MULTIPLE GINZBURG-LANDAU VORTICES PINNED BY …MULTIPLE GINZBURG-LANDAU VORTICES PINNED BY RANDOMLY...

MULTIPLE GINZBURG-LANDAU VORTICES PINNED BYRANDOMLY DISTRIBUTED SMALL HOLES ∗

L. BERLYAND† , V. MITYUSHEV‡ , AND S. D. RYAN§

Abstract. In this work a minimization problem for the magnetic Ginzburg-Landau functional ina circular domain with randomly distributed small holes is considered. We develop a new analyticalapproach for solving a non-standard boundary value problem for the magnetic field presented as afunction of the n-tuple of degrees of vortices pinned by the n holes. The key feature of this approachis that the solution is analytically derived via the method of functional equations and does not relyon periodic geometry as in previous studies. We prove the convergence of the method of successiveapproximations applied to the functional equations for arbitrary hole locations. Once establishedthe associated energy functional is minimized as a function of vortex degrees to find the effectivevorticity distribution. After verification of the method by comparison of our results to previous worksin the periodic case, we identify the striking differences due to the presence of random hole locations.Namely, the different subdomain structure, the proximity of hole vortices to the boundary, and ourapproach allows for the estimation of the fractal dimension of the interface between regions of likedegree.

Key words. Ginzburg-Landau Superconductivity, Random Media, Vorticity, Boundary ValueProblems, Iterative Functional Equations, Multiply-connected Domains, Fractal Dimension

AMS subject classifications. 30E25, 35Q56, 39B12, 78A48

1. Introduction. Superconductivity has been a primary object of interest overthe last half century. Special focus has been given to the study of type-II supercon-ductors with size comparable to the London penetration depth. Among the vast arrayof striking physical phenomena observed, the main object of interest in this work isthe pinning of vortices inside holes within the sample. The case of a periodic arrayof finitely many holes in a two-dimensional cross-section was extensively studied in[5] leading to striking analytical results. Specifically, a vortex phase separation wasexhibited by the formation of nested subdomains of the same vorticity. This workwill extend the current understanding by studying domains with an arbitrary set oflocations for each hole.

In particular, we seek to quantify the effect of randomness in the hole locations onthe vortex distribution. This case is important experimentally, since it may be easierto create such a sample through various techniques (e.g., ion bombardment or drillingholes in a superconducting sample [7, 19, 22]). The optimal vortex distribution isfound by minimizing the energy of the system also known as finding the so-calledground state. The physical systems under consideration in this work are Bose andCoulomb glasses where glassiness refers to the absence of a ground state [29]. Forexample, one can rearrange the vortex distribution to further lower the energy orthere are degenerate (infinitely many) ground states with indistinguishable energies.

The mathematical and physical literature on superconductivity and the studyof the Ginzburg-Landau (GL) energy functional is rich. Here we briefly review the

∗The work of VM and SR was partially supported by NSF grant DMS-1106666 and the work ofLB was partially supported by NSF grants DMS-1106666 and DMS-1405769.†Department of Mathematics, Pennsylvania State University, McAllister Bldg. University Park,

Pennsylvania 16802, USA ([email protected])‡Department of Computer Science and Computational Methods, Pedagogical University of Cra-

cow, Poland ([email protected])§Department of Mathematical Sciences and Liquid Crystal Institute, Kent State University, Kent,

Ohio 44240, USA ([email protected])

1

2 L. BERLYAND, V. MITYUSHEV, AND S. D. RYAN

most relevant mathematical works, which will provide the greatest comparison to ourpresent investigation. This work should be viewed as a contrast to the recent work [5]where rigorous homogenization results are derived via Γ−convergence for the effectivevorticity in a domain with periodically dispersed holes. The results of [5] were thefirst to exhibit a non-homogeneous vortex structure in the form of nested subdomainsof increasing vorticity towards the sample center. The investigation of pinning at afinite set of locations was originally carried out in [21] for the case of the simplified GLfunctional. In that work, the magnetic field is not present (the results were extendedfor the magnetic GL functional in [3, 18]). In [21], a single inclusion was consideredwith a discontinuous pinning term and the existence of n vortices of degree 1 wasestablished in the case of Dirichlet boundary conditions with degree n. An additionalresult was recently obtained for the simplified GL functional with finitely many holesas the hole size goes to zero and the GL parameter κ goes to infinity [14]. Theextension of these prior theoretical results to the full GL functional with finitely manyinclusions has recently been established in [1, 2]. In this work, we advance these recentstudies by quantifying the effect of randomness in pinning sites/hole locations on thenon-homogeneous vortex structure of type-II superconductors.

The main tool used throughout this work to derive analytical results in the ab-sence of periodicity in the hole locations is the method of functional equations. By afunctional equation, we mean an equation with a shift into the domain, i.e., the valueof a desired unknown function at one point depends on the value at other points in thedomain. It is worth noting that such a functional equation does not contain integralterms, but rather compositions with known functions that enable one to apply sym-bolic computations to get an approximate solution in an analytical form. This ideais used in this work to find the solution to a non-standard boundary value problemin R2 ∼= C by first deriving a corresponding functional equation and then solving ititeratively via the method of successive approximations [24, 25, 26].

This technique has been used extensively in the past to study boundary valueproblems for Laplace and Poisson equations [27, 28, 30]. Analogous to the presentwork, this method was used to investigate the effective conductivity of two-phasecomposites for periodic and random arrays of inclusions with different conductivitythan the background medium [4, 11, 12]. In particular, the authors developed anapproach based on the method of functional equations and concluded that the periodiccase represented a special configuration in which the effective conductivity was at aminimum and the randomness increased the effective conductivity. This work providesa nontrivial extension of the previous results [4, 11, 12] to a bounded domain. Herean additional boundary curve leads to new difficulties in solving the boundary valueproblem. To derive the functional equations, we first postulate the form of the solution(magnetic field hε) and reduce the problem to a classical Riemann-Hilbert problemfor multiply connected domains, which is then proved to be equivalent to an R-linearproblem for unbounded domains [6, 24, 26, 30]. The R-linear problem is reduced to asystem of functional equations, which is then solved by use of the method of successiveapproximations.

The main achievement of this work is the development of an analytical approachfor finding the vortex distribution that no longer relies on periodicity, but ratheris valid for an arbitrary set of given non-overlapping hole locations. The primaryassumption involves the relationship between the GL parameter and the hole radii,| log(κ)| | log(rε)| resulting in exponentially small holes in the computational do-main. In section 2, the mathematical problem is formulated; namely, the non-standard

GL VORTICES IN MULTIPLY-CONNECTED DOMAINS WITH RANDOM HOLES 3

boundary value problem for the magnetic field is stated and the energy as a functionof the nε-tuple of degrees is introduced. Next, in Section 3 the analytical methodfor solving the boundary value problem is developed where a functional equation forthe solution to order O(r2

ε) is derived. The convergence of the successive approxima-tions for the functional equations to a unique analytic solution is rigorously provenin the case of non-overlapping holes. Once established the energy functional to or-der O(r2

ε) is reformulated in terms of degrees and minimized resulting in a systemof linear algebraic equations for the minimizing set of degrees in Section 4. Finally,in Section 5 numerical simulations are implemented to solve this system of equationsand find the minimizing vortex distribution. Our results show that as the level ofrandomness in the hole locations increases the form of the vortex distribution dras-tically changes. Namely, the interface between subdomains of like degree becomesfractal-like with fractal dimension around 1.3, holes that pin vortices occur closer tothe domain boundary, and, most importantly, the critical external magnetic field forthe onset of hole vortices decreases [29]. These results effectively change the magneticfield-temperature phase plane by showing that as the randomness grows the regionwhere hole vortices form in the absence of bulk vortices grows. Our results high-light the competition in the energy between vortex-vortex, vortex-hole, and vortexboundary interactions.

2. Formulation of the problem. In this work we consider a two-dimensionalmathematical model for the circular cross-section of a cylindrical superconductingsample with exponentially small holes in order to study the effect of hole locations onthe pinning of vortices. Specifically, we investigate the effect of random inclusion/holelocations theoretically and capture the basic difference in the vortex distribution ascompared to a periodic array. In this work type-II superconductors are consideredwhere κ = λ

ξ 1 where κ is the Ginzburg-Landau parameter, λ is the Londonpenetration depth, and ξ is the coherence length. Introduce dimensionless quantities

where distances are rescaled by λ resulting in a dimensionless domain size diam(Ω)λ ∼ 1

and hole radius rε = rλ ε where ε is the average distance between holes. Henceforth,

only dimensionless quantities are used.Let Ω be the unit disk in C with boundary ∂Ω where C := C∪∞ is the extended

complex plane and define the sets D0 := z : |z| ≥ 1 and Dk := z : |z − ak| < rεrepresents the kth hole with boundary circle Tk. Also, the set D := C \

∑nk=0 Dk.

For comparison with the notation in [5] we note that D := Ωε and Dk := ωεk. withboundary circles Tk := ∂ωεk. The solutions defined herein are valid for a domaincontaining the disjoint disks Dk, which lie in the unit disk, i.e., 0 < rε < rtouching :=infr|dist(Dj ,Dk)| for j, k = 0, ..., n (j 6= k) and |ak|+rε < 1. It will be shown later that

this restriction is imposed so that the method developed will converge using successiveapproximations.

