . (9)
| |≫| |
Suppose
σ R Σ (10)
+
∈ \
where Σ is a discrete set described below. Let
h =σ logδ (11)
ext
| |
and (uε,Aε) and (u ,A ) be defined by (5) and (7), respectively.
δ δ δ δ
Then, for a sufficiently small δ, there exists an R [δ,δ+δ2] such that
δ
∈
(i) Dj = deg u ,γj coincide for all j = 1...N when Dj are defined, e.g. when uε = 0 on
δ δ R δ,ε δ 6
γj; (cid:16) (cid:17)
R
(ii) the degrees of the hole vortices Dj =deg uεδ ,γj ;
δ,ε uε R
(cid:18)| δ| (cid:19)
for any R R for which γj = ∂B(aj,R), j = 1...N are mutually disjoint and do not intersect
≥ δ R
∂Ω.
Remark 1. The set Σ includes the appropriately scaled values of the external field at which the
degree of one of the hole vortices increments by one, i.e. from d to d+1. At these threshold field
strengths, the leading order approximation of the energy is the same for both degrees d and d+1
and the degrees of the hole vortices of minimizers uε and u cannot be determined uniquely. The
δ δ
set Σ is described as follows:
N
1
Σ= Σ where Σ = σ >0 σ 1 ξ (aj) Z+ (12)
j j 0
| − ∈ 2
j=1 (cid:26) (cid:27)
[ (cid:0) (cid:1)
4
consistsofthethresholdfieldvaluesfortheholej =1...N andthefunctionξ solvestheboundary
0
value problem
∆ξ +ξ =0 in Ω,
0 0
− (13)
(ξ0 =1 on ∂Ω.
Remark 2. Notice that, since u (x) S1, there are no vortices outside of the holes and thus
δ
∈
Dj =deg u ,γj =deg u ,∂ωj (14)
δ δ r δ δ
(cid:16) (cid:17)
(cid:0) (cid:1)
for all j =1...N.
Remark 3. AswewillshowinSection5,althoughtheexternalmagneticfieldsatisfyingthebound
(11) is strong enoughto generatehole vortices,it is too weak for vorticesto appear inside the bulk
superconductor Ω , away from the boundary ∂Ω.
δ
We prove Theorem 1 in two steps. First, we consider minimizers (u ,A ) of the variational
δD δD
problem (8) in the class of S1-valued maps with the prescribed degrees, deg(u,∂ωj) = Dj, j =
δ
1...N, by setting
(u ,A ):=arg min GL [u,A] u H1(Ω ;S1),A H(Ω;R2),deg(u,∂ωj)=Dj . (15)
δD δD δ | ∈ δ ∈ δ
n o
Then the degrees Dj of the map u minimize the energy
δ δ
l (D):=GL [u ,A ] (16)
δ δ δD δD
where D = (D1,...,DN). It turns out that the function l (D) is a quadratic polynomial in
δ
D1,...,DN. Its minimum is attained at one of the integer points adjacent to the vertex of
paraboloid l (T) with T RN. We enforce the condition (10) to ensure that such minimizing
δ
∈
integer point is unique.
We then express a minimizer (uε,Aε) of GLε[u,A] as a sum of (u ,A ) and an appropriate
δ δ δ δ δ
correction term and consider a corresponding energy decomposition in the spirit of the approach
in [3] for finite-size holes. The analysis relies principally on the techniques developedin [3] and the
ball construction method [19]. Compared to [3], new challenges arise due to the presence of the
second small parameter that require additional estimates and sharper energy bounds.
3 S1-valued problem
The main goal of this section is to establish the relation between the energy of the minimizer
(u ,A ) and the degrees D of the hole vortices corresponding to u . We approximate the
δD δD δD
minimizer(u ,A ),calculateitsenergyl (D)=GL [u ,A ],andfindtheminimizingdegrees
δD δD δ δ δD δD
D =(D1,...,DN). We prove the following theorem.
δ δ δ
Theorem 2. Let (u ,A ) be a minimizer of (15) with the prescribed degrees D ZN on the
δD δD
∈
holes. Then the Ginzburg-Landau energy GL [u ,A ], expressed as a function of D, takes the
δ δD δD
following form:
N
l (D)=π Dj 2 2σ 1 ξ (aj) Dj logδ +C logδ 2+ D 2O(1) (17)
δ 0
− − | | | | | |
Xj=1h(cid:0) (cid:1) (cid:0) (cid:1) i
5
where ξ solves the boundary value problem (13), C =O(1), and D =max Dj .
