ginzburg-landau model of a liquid crystal with … · or multiply-connected random geometries [14],...

GINZBURG-LANDAU MODEL OF A LIQUID CRYSTAL WITH RANDOM INCLUSIONS

L. BERLYAND ∗ AND E. KHRUSLOV †

Abstract. We consider a Ginzburg–Landau 3D functional with a surface energy term to model a nematic liquid crystalwith inclusions. The locations and radii of the inclusions are randomly distributed and described by a set of finite dimensionaldistribution functions. We show that the presence of inclusions can be accounted for by an effective potential. Our main objectivesare: (a) to derive the sufficient conditions on the distribution functions such that the solutions converge in probability to a solutionof a homogenized deterministic problem and (b) to compute the effective potential.

Key words. Ginzburg–Landau functional, homogenization, liquid crystals, random inclusions

AMS subject classifications.

1. Introduction. Intermediate between ordered solids and amorphous liquids, liquid crystals possess bothspecial structure and unique optical properties. The simplest kind of a liquid crystal, known as a nematic,is composed of rod-like molecules exhibiting local orientational order. That is, the molecules locally orientthemselves along some preferred direction, while maintaining the ability to move around freely. The preferreddirection can vary from point to point and coincides locally with the direction of the optic axis. As for anyheterogeneous, optically uniaxial medium, the overall optical properties of a nematic material are determinedby the spatial distribution of the direction of the optic axis.

Because positions of their molecules are not constrained, nematic liquid crystals can flow like liquids. Atthe same time, spatially non-uniform orientational order can produce elastic interactions that lead to complexnematic structures with elaborate patterns and topological defects. These structures can be subsequentlymanipulated by influencing molecular orientations with external electromagnetic forces. The correspondingchanges in optical characteristics drive most of the current practical uses of liquid crystal materials.

Although in their “pure” form liquid crystals have been widely used in a variety of important applications,most notably liquid crystal displays, a significant research effort has been concentrated recently on liquidcrystal-based composites. These new materials are of considerable interest for display technologies based uponchanging the light scattering properties of composite systems via external fields. Such systems can have distortedor multiply-connected random geometries [14], such as those produced by the polymer-dispersed liquid crystals(PDLC) [13], or dispersions of agglomerations of silica spheres in a nematic host [17].

A structure of a liquid crystal-based composite depends strongly on whether or not a liquid crystal is usedas its host material. For example, in a direct nematic emulsion [14] a nematic dispersed in water separates intodistinct, nearly spherical drops. The drops have simple structures dependent on the boundary conditions at thenematic-water interface. For this reason, direct liquid crystal emulsions have been used as a model medium tostudy topological defects [14].

An inverse nematic emulsion [19] differs from a direct emulsion in that isotropic water droplets are dispersedin a nematic host. Structures of inverse emulsions are significantly more complex than direct emulsions. Ininverse nematic emulsions and, more generally, in colloid suspensions in nematic liquid crystals, the interactionsbetween foreign inclusions in a nematic host lead to formation of a variety of novel ordered and disorderedstructures. A defining impact on formation of new structures is made by both anchoring conditions on thesurfaces of inclusions and the global geometry of a liquid crystalline matrix.

The rheological effects in liquid crystals with inclusions depend crucially on the volume fraction of inclusions(see e.g., [26] ) For example, a soft solid with a significant storage modulus was obtained in [20] by mixing modelcolloidal particles with a thermotropic nematic liquid crystal. The suspended particles with very small radii,behaving as nearly-perfect hard spheres, constituted up to 20% of the mixture. The colloid-liquid-crystalcomposite described in [20] is a switchable electro-optical solid material, giving it significant advantages inhandling and processing over the conventional, free-flowing liquid crystals.

In this paper we consider a mathematical model for a class of nematic liquid crystal composites with lowvolume fraction of small randomly distributed inclusions. Within this model, a nematic liquid crystal is described

∗Department of Mathematics and Materials Research Institute, Penn State University, University Park, PA 16802([email protected])

†Mathematical Division, Institute for Low Temperature Physics and Engineering, 47 Lenin ave., 61164 Kharkov, Ukraine([email protected])

1

2 L. BERLYAND AND E. KHRUSLOV

by the Ginzburg-Landau functional (2.2) with a (positive or negative) surface energy term. We assume that boththe surface energy density and sizes of inclusions are controlled by the same small parameter ε. Randomness inthe particle’s sizes and locations is the main issue of our consideration and the main two objectives of this workare: (a) to derive the conditions on the distribution functions such that the solutions converge in probabilitymeasure to a solution of a homogenized deterministic problem and (b) to compute the effective potential.