As in [5] we consider the vortex distribution (vorticity) on the perforated domainD with a large, but finite number nε of pinning sites (holes, see Figure 2.1) where Tkrepresents the boundary of these pinning sites. In order to find this distribution onemust find the minimizers of the non-dimensional GL energy functional

GL(u,A) =1

2

∫D|∇u− iAu|2dx +

κ2

4

∫D(1− |u|2)2dx +

1

2

∫Ω

(curlA− hεext)dx (2.1)

with the complex order parameter u and the vector potential of the magnetic fieldA as the unknowns. Here dx = dx1dx2 where the double integrals correspond to the


plane domains under consideration. The external magnetic field hεext = σε2 is given

where ε > 0 is the average distance between holes (in the periodic case this is theperiod) and σ is a positive scalar constant. The domain Ω contains exponentially

small holes with centers denoted by aεk and radius rε = e−γ/ε2

for positive scalarγ resulting in rε ε. It is assumed that the hole radius is much greater than thevortex core, | log(κ)| | log(rε)|, allowing for many vortices to be trapped in onehole. This scaling was directly shown in [5] to be crucial to obtain a striking vortexphase separation into nested subdomains in the periodic setting.

Fig. 2.1. Domain with random inclusions.

Throughout this work the external magnetic field is chosen below the first criticalmagnetic field in order to suppress the formation of bulk vortices, hεext < Hc1 (seeFig. 5.2(b)). This has been rigorously proven in the case of one hole and we proceedwith this assumption for the case of multiple holes [16]. The benefit of this assumptionis that it restricts the vortices to the holes and allows for a better comparison of thetrue effect of the hole configuration on their distribution. One of the primary resultsachieved is the establishment of a new critical magnetic field for the onset of holevortices (in the absence of bulk vortices). As opposed to the bulk, vortices tend toform in holes at a lower value of the external magnetic field, because it is energeticallymore efficient [17].

As in [5], we reduce the problem of finding the vortex distribution to the harmonicmap type functional (rigorously justified in [16] in the case of one hole in a circulardomain)

Fε(u,A) =1

2

∫D|∇u− iAu|2dx+

1

2

∫Ω

(curlA− hεext)2dx, (2.2)

in the class

Mε := infFε(u,A); u ∈ H1(D,S1), A ∈ H1(Ω;R2). (2.3)

Instead, we want to recast the problem from minimizers (u,A) to minimizing energy asa function of the degree dεk on each hole. Minimizing (2.2) is equivalent to minimizingthe following energy [5]

Eε(hε) =

1

2

∫D|∇hε|2dx+

1

2

∫Ω

(hε − hεext)2dx. (2.4)

The associated Euler-Lagrange equations result in a non-standard boundary value


problem for the magnetic field−∆hε + hε = 0, in D,hε(x) = hεext, on ∂Ω,

hε(x) = Hεj , on Dj , j = 1, 2, ..., nε,

−∫Tj

∂hε

∂ν ds = 2πdεj −∫Dj h

ε(x)dx,

(2.5)

whereHεj are unknown constants, the perforated domain boundary ∂D = ∪nεj=1 (−Tj)∪

∂Ω composed of counterclockwise oriented circles and dεj represents the degree of the

vortex in the jth hole, ∂∂ν stands for the outward normal derivatives to Tj = ∂Dj , ds

is the differential of arc length. Thus, the infinite-dimensional problem of minimizing(2.2) in the class (2.3) is reduced to the finite-dimensional problem of minimizing (2.4)as a function of the nε-tuple of degrees in the class

Mε := infEε(hε); hε = hε(x; dεj) solves (2.5) for integers dεj. (2.6)

The fact that the degrees must be integers is known as quantization and providesa constraint on the minimization of the energy. The focus of the present work liesin the vortex distribution dεj corresponding to the minimizer of (2.4). Since theenergy functional is convex, there exists a unique global minimizer (ground state).This problem was first considered in [5] where the unique minimizer was obtained forthe periodic case in the homogenization limit (ε→ 0) by introducing the convex dualproblem for the original energy functional. The authors show that the minimizers ofthe convex dual problem coincide with the minimizers of the original problem. In thiswork, an approximation of this unique global minimizer is found via a linear problemobtained by deriving an analytical solution for the magnetic field using the so-calledmethod of functional equations [24, 25, 26].

Before proceeding, we introduce some notation. Let G be an arbitrary domain onthe extended complex plane. Then the Banach space of functions which are continuouson the curves ∂G with the norm ‖f‖ = max∂G |f(t)| will be denoted as C(∂G). Inaddition, consider the closed subspace CA(G) consisting of all functions analyticallycontinued into G. In this work we will consider the spaces where G = D or G =∪nk=0Dk. For simplicity in notation, let CA := CA (∪nk=0Dk).

We will proceed by deriving a solution to (2.5) in an analytical form via themethod of functional equations and then look for the minimizer of the energy numer-ically.

3. Method of solution. First, consider the solution to the problem with noholes. Let I0(|z|) denote the modified Bessel function of the first kind of order zero.The radial function

h(0)(z) =I0(|z|)I0(1)

, |z| ≤ 1, (3.1)

exactly solves the boundary value problem∆h(0) = h(0), in the unit disk Ω,

h(0) = 1, on ∂Ω.(3.2)

Begin by looking for the magnetic field hε in the form

hε = hεext

[h(0) + h(1) + h(2) + ...

]. (3.3)


It will be observed a posteriori from the results below that the series converges forsufficiently small rε since h(p) ∼ 1

| lnp rε| . The series (3.3) is associated to a cascade of

equations for h(j) valid in D [8]

∆h(1) = 0

∆h(2) = h(1)

∆h(3) = h(2)

...

so that when summed ∆hε = ∆[hεext(h(0) + h(1) + h(2) + · · · )] = hεext[h

(0) + h(1) +h(2) + · · · ] = hε. We will consider the first two terms known as a Rayleigh (lowfrequency) Approximation, which has been shown in the past to be sufficient for findinga good approximate solution in multiple-scattering problems with small obstacles(distribution of inclusions in a domain) [13, 23]. Since rε is exponentially small, eventhe first order term provides only a small correction h(1) ∼ 1/| ln(rε)| 1. Thiscorrection can be viewed as the effect of the distribution of holes and is the maincomputational focus of this work.

3.1. First approximation. The first order approximation h(1) satisfies theboundary value problem:

∆h(1)(x) = 0, x ∈ Dh(1)(x) = 0, x ∈ ∂Ω

h(1)(x) = Hk − h(0)(x), x ∈ Dk, k = 1, 2, ..., n.

(3.4)

The fourth condition in (2.5) in the considered approximation takes the form

−∫Tj

[∂h(0)

∂ν+∂h(1)

∂ν

]ds = 2πdj −

∫Dj

[h(0)(x) + h(1)(x)]dx (3.5)

for a given set of degrees d = (d1, d2, . . . , dn) ∈ Rn with dj = [hεext]−1dεj ∈ [hεext]

−1Z.

Using Green’s identity∫Tj

∂h(0)

∂ν ds =∫Dj ∆h(0)dx and ∆h(0) = h(0) in each disk Dj we

arrive at the following equation∫Tj

∂h(0)

∂νds =

∫Djh(0)dx (j = 1, 2, . . . , n). (3.6)

Then, after division by 2π (3.5) becomes

− 1

2π

∫Tj

∂h(1)

∂νds = dj −

r2ε

2Hj +

1

2π

∫Djh(0)dx, j = 1, 2, . . . , n, (3.7)

where new undetermined constants are introduced as Hj = [hεext]−1Hε

j .Before introducing the method of complex potentials, fix ε > 0 and suppress it

from the notation henceforth (e.g., hext, dk, n and r). Since we consider a two-dimensional problem, this allows for the use of complex potentials. Let z = x1 + ix2

denote a complex variable and look for a solution in the form

h(1)(z) = Re ϕ(z), z ∈ D, (3.8)


where ϕ(z) is holomorphic, hence, Re ϕ(z) is harmonic in D. The function ϕ(z) hasthe general representation (justified in [25])

ϕ(z) = ϕ(z) +

n∑k=1

Ak ln(z − ak), (3.9)

where ϕ(z) is single-valued, ln(z−ak) is multi-valued, and the constants Ak ∈ R needto be determined. The left hand side of (3.7) can then be computed explicitly by(3.8)-(3.9) [25]

1

2π

∫Tk

∂h(1)

∂νds = Ak, (3.10)

The mean value theorem for the Helmholtz equation reads as follows [9] (p. 280)∫Dkh(0)dx = πr2I0(r)h(0)(|ak|). (3.11)

Therefore, (3.7) becomes

r2

2

[Hk −

I0(r)I0(|ak|)I0(1)

]−Ak = dk, k = 1, 2, ..., n, (3.12)

and by extending (3.8) to the boundaries of the multiply connected domain one findsn+ 1 boundary conditions

Re ϕ(t) = 0, |t| = 1, (3.13)

Re ϕ(t) = Hk −I0(|t|)I0(1)

, |t− ak| = r (k = 1, 2, . . . , n). (3.14)

In order to eliminate the unknown constants, Hk, introduce the derivative ψ(z) :=ϕ′(z) and derive corresponding equations. In particular, from (3.13)-(3.14), we canrecast the problem as a Riemann-Hilbert problem for ψ(z) [10, 24, 25].