0
| | j | |
Proof. Themainideaoftheproofistoapproximatetheinducedmagneticfieldh =curlA
δD δD
as a sum of functions that depend on external magnetic field and the prescribed degrees on the
holes, respectively. First, prescribe the degrees of the order parameter
deg(u,∂ωj)=Dj, j =1...N (18)
δ
andwritedowntheEuler-Lagrangeequationfor(8)intermsoftheinducedmagneticfieldh=curlA
with the corresponding boundary conditions:
∆h+h=0, in Ω ,
δ
−
h=hext, on ∂Ω,
(19)
h=Hj, in ωδj, j =1...N,
∂hds=2πDj hdx, j =1...N.
The constants Hj area−pRri∂oωrδji ∂uνnknown and−arReωdδjefined through the solution hδD =hδ(D) of (19)
where D = (D1,...,DN) is the vector of the prescribed degrees. The energy (8) of the minimizer
(u ,A ) can be expressed in terms of h :
δD δD δD
1 1
GL [u ,A ]=GL [h ]= h 2dx+ (h h )2dx. (20)
δ δD δD δ δD δD δD ext
2 |∇ | 2 −
ZΩδ ZΩ
Decompose the solution of (19) h into
δD
h =h +h +h , (21)
δD 1 2 3
where h captures the influence of the external field h , h takes into account the hole vortices,
1 ext 2
and h is the remainder. More precisely,
3
h =h ξ , (22)
1 ext 0
where ξ solves the boundary value problem (13) in the domain Ω with no holes:
0
∆ξ +ξ =0 in Ω,
0 0
− (23)
(ξ0 =1 on ∂Ω.
The function h is defined by
2
N
h (x)= Djθ (x)φ (x) (24)
2 j j
j=1
X
where Dj are as in (18). Here
θ (x)=θ(x aj), j =1,...,N
j
−
and θ is a truncated modified Bessel function of the second kind
K (δ), x δ,
0
θ(x)= | |≤ (25)
(K0(x), x >δ.
| | | |
6
The cutoff function φ (x)=φ(x aj) C∞(R2) satisfies
j
− ∈
1, x R/4,
φ(x)= | |≤ (26)
(0, x R/2,
| |≥
withRbeingdefinedasthelargestradiusforwhichB(aj,R),j =1...N intersectneithereachother
nor the boundary ∂Ω. Here the choice of K (x) is motivated by the fact that it is a fundamental
0
solution of the equation ∆u+u=2πδ(x) in|R|2. Note that h solves the following problem:
2
−
∆h +h = N Dj[ ∆+I](θ φ ), in Ω ,
− 2 2 j=1 − j j δ
h2 =0, P on ∂Ω, (27)
h2 =D∂jhK20d(sδ)=, 2πDj DjK (δ)ωj +DjO(δ2), ojn=∂1ω.δj.,.Nj. =1...N,
Sinceforeach−jR=∂ω1δj,.∂.ν.,N thefunc−tionf (0x):|=δ[| ∆+I](θ φ )isnonzeroonlyinsidetheannular
j j j
−
regionT :=B(aj,R/2) B(aj,R/4)thatdoes notintersectanyofthe holes,the functions f , j =
j j
\
1,...,N are smoothand finite. Thus, for everyj =1,...,N, the function h has the degree Dj on
2
the hole ωj and θ φ is constant on ωj and decays to zero on ∂B(aj,R/2).
δ j j δ
Next, we show that the contribution of the remainder h =h h h to the energy is small,
3 1 2
− −
hence the interactionbetweenthe hole vorticescontributesa negligibleamountto the energy. This
provides a justification for treating each hole vortex as being independent from the other hole
vortices.
We deducethe boundaryvalue problemforh fromthe originalproblem(19),the problem(13)
3
for h =h ξ , and the expression (24) for h to obtain:
1 ext 0 2
∆h +h = N Djf (x), in Ω ,
− 3 3 − j=1 j δ
h3 =0, P on ∂Ω, (28)
h3 =Hj −hext(ξ0(x)−ξ0(aj)), on ∂ωδj, j =1...N,
∂h3ds= H ωj +DjO(δ2)+O(δ3logδ), j =1...N.
where H =H−R∂ωδhje∂νξ (aj)−eDj|jKδ|(δ) are the unknown constants. The next lemma establishes
j j ext 0 0
− −
the necessary estimates for h .