A similar problem for a deterministic geometry characterized by a small volume fraction of inclusions wasconsidered in [4], [6] ( see also [16], [25] for physical problems). It was shown that the presence of inclusions canbe accounted for by an effective potential that was computed explicitly as a function of material parametersand geometric characteristics of inclusions. Two main control parameters were introduced - the average size ofinclusions and the inverse intensity of the surface energy. The asymptotic limits when both parameters tendto zero were considered and all possible relationships between these two parameters that lead to nontrivialhomogenization limits were identified and studied. We emphasize that these relationships were not deducedfrom a specific physical problem - on the contrary, they arose in the course of the homogenization analysis ofthe model. The relevance of these relationships to liquid crystal composites is an interesting open questionsuggested by our analysis. In present paper we show that in the random setting the same scaling relationshipsleads to a nontrivial deterministic limit.

The effective potential was represented as a sum of two terms responsible for the surface and the bulk energyof a thin boundary layer around inclusions, respectively. The analytic formulas for the effective potential thatwere obtained in [6] do not require the solution of a cell problem. (cell problems for linear elliptic problems aredefined in [3].) An additional geometric condition under which the homogenization procedure was carried outin [4], [6] was that the inclusions cannot form clusters.

The presence of the surface energy term in a variational formulation of our problem implies that theminimizer (which solves the the nonlinear Ginzburg-Landau equation) is subject to Robin boundary conditionson surfaces of inclusions. For linear scalar problems (Laplace operator) a homogenization problem for perforateddomains with Robin boundary condition on boundaries of holes has been studied by several authors. In [8], [9]the case of large holes, where the homogenized operator becomes anisotropic has been considered by using themethod of mesocharacteristics. In [11], [12] several possible relations between a parameter in Robin boundarycondition and sizes of periodically arranged holes have been thoroughly studied and classified. In the same work,a version of the two-scale approach ([1], [3], [15], [21]) suitable for the analysis of Robin boundary condition onsurfaces of holes, has been developed.

The case of deterministic inclusion that remain sufficiently far apart (the ratio of the size of an inclusionto the distance between it and other inclusions is bounded) was treated in [5]. The domains were not requiredto have periodic geometry, and the surface energy term in (2.2) was not assumed to be negative. The mainconsequence of the lack of non-negativity is that there is no a priori lower bound on the energy, and this boundhas to be established independently.

Under these assumptions on the functional and the geometry of the domain, it was shown in [5] that onecan account for inclusions by an anisotropy of the homogenized differential operator and an effective potential.The potential can be viewed as an effective external field. Furthermore, it was established that a “cross-term”of the form cikj

∂uk

∂xjui is not present in the homogenized energy. However, such a cross-term may appear for

more general domains, where distances between inclusions can be much smaller than their sizes. At present thisis an interesting open question.

Finally we note that in recent works [22], [2] homogenization problems for liquid crystals with a periodicarray of polymeric inclusions in the presence of an applied magnetic field was considered.

2. Formulation of the problem and the main result. An idealized mathematical model for a liquidcrystal with spherical inclusions can be formulated as follows.

Let G be a bounded domain in R3 with a piecewise smooth boundary ∂G and Biε = B(xi

ε, aiε) be the ballsof small radii aiε centered at the points xi

ε (i = 1, ..., Nε).The small parameter ε is of order of the average distance between the nearest balls and also characterizes

the sizes of the balls Biε. We assume that Nε ∼ ε−3 and aiε = O(εα). Here α > 2, that is the balls are small

with respect to the average distance to the nearest neighbor.Let

Gε = G \Nε⋃i=1

Biε,

LIQUID CRYSTAL WITH RANDOM INCLUSIONS 3

be the perforated domain occupied by a liquid crystal. We introduce the class H1U (Gε) ⊂ H1(Gε) of vector

functions u : Gε → R3 with the trace u = U on ∂G, where U : G → R3. For simplicity we assume thatU ∈ C1(G).