Im [tψ(t)] = 0, |t| = 1, (3.15)

Im

[(t− ak)

(ψ(t) + 2

I1(|t|)akI0(1)|t|

)]= 0, |t− ak| = r (k = 1, 2, . . . , n), (3.16)

where I1(x) = ddxI0(x) is the modified Bessel function of the first kind with order 1.

The details can be found in Appendix 6.1.Thus, from (3.13)-(3.14) through differentiation we have (3.15)-(3.16), which

achieves two goals: (i) eliminating the unknown constants Hk and (ii) the functionψ(z) is now single-valued by (3.9)

ψ(z) = ϕ′(z) = ϕ′(z) +

n∑k=1

Akz − ak

. (3.17)

The system (3.15)-(3.16) is a classical Riemann-Hilbert problem and we proceedto look for a solution in the class of single-valued functions. Instead of addressing thisdirectly, in the next section a lemma is proven to show that (3.15)-(3.16) is equivalentto an R-linear problem studied in the previous works [24, 25, 26] in the case of anunbounded domain. The R-linear conjugation condition is a condition when the limitvalues W and Z of analytic functions from different sides of a curve are related bythe R-linear equation W = aZ + bZ + c (see for instance (3.18) below) .


3.2. Riemann-Hilbert problem. The main focus of this section will be findingan analytic solution to the Riemann-Hilbert problem (3.15)-(3.16). This is a classicalproblem that has been studied extensively [6, 24, 25]. Reformulate (3.15) as a so-calledR-linear problem (the RHS is an R-linear combination of the unknown functions) firstfor the unit circle

tψ(t) = tψ0(t) + tψ0(t) + β0, |t| = 1, (3.18)

where β0 ∈ R. In addition, reformulate (3.16) as the following R-linear condition onthe inclusions for k = 1, ..., n

(t− ak)

(ψ(t) +

2I1(|t|)akI0(1)|t|

)= (t− ak)ψk(t) + (t− ak)ψk(t) + βk, for |t− ak| = r,

(3.19)where βk are undetermined real constants. To solve (3.18)-(3.19), one must find afunction ψk(z) analytic in |z−ak| < r and continuous up to the boundary |z−ak| ≤ r(ψ0(z) in |z| ≥ 1) such that (3.18)-(3.19) is satisfied. The equivalence of (3.15)-(3.16)and (3.18)-(3.19) was proven in [26] for unbounded domains and a similar result isobtained here for bounded domains.

Lemma 3.1.(i) If ψ(z), ψk(z), and ψ0(z) are solutions of (3.18)-(3.19), then ψ(z) satisfies(3.15)-(3.16).(ii) If ψ(z) is a solution of (3.15)-(3.16), there exists a function ψ0(z) ∈CA(D0), ψk(z) ∈ CA(Dk), and real constants βk such that the R-linear condi-tions (3.18)-(3.19) are satisfied.

Proof. The proof of (i) is obvious. First assume (3.18)-(3.19), then take theimaginary part to get (3.15)-(3.16). Conversely, let ψ(z) satisfy (3.15)-(3.16). Thefunctions

Ψ0(z) :=β0

2+ zψ0(z), Ψk(z) :=

βk2

+ (z − ak)ψk(z), k = 1, ..., n (3.20)

can be uniquely determined respectively from the Schwarz problem for the unit diskD0 and for the kth disk Dk [15, 25]

2ReΨ0(t) = Retψ(t), |t| = 1, (3.21)

2ReΨk(t) = Re

(t− ak)

(ψ(t) + 2

I1(|t|)I0(1)|t|

ak

), |t− ak| = r. (3.22)

These problems have a unique solution for simply connected domains even though apriori the solution can vary by an additive imaginary constant, since ImΨ0(0) = 0and ImΨk(ak) = 0 from (3.20) [15]. Therefore, the functions ψk(z) and the constantsβk are uniquely determined in terms of Ψ(z) for k = 0, 1, ..., n.

As a solution of the Schwarz problem for |z| > 1, Ψ0(z) is analytic at∞. Thus, theanalytic function ψ0(z) has a zero of second order at infinity, since Ψ0(z) = β0

2 + c1z +

c2z2 + ... as z →∞, then ψ0(z) = 1

z

[Ψ0(z)− β0

2

]. Thus, by (3.20) ψ0(z) = c1

z2 + c2z3 + ...,

as z →∞.For use in later computations, divide (3.18) by t to find

ψ(t) = ψ0(t) +1

t2ψ0(t) +

β0

t, |t| = 1. (3.23)


Observe that tt = 1

t2 on the unit circle. For the boundary equations on the inclusions,divide (3.19) by (t− ak) to find

ψ(t) + 2I1(|t|)akI0(1)|t|

= ψk(t) +

(r

t− ak

)2

ψk(t) +βk

t− ak, |t− ak| = r,

where we use the fact that t−akt−ak = r2

(t−ak)2 . Then rearrange the terms to obtain

ψ(t) = ψk(t) +

(r

t− ak

)2

ψk(t) +βk

t− ak− 2

I1(|t|)akI0(1)|t|

, |t− ak| = r, k = 1, 2, ..., n.

(3.24)

3.3. Reduction to functional equations. Any function Fk(t) that is Holdercontinuous on Tk can be represented in the form of a difference of the limit values offunctions analytic in the interior and exterior of Tk by Sochocki’s formulas [15, 25]

Fk(t) = F+k (t)− F−k (t), |t− ak| = r. (3.25)

These functions are analytic in |z− ak| < r and in |z− ak| > r are given by a Cauchy

integral. We now are interested in the case Fk(t) = 2I1(|t|)akI0(1)|t| on |t−ak| = r. Introduce

the functions

Fk(z) =akπi

∫Tk

I1(|t|)I0(1)|t|

dt

t− z. (3.26)

as Cauchy integrals. Also, introduce the inversion transformation

t∗(k) :=r2

t− ak+ ak (3.27)

with respect to the circle |t− ak| = r where t = r2

t−ak + ak on ∂Dk.

Now we derive an equation for an analytic function ψ(z) valid for z ∈ D. The

main trick is to introduce an auxiliary function in all parts of the complex plane Csimilar to ideas presented in [4, 24, 25, 26]

Φ(z) =

ψk(z)−

∑m 6=k

(r

z−am

)2

ψm(z∗(m))−1z2ψ0(z∗(0))−

∑m 6=k

βmz−am − Fk(z), |z − ak| ≤ r

ψ0(z)−n∑

m=1

(r

z−am

)2

ψm(z∗(m))−n∑

m=1

βmz−am + β0

z, |z| ≥ 1

ψ(z)−n∑

m=1

(r

z−am

)2

ψm(z∗(m))−1z2ψ0(z∗(0))−

n∑m=1

[βmz−am − Fm(z)

], z ∈ D.

(3.28)

where z∗(m) is defined in (3.27) and z∗(0) is inversion with respect to ∂D0 (take a0 = 0

and r = 1 in (3.27)). One can check that Φ(z) is analytic in every subdomain of Cconsidered; namely, |z− ak| < r (k = 1, 2, ..., n), |z| > 1, D, and is continuous in theirclosures. While previously studied for unbounded domains the addition of the termψ0(z) accounting for the bounded domain D adds additional difficulty to establishing

the desired results. The next step is to show that Φ(z) is in fact analytic in C.Calculate the jumps of Φ(z) on each component of ∂D. First, consider the unit

circle and let “+” indicate the interior (to the left of the boundary curve) and “−”denote the exterior (to the right). More precisely, Φ+(t) := limz→t,|z|<1 Φ(z) and


Φ−(t) := limz→t,|z|>1 Φ(z). Observe that by (3.23) the difference on the boundary ofthe outer domain is zero,

Φ+(t)− Φ−(t) = ψ(t)− ψ0(t)− β0

t− 1

t2ψ0(t) = 0, |t| = 1. (3.29)

Observe in the third term there is no inversion in the unit circle, where t = t∗(0) is

used. In addition, by (3.24) and (3.26) the difference on the inclusion boundaries isalso zero

Φ+(t)−Φ−(t) = ψk(t)−ψ(t) +

(r

t− ak

)2

ψk(t) +βk

t− ak−Fk(t) = 0, |t− ak| = r.

(3.30)

From the Principle of Analytic Continuation, since there are no jumps and eachpiece is analytic, then the function Φ(z) is analytic on C. Now, consider Φ(z) andusing ψ0(∞) = 0 we find Φ(∞) = 0, since all the other terms in the definition of Φ(z)

go to zero as z →∞. Since Φ(z) is analytic in C, then, by Liouville Theorem, it mustbe a constant and Φ(z) ≡ 0 everywhere.

Therefore, we can set each part of the definition of Φ(z) equal to zero and find asystem of functional equations for ψk, (k = 0, 1, 2, ..., n):

ψk(z) =∑m 6=k

(r

z − am

)2

ψm(z∗(m)) +1

z2ψ0(z∗(0)) +

∑m 6=k

βmz − am

+ Fk(z), |z − ak| ≤ r

(3.31)

ψ0(z) =

n∑m=1

(r

z − am

)2

ψm(z∗(m)) +

n∑m=1

βmz − am

− β0

z, |z| ≥ 1.