3
e
Lemma 1. The solution h of (28) satisfies the following estimates:
3
h3 L∞(Ω) C1δ logδ 2+C2 D , (29)
k k ≤ | | | |
h3 L∞(Ω) C1 logδ 2+C2 D logδ , (30)
k∇ k ≤ | | | || |
∂h
3 C logδ +C D on ∂ωj for all j =1...N. (31)
∂ν ≤ 1| | 2| | δ
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof. We beginby splitti(cid:12)(cid:12)ng (2(cid:12)(cid:12)8)into severalsubproblems. First,let η = Nj=1Djηj be a solution
of the nonhomogeneous equation in (28), where η solves
j
P
∆η +η = [ ∆+I](θ φ ) , in Ω,
− j j − − j j 1Tj (32)
(ηj =0, on ∂Ω,
7
for everyj =1,...,N. Here η , j =1,...,N are smoothanddo notdepend onδ. Next, introduce
j
η that both solves the homogeneous equation and satisfies the conditions on ∂ωj in (28) to give
0 δ
∆η +η =0, in Ω ,
0 0 δ
−
η =0, on ∂Ω, (33)
 0
η0 =−hext(ξ0(x)−ξ0(aj))−(η(x)−η(aj)), on ∂ωδj, j =1...N.
Note that, bythe Maximum Principle,
η0 L∞ Cδ( logδ +max Dj ). (34)
k k ≤ | | j | |
Lemma 6 provides the estimate on the gradient of η of the form
0
η0 L∞ C( logδ +max Dj ). (35)
k∇ k ≤ | | j | |
The remainder ζ =h N η solves the following system:
3− j=0 j
P
∆ζ+ζ =0, in Ω ,
δ
−
ζ =0, on ∂Ω,
(36)
ζ =cj,∂ζds= ωj c +Aj, ojn=∂1ω.δj.,.Nj, =1...N,
where c =H η(aj)a−reRu∂ωnδjkn∂oνwn co−ns|taδn|tjs andδAj = D O(δ)+O(δlogδ) is an error. The first
j j− δ | |
threeequationsin(36)setuptheboundaryvalueproblemforζ withtheunknownboundaryvalues
c . Thefourethlinein(36)givesthe systemofN equationsforN unknownsc . Sincethe boundary
j j
value problem for ζ is linear, we start with the estimates for the basis functions ζ that solve the
i
problem
∆ζ +ζ =0, in Ω ,
i i δ
−
ζ =0, on ∂Ω, (37)
 i
ζi =δij, on ∂ωδj, j =1...N,
for every i = 1,...,N. Then, using representation ζ = iciζi, we will solve the linear system for
c .
i
P
Weusethemethodofsub-andsupersolutionstogetestimatesforζ . BytheMaximumPrinciple,
i
we have that 0 ζ 1 for every i = 1,...,N. In the case of a radially symmetric domain with
i
≤ ≤
one hole at the center, the solutions of (37) are the modified Bessel functions. We show that they
provide a good approximation for ζ . First, fix i 1...N and construct a supersolution for ζ .
i i
∈
Take R >0 such that Ω B(ai,R ) and set
max max
∈
K |x−ai|
ζsup = 0 Rmax . (38)
i (cid:16) (cid:17)
K δ
0 Rmax
(cid:16) (cid:17)
The function ζsup is strictly positive in Ω , equals 1 on ∂ωi, and has [ ∆+I]ζsup =0. Therefore
i δ δ − i
8
it satisfies
∆ζsup+ζsup =0 in Ω ,
− i i δ
ζsup >0 on ∂Ω,
 i (39)
ζisup =1 in ωδi,
ζsup >0 in ωj, j =i, j =1...N,
i δ 6
and is thus a supersolution. This yields the bound
0 ζ ζsup in Ω, i=1...N. (40)
≤ i ≤ i
Next, we constructa subsolution. Take R >0 such that B(ai,2R ) Ω for every i=1...N
min min δ
∈
and set
K |x−ai|
ζsub = 0 Rmin (41)
i (cid:16) (cid:17)
K δ
0 Rmin
(cid:16) (cid:17)
The Bessel function is a fundamental solution of [ ∆+I]u = δ(x) and it is decreasing, therefore
−