Consider the variational problem

Eε[u] → min, u ∈ H1U (Gε)(2.1)

for the Ginzburg–Landau functional with a surface energy contribution on Siε

Eε[u] ≡ k

∫Gε

|∇u|2 +∫Gε

(|u|2 − 1)2 + q∑

i

∫Si

ε

(1 + κε(ν, u)2).(2.2)

Here Siε = ∂Bi

ε and ν is the unit normal vector to Siε (for the sake of definiteness we assume that ν is directed

into the domain Gε). The quantity (ν, u) =3∑

i=1

νiui is the scalar product in R3 and k, q and κε are given

parameters.It follows from standard analysis [18] that there exists at least one global minimizer uε ∈ H1

U (Gε) of theproblem (2.1)–(2.2). One can show that under certain conditions on the parameters k, q, κε, the sizes ofthe domain G, and the balls Bi

ε, the minimizer is unique. However, generally, there could be more than oneminimizer. Further, the minimizers of problem (2.1)–(2.2) exist even if the balls intersect, i.e. when the surfaceterm in (2.2) is not defined at the points of intersection of the spheres Si

ε (the surface measure of the set of suchpoints is zero).

The minimizers of the problem (2.1)–(2.2) describe the equilibrium state of a liquid crystalline mediumoccupying the domain Gε.

The direction of the vector-valued minimizer uε determines the average direction of the liquid crystalmolecules in the neighborhood of the point x and its magnitude determines, roughly speaking, the fraction ofthe molecules in a neighborhood of a point x oriented along the preferred direction of u (the orientational rate).The parameters k, q, κε characterize the materials properties of the liquid crystal and interfacial effects betweenthe liquid crystal and the inclusions. These parameters satisfy the following conditions: 0 < k < ∞, 0 ≤ q < ∞,and −1 ≤ κε < ∞.

In this paper we study the asymptotic behavior of the minimizers of problem (2.1)–(2.2) as ε → 0, whenthe number of the balls Nε tends to infinity, their radii tend to zero, and the locations of the balls in G andtheir radii are random.

More precisely we assume that the centers xiε of the balls Bi

ε and their radii aiε are defined by the set ofs–partial distribution functions

f εs(x1, ..., xs; a1, ..., as) : (G)s × [0,∞)s → [0,∞) (s = 1, 2, ..., Nε).

The probability of finding the location of the centers and the radii of a group of s balls in (xi, xi + dxi),(ai, ai + dai), where i = 1, ..., s is

f εs(x1, ..., xs; a1, ..., as)dx1...dxsda1...das.

These functions satisfy the conditions of symmetry, normalization and, concordance which follow from theirprobabilistic interpertation (see, e.g. [10]).

f εs(x1, ..., xk, ..., xl, ..., xs; a1, ..., ak, ..., al, ..., as) = f ε

s(x1, ..., xl, ..., xk, ..., xs; a1, ..., al, ..., ak, ..., as);

∫G

∞∫0

. . .

∫G

∞∫0

f εs(x1, ..., xs; a1, ..., as)da1dx1...dasdxs = 1, s = 1, ..., Nε;

∫G

∞∫0

f εs(x1, ..., xs; a1, ..., as)dasdxs = f ε

s−1(x1, ..., xs−1; a1, ..., as−1), s = 2, ..., Nε.


The distribution functions generate the probability measure Pε in the probability space Ωε. The points ωε of thisspace are in one–to–one correspondence with the random sets B(ωε) =

⋃i Bi

ε in G [23]. For any realization of theset B(ωε) there exists at least one minimizer uε(x, ωε) of problem (2.1)–(2.2) in the domain G(ωε) = G \B(ωε).Let us denote by M(ωε) the set of the minimizers which correspond to ωε ∈ Ωε and consider in the space Ωε

the random variable

ρ(ωε) = maxM(ωε)

∫G(ωε)

|uε(x, ωε) − u(x)χ(x, ωε)|2dx,(2.3)

where u is a vector-valued function u ∈ H1(G) and χ(x, ωε) is the characterisic function of the subdomainG(ωε) ⊂ G.