(3.32)

Once the ψk are obtained then ψ(z) can be defined in the perforated domain, D, asfollows

ψ(z) =

n∑m=1

(r

z − am

)2

ψm(z∗(m)) +1

z2ψ0(z∗(0)) +

n∑m=1

[βm

z − am− Fm(z)

], z ∈ D.

(3.33)

3.4. Convergence of the method of successive approximations. Now thatthe functional equations (3.31)-(3.32) have been derived, we address the issue of con-vergence of successive approximations to a unique solution. Therefore, we obtain thefunction ψ(z) defined for z ∈ D needed to define the solution to the first approximationproblem, h(z). The following Lemma and Theorem are used to rigorously prove theconvergence to a unique solution. The Lemma will address the “homogeneous” casein the sense that the terms with βk and Fk(z) are not present in (3.31)-(3.32). Thiswill then be used to prove the necessary Theorem for convergence to a unique analyticsolution in the original system. The same approach applied to a similar problem for


an unbounded domain was considered in [6, 24, 25, 26].Lemma 3.2. Consider the system of functional equations (3.31)-(3.32) with re-

spect to the functions ψk(z) analytic in Dk for k = 0, 1, 2, ..., n where r < rtouching

ψk(z) =

n∑m=1m 6=k

(r

z − am

)2

ψm(z∗(m)) +1

z2ψ0(z∗(0)), |z − ak| ≤ r, (3.34)

ψ0(z) =

n∑m=1

(r

z − am

)2

ψm(z∗(m)), |z| ≥ 1. (3.35)

This system has only the trivial solution.Proof. Let ψm(z) (m = 0, 1, 2, ..., n) be a solution to the system (3.34)-(3.35).

Then the righthand side of (3.34) implies that the function ψk(z) is analytic in |z −ak| ≤ r (k = 1, 2, ..., n), since each ψm(z) obtained in Lemma 3.1 is analytic insideDm. Also, the righthand side of (3.35) implies that ψ0(z) is analytic in |z| ≥ 1.Introduce the auxiliary function

γ(z) :=

n∑m=1

(r

z − am

)2

ψm(z∗(m)) +1

z2ψ0(z∗(0)), (3.36)

which is analytic in the closure of D. Then for t ∈ ∂Dk, using (3.34), we find

γ(t) = ψk(t) +

(r

t− ak

)2

ψk(t). (3.37)

Thus, we have the following R-linear problem

t− akr

γ(t) =t− akr

ψk(t) +t− akr

ψk(t) = 2Re

t− akr

ψk(t)

, (3.38)

which implies that γ(t) has zero increment along each circle |t− ak| = r

[γ(t)]|t−ak|=r :=

∫|t−ak|=r

dγ(t) = 2

∫|t−ak|=r

d

[r

t− akRe

t− akr

ψk(t)

]= 0.

The last equality is due to the fact that rt−akRe

t−akr ψk(t)

has zero increment along

the circle |t− ak| = r, since ψk has zero increment as an analytic function in a simplyconnected domain. Hence, (3.38) implies

Im

[t− akr

γ(t)

]= 0, t ∈ ∂Dk, k = 1, ..., n. (3.39)

Along similar lines we derive the boundary condition

Im tγ(t) = 0, |t| = 1. (3.40)

The boundary value problem (3.39)-(3.40) has only the zero solution in the class ofanalytic single-valued functions in D [26]. Then using (3.37) with γ(t) ≡ 0 one findsfor k = 1, 2, ..., n

Re

[t− akr

ψk(t)

]= 0, |t− ak| = r. (3.41)


We can conclude that ψk(z) ≡ 0 in Dk for k = 1, 2, ..., n, since this Riemann-Hilbertproblem (3.41) has only the zero solution in the simply connected domain Dk [25]. Ifψk(z) ≡ 0, then by its definition (3.35) we have ψ0(z) ≡ 0. Therefore, the system(3.34)-(3.35) has only the trivial solution in CA.

Now that Lemma 3.2 for the homogeneous equations is established, we turn tothe desired non-homogeneous case. This result relies heavily on the previous Lemma.

Theorem 3.3. Consider the system of functional equations with respect to thefunctions ψk(z) analytic in |z − ak| < r and continuous in |z − ak| ≤ r for k =0, 1, 2, ..., n. Let f ∈ A and r < rtouching:

ψk(z) =

n∑m=1m6=k

(r

z − am

)2

ψm(z∗(m)) +1

z2ψ0(z∗(0)) + fk(z), |z − ak| ≤ r (3.42)

ψ0(z) =

n∑m=1

(r

z − am

)2

ψm(z∗(m)) + f0(z), |z| ≥ 1. (3.43)

where

f(z) :=

fk(z) =

n∑m6=k

βmz−am + Fk(z), |z − ak| ≤ r,

f0(z) =n∑

m=1

βmz−am −

β0

z , |z| ≥ 1,(3.44)

with fixed βk ∈ R for k = 0, 1, ..., n. Equations (3.42)-(3.43) have a unique solutionΨ ∈ CA where Ψ(z) := ψk(z) for |z − ak| ≤ r (k = 1, 2, ..., n) and Ψ(z) := ψ0(z) for|z| ≥ 1. The solution can be found by the method of successive approximations, wherethe approximations converge in CA.

Proof. Write the system (3.42)-(3.43) on the boundary of the disks Dk denotedTk in the form of a system of Cauchy integral equations, respectively

ψk(t) =∑m6=k

(r

t− am

)21

2πi

∫Tm

ψm(τ)

[1

τ − t∗(m)

]dτ

+1

t21

2πi

∫Tm

ψ0(τ)

[1

τ − t∗(0)

]dτ + fk(t), (3.45)

ψ0(t) =

n∑m=1

(r

t− am

)21

2πi

∫Tm

ψm(τ)

[1

τ − t∗(m)

]dτ + f0(t). (3.46)

The system (3.45)-(3.46) can be written as an equation in the space C (∪nk=0Tk) inthe following form

Ψ = AΨ + f. (3.47)

The integral operators in (3.45)-(3.46) are compact in C(Tk) [20] and the operators

of complex conjugation and multiplication by(

rt−am

)2

are bounded in C(Tk). Thus,

the composition of the compact and bounded operators results in a compact operator,and therefore A must be compact in C := C (∪nk=0Tk). If Ψ(z) is a solution of (3.47),then we have Ψ ∈ CA following from the properties of Cauchy integrals and the factthat f ∈ CA. Therefore, (3.47) and (3.42)-(3.43) are equivalent in CA as long as


f ∈ CA. By Lemma 3.2 the homogeneous equation Ψ = AΨ has only the trivialsolution. Thus, the Fredholm theorems then imply that (3.47) or equivalently thesystem (3.42)-(3.43) has a unique solution [32, 33].

Remark 3.4. We establish compactness in CA, since CA is a closed subspaceof C. Also, observe that the restriction of r < rtouching is crucial. If r = rtouching,then A is no longer compact in C since the Cauchy operators (3.45)-(3.46) can havesingular points in 1

τ−t∗(m)

for τ ∈ Tm and t ∈ Tk though k 6= m.

Now we prove the convergence of the method of successive approximations. Sincethe operator A is compact in C, then it is sufficient to prove convergence of thesuccessive approximations for the homogeneous equation

Ψ = λAΨ, (3.48)

with |λ| ≤ 1. By the Successive Approximation Theorem one must prove the in-equality ρ(A) < 1, where ρ(A) is the spectral radius of the operator A. Recall thatρ(A) := supx∈σ(A) |x|, where σ(A) is the spectrum of A [20], hence, σ(A) is the set of

λ−1 for which equation (3.48) has nontrivial solutions. This inequality is satisfied iffor all complex numbers λ such that |λ| ≤ 1, the equation (3.48) has only the trivialsolution. It is worth noting that the spectrum σ(A) contains a countable number ofeigenvalues σj accumulating at zero [20] such that |σ1| ≥ |σ2| ≥ .... Therefore, theproof that |λ| ≤ 1 will yield ρ(A) = |σ1| < 1.

Equation (3.48) can then be rewritten in the extended form

ψk(z) = λ

n∑m=1m6=k

(r

z − am

)2

ψm(z∗(m)) +1

z2ψ0(z∗(0))

, |z − ak| ≤ r, (3.49)

ψ0(z) = λ

[n∑

m=1

(r

z − am

)2

ψm(z∗(m))

], |z| ≥ 1. (3.50)

Consider the case when |λ| < 1. Introduce the following auxiliary function, whichis analytic in the closure of D

Γ(z) = λ

[n∑

m=1

(r

z − am

)2

ψm(z∗(m)) +1

z2ψ0(z∗(0))

].

Then Γ(z) and ψk(z) satisfy the following R-linear problem for k = 1, 2, ..., n

Γ(t) = ψk(t) + λ

(r

t− ak

)2

ψk(t∗(k)), |t− ak| = r. (3.51)

By Theorem 1 in [6], this R-linear problem has a unique solution, ψk(z), which is the

unique solution of the system (3.49) when

∣∣∣∣λ( rt−ak

)2∣∣∣∣ < 1 for |t− ak| = r. Through

(3.50) this defines a unique solution ψ0(z). This occurs when |λ| < 1 and thus, thetrivial solution is the only solution in this case.

Consider λ = e2iθ(0 ≤ θ < 2π) so that |λ| = 1 and follow the idea presented in[24]. Substitute ψk(z) = eiθωk(z) into the system (3.49)-(3.50), which is reduced tothe same system with λ = 1. Thus, an immediate application of Lemma 3.2 provesthat ωk(z) = ψk(z) = 0. Hence, ρ(A) < 1. This inequality proves the lemma.