ζsub is negative outside B(ai,R ). Thus it satisfies
i min
∆ζsub+ζsub =0 in Ω ,
− i i δ
ζsub <0 on ∂Ω,
 i (42)
ζisub =1 in ωδi,
ζsub <0 in ωj, j =i, j =1...N,
i δ 6
and is thus a subsolution. This, together with (40), implies that
max(0,ζsub) ζ ζsup, (43)
i ≤ i ≤ i
for everyi=1...N, giving a very sharpdescription of the behavior of ζ near ith hole. Note that,
i
for x ∂ωi, we have
∈ δ L ∂ζsub ∂ζsup L
1 i (x) i (x) 2 (44)
δlogδ ≤ ∂ν ≤ ∂ν ≤ δlogδ
with L ,L >0, therefore
1 2
∂ζ 1
i(x) on ∂ωi. (45)
∂ν ∼ δlogδ δ
Toestimatethenormalderivativeofζ on∂ωj forj =iweneedabettersupersolutionthatcaptures
i δ 6
the appropriate Dirichlet boundary conditions. Outside of B(ai,R ), we have
min
K Rmin
ζ (x) 0 Rmax C logδ −1. (46)
| i |≤ K (cid:16) δ (cid:17) ≤ R| |
0 Rmax
(cid:16) (cid:17)
Construct ζsup that solves the following conditions:
ij
∆ζsup+ζsup =0 in B(aj,R ) B(aj,δ),
− ij ij min \
ζsup =C logδ −1 on ∂B(aj,R ), (47)
 ij R| | min
ζsup =0 on ∂ωj.
ij δ

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This problem is radially symmetric in B(aj,R ) B(aj,δ). The function
min
\
ζsup =C I (r)+C K (r), r= x aj (48)
ij 1 0 2 0 | − |
with
C logδ −1 and C logδ −2. (49)
1 2
∼−| | ∼| |
satisfies(47)becausethe modifiedBesselfunctionsI andK behaveas1and logr, respectively,
0 0
−
near the origin. Therefore
∂ζ ∂ζsup C
0 i ij = ij on ∂ωj. (50)
≤ ∂ν ≤ ∂ν δ logδ 2 δ
| |
As a result
∂ζ C
i
ds . (51)
Z∂ωδj(cid:12)∂ν(cid:12) ≤ |logδ|2
(cid:12) (cid:12)
foralli=j. Combiningtheestimatesonth(cid:12)ebe(cid:12)haviorofζ on∂ωi in(45)with(51)andestimating
6 (cid:12) (cid:12) i δ
the constants c using the fourth equation in (36) we find:
i
∂ζ ∂ζ ∂ζ
πδ2 c + Aδ ds c ids c jds
| i| (cid:12)(cid:12) i(cid:12)(cid:12)≥(cid:12)(cid:12)(cid:12)(cid:12)Z∂ωδi ∂ν (cid:12)(cid:12)(cid:12)(cid:12)≥(cid:12)(cid:12)(cid:12)(cid:12) iZ∂Cωδi ∂ν N(cid:12)(cid:12)(cid:12)(cid:12)−Xj6=i(cid:12)(cid:12)(cid:12)(cid:12)CjZ∂ωδi ∂ν (cid:12)(cid:12)(cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) 1 (cid:12) (cid:12) 2 (cid:12)
c c (52)
≥| i| logδ − | j| logδ 2
| | j6=i | |
X
or
N
C C
c 1 πδ2 c 2 Aδ , (53)
| i| logδ − − | j| logδ 2 ≤ i
(cid:18)| | (cid:19) j6=i | |
X (cid:12) (cid:12)
(cid:12) (cid:12)
with some positive C ,C >0 for all i=1...N. The coefficient matrix is a small perturbation of
1 2
the identity matrix, up to the factor C logδ −1. This allows us to conclude that
1
| |
c D O(δlogδ)+O(δlog2δ) (54)
i
| |≤| |
for all i=1...N. Let
c =max c . (55)
i j
j | |
Then
−1
C C
c Aδ 1 πδ2 (N 1) 2 D O(δlogδ)+O(δlog2δ), (56)
| i|≤ i logδ − − − logδ 2 ≤| |
(cid:18)| | | | (cid:19)
(cid:12) (cid:12)
hence (cid:12) (cid:12)
kζkL∞(Ωδ) ≤ |cj|≤C1|D|δ|logδ|+C2δ|logδ|2. (57)
j
X
The statement of the lemma for
N N
h =η + Djη + c ζ (58)
3 0 j j j
j=1 j=1
X X
then follows once we combine the estimates above.
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