We will show that under some conditions on the distribution functions f ε1(x; a) and f ε

2(x1, x2; a1, a2) andwith the appropriate choice of the vector function u the random variable (2.3) converges to zero in probabilityPε as ε → 0, i.e.

limε→0

Pεωε ∈ Ωε : ρ(ωε) < δ = 1(2.4)

for any δ > 0.We now introduce the limiting (homogenized) vector function u and the conditions on the one-point and

two-point distribution functions f ε1 and f ε

2 for which the convergence takes place.First, we assume that these functions have the form :1) f ε

1(x; a) = ε−αf(x; ε−αa);2) f ε

2(x1, x2; a1, a2) = f ε1(x1; a1)f ε

1(x2; a2),where f(x; r) ∈ L∞(G × [0,∞)) is some nonnegative function normalized by 1 in L∞(G × [0,∞)) with a

compact support G′ × [a0, A0] in G × [0,∞) (G′ ⊂ G, 0 < a0 < A0 < ∞).We also assume that the parameter κε which characterizes the properties of the surfaces Si

ε of the balls Biε

has the form3) κε = κ0ε

β, where κ0 ∈ [−εβ,∞), β ∈ (−∞,∞). We set gε = qκε ≡ gεβ, then g ∈ (−∞,∞).Moreover, if g < 0 we assume that

A0 < k|g|−1,(2.5)

where A0 is a number which defines the diameter of the support of the function f(x; r) (see condition 1)), kand g = qκ0 are the parameters of the functional (2.2).

This choice of scaling 3) and the condition (2.5) were introduced in [6] for a deterministic model (seeintroduction).

Next we define the limiting vector function u, which appears in the definition (2.3).We set

p(x) =

∞∫0

pαβ(a)f(x; a)da,(2.6)

where f(x; a) is the function from the condition 1) and the functions pαβ(a) are defined as follows :

pαβ(a) =

4π3 ga2 for 2 < α < 3, β = 3 − 2α;

12π(22g2a3+45k2ga2)5(9k+5ga)2 for α = 3, β = −3;

264125πka for α = 3, β < −3;

0 for α, β ∈ Λg.

(2.7)

The sets Λg in (2.7) are defined as follows

Λg = 2 < α ≤ 3, β > 3 − 2α⋃

α > 3,−∞ < β < ∞


if g ≥ 0,

Λg = 2 < α ≤ 3, β > 3 − 2α⋃

α > 3, β ≥ −α.

if g < 0.Also the potential p(x) satisfies the following condition

p(x) ≥ min

1,1λ0

− k

,(2.8)

where λ0 = λ0(G) is the minimal eigenvlaue of the operator −∆ in G with the homogeneous boundary conditionon ∂G. Both conditions (2.7) and (2.8) were introduced in an analogous deterministic problem in [7] and werefer the reader to [7] for further details.

Consider a variational problem

Ep[u] → min, u ∈ H1U (G)(2.9)

for the functional

Ep[u] ≡ k

∫G

|∇u|2dx +∫G

(|u|2 − 1)2 +∫G

p(x)|u|2dx,(2.10)

where the function p(x) is defined by (2.6), (2.7) and satisfies the inequality (2.8).As it had been pointed out previously, there exists the unique global minimizer u of the problem (2.9)–(2.10).

The function u enters into the definition of the random variable (2.3).The main result of the paper is the followingTheorem 2.1. Let the conditions 1)–3), and the inequalities (2.5) (for g < 0) and (2.8) hold.Then the random variable ρ(ωε) (2.3) defined using the minimizers uε(x, ωε) and u(x) of problems (2.1)–

(2.2) and (2.9)–(2.10), respectively, converges to zero in probability (i.e. in the sense of (2.4)) as ε → 0.REMARK 2.1. Theorem 2.1 states the conditions under which all random minimizers of a stochastic problem

(2.1)–(2.2) converge in probability in the space L2(Gε) to a nonrandom vector function u(x) and this functionis the unique minimizer of a deterministic problem (2.9)–(2.10).