Remark 3.5. The successive approximations converge to the unique solution of

(3.47), Ψ(z) =∞∑j=0

Ajf where Ψ := ψk in Dk for k = 0, 1, ..., n.


Observe that the limit function in the successive approximations is analytic in z.Theorem 3.3 shows that the method of successive approximations converges uniformlyin z to a unique solution. Thus, the resulting limit is analytic in z, since it is theuniform limit of analytic functions in the parameter r2 such that 0 < r2 ≤ r2

c for somer2c < r2

touching. The case of r = 0 is more subtle and must be handled separately.

Consider (3.32) for ψ0, when r = 0 one finds that ψ0(z) =n∑

m=1

βmz−am −

β0

z . As already

noted, it will be seen a posteriori in (3.62) that βm ∼ 1| ln(r)| → 0 as r → 0 (as does

β0, since β0 =∑nm=1 βm). In the next section, we prove that ψ0(z) → 0 as r → 0

and as a result one finds that (3.33) implies that ψ(z) ≡ 0 in the limiting case r = 0.Thus, Ψ(z) ≡ 0, which is trivially analytic in z.

3.5. Approximate solution of functional equations. In order to obtaintractable results, we seek a solution ψ(z) of (3.33) to order O(r2). Since r is ex-ponentially small, it is expected that this will be sufficient to observe the qualitativebehavior of the full solution. The first term in (3.33) is of order O(r2) since ψk is oforder O(1) and the coefficient is of order O(r2). We now consider the asymptotics of

the second term ψ0

(1z

)to determine which of its components will contribute to the

solution at an order lower than O(r2). First, use (3.32) to find the form of ψ0(z)

ψ0(z) =

n∑m=1

(r

z − am

)2

ψm(z∗(m)) +

n∑m=1

βmz − am

− β0

z, |z| ≥ 1.

Then estimate 1z2ψ0 (z∗0) = 1

z2ψ0

(1z

)up to order O(r4):

1

z2ψ0

(1

z

)=

1

z2

[n∑

m=1

r2

( 1z − am)2

ψm

(r2

z − am+ am

)+

n∑m=1

βm1z − am

− β0z

].

(3.52)

Since ψm(z) is analytic, take a Taylor expansion

ψm

(r2

z − am+ am

)= ψm(am) + ψ′m(am)

r2

z − am+O(r4)

Thus, after substitution of this Taylor expansion into (3.52) we obtain

1

z2ψ0

(1

z

)=

n∑m=1

r2

(1− amz)2ψm(am) +1

z

[n∑

m=1

βm1− amz

− β0

]+O(r4).

Recall that the function ψ0(z) has a second order zero at infinity; hence, 1z2ψ0

(1z

)is

analytic at z = 0. This implies that β0 =∑nm=1 βm. Therefore,

1

z2ψ0

(1

z

)=

n∑m=1

r2

(1− amz)2ψm(am) +

n∑m=1

βmam1− amz

+O(r4).

In computing the distribution of degrees we will only be concerned with approximatingψ(z) to order O(r2), and in this case we take

1

z2ψ0

(1

z

)=

n∑m=1

βmam1− amz

+O(r2). (3.53)


Remark 3.6. A priori we do not know that βk are bounded in r, it will beobserved later in (3.62).

3.5.1. Computation of ψ(z). We have established the convergence to a uniquesolution. We now want to express ψ(z) to order O(r2) and then determine the desiredsolution h(z). Once obtained we will plug the Rayleigh approximation hε ≈ hext(h(0)+h(1)) into the energy functional and minimize with respect to the n-tuple of degrees.Using (3.33), find ψ(z) up to O(r2)

ψ(z) =

n∑m=1

βm

[1

z − am+

11am− z

]+O(r2), z ∈ D. (3.54)

Integrate this equation to find the desired function ϕ(z) up to an arbitrary real con-stant c0

ϕ(z) = c0 +

n∑m=1

βm

[ln(z − am)− ln

(1

am− z)]

, z ∈ D. (3.55)

Observe that the second term in the summand is analytic in D (since 1am

/∈ D). Thusby matching coefficients of ln in (3.8) and (3.55) one sees that

Ak = βk for k = 1, 2, ..., n. (3.56)

Take the real part of (3.55)

Reϕ(z) = c0 +

n∑m=1

βm ln

∣∣∣∣∣ z − am1am− z

∣∣∣∣∣ , z ∈ D. (3.57)

Using the boundary conditions (3.14) obtain c0 and βm (with t = ak + reiθ):

Hk −I0(|t|)I0(1)

= c0 + βk lnr|ak|

1− |ak|2+∑m6=k

βm ln

∣∣∣∣ (ak − am)am1− amak

∣∣∣∣+O(r). (3.58)

We use the fact that ln(c1 + r) = ln(c1) +O(r). Hence, up to O(r) we have for eachk = 1, 2, . . . , n

c0 + βk lnr|ak|

1− |ak|2+∑m 6=k

βm ln

∣∣∣∣ |am|2 − amak1− amak

∣∣∣∣ = Hk −I0(|ak|)I0(1)

. (3.59)

Along similar lines, by use of (3.13), we obtain c0 +∑nm=1 βm ln

∣∣∣ eiθ−am1am−eiθ

∣∣∣ = 0 where

t = eiθ. The constant c0 must be real as a constant of integration; therefore, we

calculate the average for θ ∈ [0, 2π) of ln∣∣∣ eiθ−am1am−eiθ

∣∣∣ and obtain the approximate value

c0 = −n∑

m=1

βmln |am|1 + nr

. (3.60)

Substitute (3.60) into (3.59) to express each Hk (k = 1, 2, . . . , n) through a linearcombination of βm (m = 1, 2, . . . , n)

Hk =I0(|ak|)I0(1)

+ βk

[ln

r|ak|1− |ak|2

− ln |ak|1 + nr

]+∑m6=k

βm

[ln


∣∣∣∣− ln |am|1 + nr

].

(3.61)


It is worth noting that the main term in (3.61) is ln r as r → 0. The latter fact followsfrom (3.63) and boundedness of Hk as r → 0. Note that the problem (3.4) in thelimiting case when r = 0 yields the standard Dirichlet problem for the unit disk onhε for which the constants Hk = [hεext]

−1hε(ak) are bounded.The system of linear algebraic equations (3.61) for βk has a dominant diagonal

part for small r. This implies that the determinant of (3.61) is not small, hence,(3.61) has a unique solution. Thus, a well-defined numerical solution must exist forthis system. If we consider the terms at O(ln r) in (3.61) we find

βk ∼1

| ln r|. (3.62)

This approximation is not sufficient for further investigation, but it shows the order ofβk used above. We now describe the convergence of this series following [23] where lowfrequency approximations are considered in the small parameter ka where k is the wavenumber and a is the radius of the inclusion. In this work, we focus on an equivalent“Rayleigh Approximation” with first correction hε ≈ hext(h

(0) + h(1)) defined in [13]as the first nontrivial terms in the low frequency expansion. Analogously, here k =1 and a = 1

| ln r| so that ka 1. Observe that since the hole radius is assumed

to be exponentially small and the first correction h(1) ∼ 1| ln r| , then the Rayleigh

approximation already provides a solution with precision O(

1| ln r|

).

It follows from (3.12) and (3.56) that dk is related to Hk and βk by equation

dk =r2

2

[Hk −

I0(r)I0(|ak|)I0(1)

]− βk. (3.63)

Now that the first approximation h(1) is well-defined, in the next section, wecompute the corresponding energy.

4. Computation of the energy functional. Consider Eε(h(z, dεk)) definedin (2.4). Let E = 1

2 (I1 + I2) with I1 =∫D |∇h

ε|2dx and I2 =∫

Ω(hε − hext)2dx. From

Green’s formula and (2.5) we find

I1 = −∫Dhε∆hεdx+

∫∂Dhε∂hε

∂νds

= −∫D

(hε)2dx+

∫∂Ω

hext∂hε

∂νds−

n∑k=1

Hεk

∫∂Dk

∂hε

∂νds

and compute I2 =∫D(hε − hext)2dx + πr2

∑nk=1(Hε

k − hext)2. Combining the expres-sions for I1 and I2 one obtains

I1 + I2 = h2ext|D| − 2hext

∫Dhεdx + hext

∫∂Ω

∂hε

∂νds

−n∑k=1

Hεk

∫∂Dk

∂hε

∂νds+ πr2

n∑k=1

(Hεk − hext)

2. (4.1)

Now substitute into (4.1) the Rayleigh approximation for hε

hε ≈ hεext(h(0) + h(1)), Hεk = hextHk.


Hereafter, the approximate equality above and its validity hold in the sense of theRayleigh Approximation explained at the end of the previous section. We have

h−2ext(I1 + I2) ≈ |D| − 2

∫D(h(0) + h(1))dx +

∫∂Ω

[∂h(0)

∂ν+∂h(1)

∂ν

]ds (4.2)

−n∑k=1

Hk

∫Tk

[∂h(0)

∂ν+∂h(1)

∂ν

]ds− πr2

n∑k=1

(Hk − 1)2.

Using (3.6) we obtain the estimation∫Tk

∂h(0)

∂ν ds = O(r2). Neglecting terms of order

O(r2) we select the varing part of (4.2), i.e., depending on the parameters βk

J = −2

∫Dh(1)dx +

∫∂Ω

∂h(1)

∂νds−

n∑k=1

Hk

∫Tk

∂h(1)

∂νds (4.3)

The function h(1)(x) is harmonic in D. Hence, the integral of its normal derivativeover the boundary of D vanishes:∫

∂Ω

∂h(1)

∂νds =

n∑k=1

∫Tk

∂h(1)

∂νds (4.4)

We now find the integral using (3.8), (3.55), and (3.60)∫Dhdx =

n∑m=1

βm

(Jm −

|D|1 + nr

ln |am|), (4.5)

where

Jm :=

∫D

ln

∣∣∣∣∣ z − am1am− z

∣∣∣∣∣ dx =

∫|z|<1

ln

∣∣∣∣∣ z − am1am− z

∣∣∣∣∣ dx−n∑k=1

∫Dk

ln

∣∣∣∣∣ z − am1am− z

∣∣∣∣∣ dx, (4.6)

with z = x1 + ix2, dx = dx1dx2. Substitution of∫Tk

∂h(1)

∂νds = 2πβk

into (4.3) yields

J = 2

n∑k=1

βk

[π(1−Hk)− Jk + |D| ln |ak|

1 + nr

]. (4.7)

5. Numerical results. After substitution of (3.61) into (4.7) we find

J = 2

n∑k=1

βk

π − π I0(|ak|)

I0(1)− πβk

[ln

r|ak|1− |ak|2

− ln |ak|1 + nr

](5.1)

−∑m6=k

πβm

[ln


∣∣∣∣− ln |am|1 + nr

]− Jk + |D| ln |ak|

1− nr

+O(r).

One can see that J = J [β1, β2, . . . , βn] is a quadratic form in β1, β2, . . . , βn. Thus, inorder to find the minimum of (4.7), we take the gradient with respect to β1, β2, . . . , βn


and equate it to zero. This yields a system of linear equations, which can then besolved numerically.

0 =∂J

∂βi= 2

π − π I0(|ai|)

I0(1)− 2πβi

[ln

r|ai|1− |ai|2

− ln |ai|1 + nr

]− Ji + D ln |ak|

1nr(5.2)

−∑m 6=i

πβm

[ln

∣∣∣∣ |am|2 − amai1− amai

∣∣∣∣− ln |am|1 + nr

]−∑k 6=i

πβk

[ln

∣∣∣∣ |ai|2 − aiak1− aiak

∣∣∣∣− ln |ai|1 + nr

].

This forms a linear system of the form Aβ = r where

Aik = −2π

[ln

∣∣∣∣ |ak|2 − akai1− akai

∣∣∣∣− ln |ak|1 + nr

]+

[ln


∣∣∣∣− ln |ai|1 + nr

](5.3)

Aii = −4π

[ln

r|ai|1− |ai|2

− ln |ai|1 + nr

], ri = −2

π − π I0(|ai|)

I0(1)− Ji + D ln |ak|

1nr

. (5.4)

Satisfying this system is only a necessary condition, we now distinguish whetherthe solution as a critical point is the unique global minimizer using the Hessian. Wehave a true global minimum if the Hessian is positive definite.

∂2J

∂βi∂βp= −2π

2δip

[ln

r|ai|1− |ai|2

− ln |ai|1 + nr

](5.5)

−∑k 6=i

δkp

[ln

∣∣∣∣ |ak|2 − akai1− akai

∣∣∣∣− ln |ak|1 + nr

]−∑k 6=i

δkp

[ln


∣∣∣∣− ln |ai|1 + nr

].

Thus, the Hessian Matrix M has the form:

Mii = −4π

[ln

r|ai|1− |ai|2

− ln |ai|1 + nr

], (5.6)

Mip = −[ln

∣∣∣∣ |ap|2 − apai1− apai

∣∣∣∣− ln |ap|1 + nr

]+

[ln

∣∣∣∣ |ai|2 − aiap1− aiap

∣∣∣∣− ln |ai|1 + nr

]. (5.7)

Since the Hessian is symmetric it is sufficient to show that if it contains positivepivots, then the Hessian will be positive definite. This was checked numerically andit was observed that all pivots are indeed positive. This follows from the observationthat the matrix M has dominate positive diagonal elements Mii for sufficiently smallr. Therefore, M is symmetric with positive pivots so it is therefore positive definite.This can also be seen by observing that − ln(r|ai|) is the dominant (positive) termand occurs only on the diagonal . Thus, the off diagonal terms can be viewed as asmall perturbation from the identity matrix, δip, which is positive definite. Since wehave a positive definite Hessian it guarantees we have found at least a local minimizer.Also, knowing the energy functional is convex, the local minimizer is in fact a globalminimizer.

Once the linear system for d is solved, it must then be multiplied by the ex-ternal magnetic field hext and made an integer, dint = [hextd] (quantization). Thisprocedure will give the solution with accuracy O(r) due to the approximation of thelinear system for c0, βk in (3.58). When the minimizer β∗1 , β

∗2 , . . . , β

∗n of (4.7) is found,

the minimizer d∗1, d∗2, . . . , d

∗n is calculated by formulas (3.63) and (3.61). In the next

section, the vortex distribution is computed using the above method and system pa-rameters such as the applied magnetic field and the randomness in the distributionof holes will be investigated.


As r → 0, we see from (3.63) that dk ≈ −βk. Also, as r → 0 βk ∼ 1| ln r| → 0. Thus,

the energy is minimized when dk ≡ 0 and the resulting effective vorticity is identicallyzero. This makes sense, because as r → 0 with fixed n and ε the hole sizes shrink tozero, but the hole concentration also goes to zero. As this occurs the energy becomesmuch greater and cannot support the hole vortices resulting in a homogenized vorticityof zero. This is consistent with the previous theory since the external magnetic field istaken below the first critical value resulting in the suppression of bulk vortices. Thus,in the absence of holes in the limit and the absence of bulk vortices one should expectthe resulting vortex distribution to be identically zero.

5.1. Periodic case with varying external magnetic field. To justify themethod developed in this work, we begin by considering the case of a periodic arrayof holes separated by distance ε. To verify the analytical results derived above, wecompare the results of our simulations for a periodic array of holes to the resultsobtained in [5, 17]. The results of our numerical simulations verify the method andthe fact that the Rayleigh approximation provides the qualitatively correct picture inthe periodic case. In addition, the simulations reveal a nested structure with a vortexphase separation (see Figure 5.1). The formation of nested subdomains at variouslevels of the external magnetic field is consistent with the theoretical predictions of[5, 17]. Each of the nested subdomains takes the form of the original domain due tothe symmetry in the array of holes.

This system displays many interesting features, one of which is that the numberof subdomains is not a continuous spectrum. By this we mean that the subdomainsof like vorticity do not change instantaneously with a change in the external mag-netic field. Rather the vortex distribution remains constant for a range of valuesuntil reaching the next critical value corresponding to a formation of a subdomain ofhigher vorticity at the domain center. This next subdomain only forms at certain res-onant values of the external magnetic field as first proven rigorously in [5]. Also, thesimulations show that each vortex subdomain containing hole vortices of like degreeis completely surrounded by a subdomain of lower degree. This structure dependscrucially on the assumption of exponentially small holes, | log(κ)| | log(r)| as firstasserted in [17]. Before proceeding to the case of arbitrary hole locations we firstinvestigate what is known as the “shaking” geometry where each hole in the periodicarray is perturbed by a small amount in a randomly chosen direction.

Fig. 5.1. External magnetic field hext = σε2

, with σ = 2πj + (j − .5)γ and a) j = 2, b) j = 50,c) j = 500 respectively.


5.2. Shaking geometry case with varying external magnetic field. Theshaking geometry was first studied in the context of deriving an effective conductivityfor a two-phase composite material [4]. In that work it was shown that the periodiccase is special in that it is the arrangement that produces an extremal value for theeffective conductivity in the composite. Similarly in superconductors, it was shownthat the periodic case of holes produces extremal values for the fractal dimension ofthe interface between the region of degree zero and the region of degree one [29].

To define the shaking geometry consider a periodic lattice of circles of radiusα < ε

2 in the sample domain. The centers of the holes ak are placed at randomlocations inside each (former) periodic cell of size ε according to a uniform distributionin this circle (see Figure 5.2a)). The points ak are considered to be the centers of eachcircular inclusion (blue disks in Figure 5.2a)). Due to the construction the number ofholes in the computational domain remains the same as well as the average distancebetween, ε. All inclusions have the same radius, which ensures that each remainsstrictly inside the periodicity cell. Thus, the parameter d = α/ε is a dimensionlessquantitative measure of the degree of disorder for the shaking geometry (e.g., if d = 0then circular inclusions form a perfect periodic lattice).

Fig. 5.2. a) Illustration of “shaking” geometry, the circles at the center of each cell are theregions where the hole locations have equal probability of occuring. (b) HT−phase plane showingregions where hole vortices appear.

The results are as one might expect: the vortex distribution looks like a small per-turbation from the periodic case where the nested structure of subdomains at variouslevels of the external magnetic field remains visible; however, the interface becomesless smooth, see Figure 5.3. This case provides some evidence of what is to come as therandomness in hole locations increases; namely, one sees that the interface becomesmore jagged or fractal. In the next section, the vortex distribution is investigatedas the distribution of hole locations becomes increasingly more random (disordered).First, the procedure for quantifying randomness and choosing the locations of theholes is introduced.

5.3. Algorithm for the transition from periodic to random. We nowemploy a method for considering the transition from periodic to random developedfirst in [11, 12]. Let the centers ak ∈ C be random variables distributed so that thedisks Dk = z ∈ C : |z−ak| < r for k = 1, 2, ..., n form a set of uniformly distributednon-overlapping disks. This uniform non-overlapping distribution will be denoted byU , i.e., the points a1, ..., an are considered as identically distributed random variableswith the restriction that the holes should not overlap (e.g., |am − ak| > 2r for m 6= kwith m, k = 1, 2, ..., n).

Now the constructive procedure based on random walks of each hole location ak


Fig. 5.3. Results for the “shaking” geometry where the external magnetic field hext = σε2

, withσ = 2πj + (j − .5)γ and a) j = 2, b) j = 50, c) j = 500 respectively.

is described. First, place the centers into a periodic array of regular nodes in the disk|z| ≤ 1. Choose a constant ρ > 0 such that ρ < mink 6=m |ak − am| − 2r. Then take arandomly chosen direction φ ∈ [0, 2π) and translate ak a distance ρ in this directionso that each center obtains new coordinates

a′k = ak + ρeiφk = (xk + ρ cos(φk), yk + ρ sin(φk)) .

Repeat this M times where a center is moved if |a′k − am| ≥ 2r, if not, then thecenter ak does not move at this step and it is called “blocked”. Since the radii of theholes are exponentially small the probability of a center being blocked is almost zero.After a sufficiently large number of walks (≈ 80) the obtained location of the center isconsidered a statistical realization of the random distribution U [11, 12]. The typical

value for ρ used in the simulation results presented is ρ := 15

(ω1√n− 2r

)where ω1 is

the domain size and n is the number of holes. If all k = 1, ..., n centers move at a givenstep it is called a cycle. We can view this process of generating a random walk asrepeating the process of forming the shaking geometry a given number of times. Theshaking geometry corresponds to the case M = 1 and the periodic array correspondsto the case M = 0. Thus, by introducing these random walks, we now have a way toquantify the amount of randomness in the system according to a spectrum betweenperiodic and Poisson distributions of holes.

Fig. 5.4. Results for a random geometry (M = 80) where the external magnetic field hext = σε2

,with σ = 2πj + (j − .5)γ and a) j = 2, b) j = 50, c) j = 500 respectively.


5.4. Random geometry case with varying external magnetic field. Usingthe algorithm defined in the previous section, the effect of randomness on the effectivevortex distribution is investigated. As the number of steps in the random walk, M ,increases we consider the amount of randomness to have increased until around M =80 where the distribution is statistically random [11, 12]. The resulting set of centersa′k will automatically fit a Poisson distribution by construction (cf. [11, 12, 31]).

Due to the increasing randomness in the hole locations we no longer have cen-tralized nested subdomains. Rather, one observes that the interface between thesubdomains becomes less smooth and the domains of like vorticity have drasticallydifferent structures, see Figure 5.4. Our results still show a continuous transition ofdegrees (no big jumps in vorticity) radially outward from the domain center. Due torandomness the subdomains no longer have a clearly defined shape or form revealingthat each is a function of the distribution of holes. It was observed in the periodic caseabove that the subdomains keep the essential structure of the entire sample domain(e.g., a convex domains imply convex subdomains), but this is no longer the case inthe random setting. We also observe that the number of hole vortices of like degreeremain approximately the same as the periodic case suggesting that the associatednumbers of each degree are independent of the distribution of holes, but its particularstructure is not.

Fig. 5.5. Effective vortex distribution with increasing randomness a) M = 0 the periodic case,b) M = 40, and c) M = 80. This illustrates the increasing fractal dimension, D, of the interfacebetween the subdomain of degree 0 (blue) and degree 1 (red) with, the parameter M , the measure ofrandomness.

As reported in our prior work [29] many interesting results are revealed as com-pared to the periodic setting. Namely, as randomness increases the interface becomesmore fractal, the subdomains containing vortices move closer to the sample boundary,and the critical magnetic field for the onset of hole vortices decreases (see Figure 5.5and [29] for illustrations). We use the Box-counting method to determine the fractaldimension stopping at box sizes roughly equivalent to inter-hole distance. In addition,new data obtained from the methods developed herein allows one to estimate the frac-tal dimension, D = limN→∞DN (see Figure 5.6). Since the distribution of vorticescrucially depends on the number of holes, we choose the magnetic field for each datapoint to be such that the nonzero degree regions (none-blue regions in Figure 5.5) areapproximately the same size.

Remark 5.1. The theoretical fractal dimension, D, is calculated as the limit ofDN when the number of holes N goes to infinity.

The results of our studies fundamentally change how the traditional magnetic


field–temperature phase diagram is viewed in the presence of holes by investigatingthe delicate balance between vortex-vortex, vortex-defect, and confinement within afinite system in determining the effective vortex distribution, see Figure 5.2b). Inaddition, this work was carried out under a strong assumption on the relationshipbetween the hole radius, r, and average distance between holes, ε; namely, r = e−γ/ε

2

.It remains to study the corresponding effect on the vortex distribution if there is adifferent relationship such as r = e−γ/ε and this may be the subject of future work.

0 5000 10000

No. of Holes

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

D, F

racta

l dim

ensi

on

Fig. 5.6. A sequence of fractal dimension computations, DN , as the number of holes N in-creases. Around N = 300 the behavior begins to change. This plot gives an estimation of the fractaldimension D of the lmit curve as N goes to infinity.

Acknowledgment. The authors thank V. M. Vinokur for useful discussions onthe physical and experimental implications of this work. The authors would alsolike to thank Wojciech Nawalaniec for the algorithm used to study the transitionfrom periodic to random hole locations in the numerical simulations. The work ofVM and SR was partially supported by NSF grant DMS-1106666 and the work ofLB was partially supported by NSF grants DMS-1106666 and DMS-1405769. Theproblem was conceived and part of the work was done while V. Mityushev was visitingthe Department of Mathematics at the Pennsylvania State University. The work ofSR was mostly done while at the Pennsylvania State University, but at the time ofsubmission SR moved to the Department of Mathematical Sciences and the LiquidCrystal Institute at Kent State University.

REFERENCES

[1] S. Alama and L. Bronsard, “Pinning effects and their breakdown for a Ginzburg-Landau modelwith normal inclusions”, J. Math. Phys. 46 pp. 39 (2005).

[2] S. Alama and L. Bronsard, “Vortices and pinning effects for the Ginzburg-Landau model withmultiply connected domains”, Comm. Pure Appl. Math. 59 pp. 36-70 (2006).

[3] H. Aydi and A. Kachmar, “Magnetic vortices for a Ginzburg-Landau type energy with discon-tinuous constraint. II”, Commun. Pure Appl. Anal. 8 pp. 977-998 (2009).

[4] L. Berlyand and V. Mityushev, “Generalized Clausius-Mosotti formula for random compositewith circular fibers”, Journal of Statistical Physics 102 pp. 115-145 (2001).

[5] L. Berlyand and V. Rybalko, “Homogenized description of multiple Ginzburg-Landau vorticespinned by small holes”, special issue NHM 8:1 pp. 115-130 (2013).


[6] B. Bojarski and V. Mityushev, “R-linear problem for multiply connected domains and alternatingmethod of Schwarz”, Journal of Mathematical Sciences 189:1 (2013).

[7] L. Civale et al. “Vortex confinement by columnar defects in Y Ba2Cu3O7 crystals: Enhancedpinning at high fields and temperatures. Phys. Rev. Lett. 67, pp. 648-651 (1991).

[8] D. Colton and R. Kress, “Integral Equation Methods in Scattering Theory”, John Wiley & Sons(1983).

[9] R. Courant and D. Hilbert, “Methods of Mathematical Physics”, Wiley-VCH Verlag GmbH &Co. KGaA, v.2 (1962).

[10] R. Czapla, V. Mityushev, and N. Rylko, “Conformal mapping of circular multiply connecteddomains onto slit domain”, ETNA 39 pp. 286-297 (2012).

[11] R. Czapla, W. Nawalaniec, V. Mityushev, “Effective conductivity of random two-dimensionalcomposites with circular non-overlapping inclusions”, Computational Materials Science 63pp. 118-126 (2012).

[12] R. Czapla, W. Nawalaniec, V. Mityushev, “Simulation of representative volume elements forrandom 2D composites with circular non-overlapping inclusions”, Theoretical and AppliedInformatics, 24:3 pp. 227-242 (2012). DOI: 10.2478/v10179-012-0014-3.

[13] G. Dassios and R. Kleinman, “Low Frequency Scattering”, Oxford Mathematical Monographs(2000).

[14] M. Dos Santos and O. Misiats, “Ginzburg-Landau model with small pinning domains”, Netw.Heterog. Media 6:4 pp. 715-753 (2011).

[15] F. D. Gakhov, “Boundary Value Problems”, Dover Publications Inc. (1990).[16] L. Berlyand, D. Golovaty, O. Iaroshenko, and V. Rybalko (In Preparation) (2014).[17] O. Iaroshenko, V. Rybalko, V. M. Vinokur, and L. Berlyand, “Vortex separation in mesoscopic

superconductors”, Scientific Reports 3 1758 (2013).[18] A. Kachmar, “Magnetic vortices for a Ginzburg-Landau type energy with discontinuous con-

straint”, ESAIM Control Optim. Calc. Var. 16 pp. 545-580 (2010).[19] M. Konczykowski et al. “Effect of 5.3-GeV Pb-ion irradiation on irreversible magnetization in

Y-Ba-Cu-O crystals”, Phys. Rev. B 44, pp. 7167-7170 (1991).[20] M. A. Krasnosel’skii, Ja. B. Rutiticki, V. Ja. Stecenko, G. M. Vainikko, P. P. Zabreiko, “Ap-

proximate Solutions of Operator Equations”, Walters - Noordhoff Publ., Groningen (1972).[21] L. Lassoued and P. Mironescu, “Ginzburg-Landau type energy with discontinuous constraint”,

J. Anal. Math. 77 pp. 1-26 (1999).[22] G. P. Lousberg, M. Ausloos, P. Vanderbemden, and B. Vanderheyden, “Bulk high-Tc super-

conductors with drilled holes: How to arrange the holes to maximize the trapped magneticflux”, Supercond. Sci. Technol. 21 pp. 025010 (2008).

[23] P. Martin, “Multiple Scattering: Interaction of time-harmonic waves with N obstacles”, Cam-bridge University Press pp. 284-285 (2006).

[24] V. Mityushev “Scalar Riemann-Hilbert problem for multiply connected domains”, FunctionalEquations in Mathematical Analysis, Springer, ed. T M. Rassias and J. Brzdek (2012).

[25] V. Mityushev and S. V. Rogosin, “Constructive Methods for Linear and Nonlinear BoundaryValue Problems for Analytic Functions”, Chapman and Hall/CRC (1999).

[26] V. Mityushev,“Riemann-Hilbert problems for multiply connected domains and circular slitmaps”, Computational Methods and Function Theory 11:2 pp. 575-590 (2011).

[27] V. Mityushev and P. Adler, “Longitudinal permeability of a doubly periodic rectangular arrayof circular cylinders II”, ZAMP 53 pp. 486-517 (2002).

[28] V. Mityushev and P. Adler, “Longitudinal permeability of a doubly periodic rectangular arrayof circular cylinders I”, ZAMM 82:5 pp. 335-345 (2002).

[29] S. D. Ryan, V. Mityushev, V. M. Vinokur, and L. Berlyand, “Rayleigh approximation for groundstates of the Bose and Coulomb glasses”, Scientific Reports 5 pp. 7821 (2015).

[30] N. Rylko, “Structure of the scalar field around unidirectional circular cylinders”, Proc. R. Soc.A 464 pp. 391-407 (2008).

[31] W. W. Sampson, “Modelling Stochastic Fibrous Materials with Mathematica”, Springer (2009).[32] A. N. Tikhonov and A. A. Samarshii, “Equations of Mathematical Physics”, Dover Publications

(2011).[33] V. S. Vladimirov, “Equations of mathematical physics”, translated from Russian by Eugene

Yankovsky, Mir, Moscow, (1984).

6. Appendix.

6.1. Derivation of the Riemann-Hilbert Problem. In this appendix, theRiemann-Hilbert problem (3.15)-(3.16) is derived from (3.13)-(3.14). First, considerthe relation (3.13) on |t| = 1. The parameterization t = exp(is) with 0 ≤ s < 2π


can be introduced on the unit circle. Then the tangent derivative is treated as thederivative d

ds = dtds

ddt + dt

dsddt with dt

ds = it and dtds = − it . This yields the differentiation

rule on |t| = 1 [10]:

d

ds= it

(d

dt− 1

t2d

dt

)(6.1)

Application of (6.1) to (3.13) gives

ψ(t)− 1

t2ψ(t) = 0, |t| = 1 (6.2)

that is equivalent to (3.15). Similar arguments yield the differentiation rule on |t −ak| = r [10]:

d

ds= i(t− ak)

[d

dt−(

r

t− ak

)2d

dt

]. (6.3)

Moreover,

dt

ds= i(t− ak),

dt

ds= −i(t− ak), (6.4)

where t = ak + r exp(is) on the circle |t− ak| = r.Application of the differential operator (6.3) to (3.14) yields

ψ(t)−(

r

t− ak

)2

ψ(t) = − 2

I0(1)

[dI0(|t|)dt

−(

r

t− ak

)2dI0(|t|)dt

](6.5)

Equation (6.5) can be written in the form

ψ(t)−(

r

t− ak

)2

ψ(t) = −2I1(|t|)I0(1)

d|t|dt

[1−

(r

t− ak

)2dt

dt

](6.6)

It follows from (6.4) that dtdt = −r−2(t− ak)2. Then (6.6) becomes

ψ(t)−(

r

t− ak

)2

ψ(t) = −4I1(|t|)I0(1)

d|t|dt. (6.7)

Using rt−ak

= t−akr calculate

d|t|dt

=1

2|t|d|t|2

dt=

1

2|t|d

dt

[t

(r2

t− ak+ ak

)]=

1

2|t|

[t− t

(r

t− ak

)2]. (6.8)

Substitute (6.8) into (6.7) and multiply the result by t−akr

t− akr

ψ(t)− r

t− akψ(t) = −2

I1(|t|)I0(1)|t|

[(rt

t− ak

)− rt

t− ak

]. (6.9)

Take the imaginary part

Im

[t− akr

ψ(t)

]= 2

I1(|t|)I0(1)|t|

Imrt

t− ak. (6.10)

The latter relation is equivalent to (3.16), since Im rtt−ak = −Im t−akr ak.


6.2. Components of the energy functional. In this appendix the compo-nents of the energy Jm defined in (4.6) are computed. The following integrals can beverified by Mathematica c© (directly or by Taylor expansion in r where z = am + reiθ)∫

|z|<1

ln

∣∣∣∣∣ z − am1am− z

∣∣∣∣∣ dx1dx2 = π ln

∣∣∣∣|am|2 − am∣∣∣∣+ π ln

∣∣∣∣ amam − 1

∣∣∣∣∫∂Dk

ln

∣∣∣∣∣ z − am1am− z

∣∣∣∣∣ dx1dx2 =π

2r2

[−1 + 2 ln

(r|am|

||am|2 − 1|

)]∫∂Dk

ln

∣∣∣∣∣ z − am1am− z

∣∣∣∣∣ dx1dx2 =π

2r2 ln

∣∣∣∣ |am|2|ak − am|2

(1− xkxm − iykxm + ixkym − ykym)2

∣∣∣∣ (k 6= m)

where aj = xj + iyj for j = 1, 2, ..., n. Therefore, from (4.6)

Jm = π ln

∣∣∣∣|am|2 − am∣∣∣∣+ π ln

∣∣∣∣ amam − 1

∣∣∣∣−π2 r2

[−1 + 2 ln

(r|am|

||am|2 − 1|

)]− πr2 ln

r

|am|

−n∑

k 6=m

π

2r2 ln

∣∣∣∣ |am|2|ak − am|2

(1− xkxm − iykxm + ixkym − ykym)2

∣∣∣∣ . (6.11)

The third term in (6.11) contains interactions between inclusions |ak−am| and provesto be one of the key differences between the periodic and random cases.

To develop an intuition for how the energy behaves, take some of the main con-tributors from the energy (4.7) and use (6.11) to compute an approximate energy. Theintegral Jm from (6.11) can be estimated up to O(r). Let nr be sufficiently small.Then

Jk = π ln

∣∣∣∣|ak|2 − ak∣∣∣∣+ π ln

∣∣∣∣ akak − 1

∣∣∣∣+O(r) = 2π ln |ak|+O(r). (6.12)

Moreover the last term from (4.7) can be estimated as follows

|D| ln |ak|1 + nr

= ln |ak|+O(r). (6.13)

Taking the main term in (3.61) we obtain

Hk =I0(|ak|)I0(1)

+ βk ln r +O(1/| ln r|). (6.14)

Substitution of (6.12)-(6.14) into (4.7) yields

J = −2π

n∑k=1

βk

[βk ln r +

I0(|ak|)I0(1)

+ ln |ak| − 1

]+O(1/| ln r|). (6.15)

The minimum of (6.15) is attained for

β∗k =1

2 ln r

[1− ln |ak| −

I0(|ak|)I0(1)

]. (6.16)

Since ln r < 0, ln |ak| < 0 and 1 − I0(|ak|)I0(1) > 0 for 0 < |ak| < 1, all β∗k are negative.

Therefore, dk ≥ 0 for sufficiently small r by (3.63).


As |ak| → 0, − ln |ak| → ∞ and I0(|ak|) → 1. Thus, βk → −∞ resulting inincreasing positive degrees dk for hole locations closer to the center. This approxi-mation illustrates the physical result that the homogenized vorticity increases as theak tend to zero giving the intuition for the nested structures observed in the figuresthroughout this work. Thus, inclusions with center near the origin have higher degreethan those near the boundary.

Date post:	12-Aug-2020
Category:	Documents
Upload:	others
View:	5 times
Download:	0 times

MULTIPLE GINZBURG-LANDAU VORTICES PINNED BY …MULTIPLE GINZBURG-LANDAU VORTICES PINNED BY RANDOMLY...

Documents