REMARK 2.2. Condition 1) (scaling) defines the characteristic dimensions of the balls (inclusions) Biε(ωε)

in the probabilistic sense. The assumption that the function f(x; r) has a compact support G′ × [a0, A0] inG× [0,∞) is made for the sake of simplicity only. It can be dropped, but then the proof of Theorem 2.1 becomesmuch more technical.

Condition 2) means that the balls Biε(ωε) are ”pairwise independent”. While this condition admits a possi-

bility of their intersection, i.e. Biε(ωε)

⋂Bj

ε (ωε) = ∅ (which is not physical), it follows from Lemma 4.1 that theprobability of realizations with intersections tends to zero as ε → 0. This condition can be relaxed to a weakercondition of ”pairwise almost indepedence”.

Condition 3) defines the character and the strength of the orientation of the liquid crystal orientations onthe surfaces of the inclusions. They ”prefer” to be orientated along the normal vector when g < 0 and along thetangent vector when g > 0.

Finally, the inequality (2.8) guarantees the uniqueness of the solution of the problem (2.9)–(2.10) (see [6]).The proof of Theorem 2.1 is based on the main theorem from [6], where an analogous deterministic problem

was considered. The theorem from [6] is proved under deterministic conditions on the distribution of the ballsBi

ε and their radii. These conditions are presented below (see Theorem 3.1 in the Section 2). In the Section 3we show that if the distribution functions satisfy the conditions stated above, then the conditions of Theorem3.1 hold ”in probabilistic sense” (and for α > 2). In the Section 4 we use this fact and Theorem 3.1 to proveTheorem 2.1.

3. Deterministic distribution of the balls. For convenience of the reader we now present an outlineof the results of the paper [6] which will be used below. Let us consider problem (2.1)–(2.2) in a deterministicdomain Gε = G \ ⋃Nε

i=1 Biε, where Bi

ε (i = 1, ..., Nε) are the balls centered at given points xiε (i = 1, ..., Nε) of

radii aiε. We assume that the following dependences of the parameters of the problem on ε holda1) gε = qκε ≡ gεβ, where β, g ∈ (−∞,∞).


a2) a0εα ≤ aiε ≤ A0ε

α, α > 1, 0 < a0 < A0 < ∞, also, if g < 0 then A0 < k|g|−1.Next, we introduce the following notations

Riε = dist(xiε,⋃j =i

xjε

⋃∂G) = minmin

j =i|xj

ε − xiε|, dist(xi

ε, ∂G)(3.1)

and

biε = biε(α, β) = |gε| a2

iε for 1 < α ≤ 3, β ≥ 3 − 2α and α ≥ 3, β ≥ −α;aiε for α ≥ 3, β < −α.

(3.2)

We assume that the balls Biε are located in the domain G and can not form clusters so that the following

conditions hold:a3) Riε ≥ aκ

iε for some κ (2/α < κ < 1),a4) for some σ (3/2 < σ ≤ 2), we have

Nε∑i=1

bσiε

R3(σ−1)iε

< Cσ,

where Cσ is a constant independent of ε.Introduce a generalized function :

pε(x) =Nε∑i=1

(pεsi + pε

vi)δ(x − xiε),(3.3)

where pεsi and pε

vi are the specific surface and boundary layer energies, respectively, for the i–th ball, defined asfollows:

pεsi =

4π3 gεa

2iε for 1 < α < 3, β = 3 − 2α;

4π3 gεa

2iε

(9k)2

(9k+5gεaiε)2 for α = 3, β = −3;

0 for (α, β) ∈ Λ+ \ (1 < α ≤ 3, β = 3 − 2α) ,

(3.4)

and

pεvi =

264π5(9k+5gεaiε)

2 k(gεaiε)2aiε for α = 3, β = −3;

264125πkaiε for α = 3, β < −3;

0 for (α, β) ∈ Λ+ \ (α = 3, β ≤ −3) ,

(3.5)

where

Λ+ = 1 < α ≤ 3, β ≥ 3 − 2α⋃

α ≥ 3, −∞ < β < ∞.Suppose that there exists a limit in a weak topology D′(G)

b) w − limε→0 pε(x) = p(x), where p ∈ L∞(G) satisfies inequality (2.8).The following theorem is proved in [6].Theorem 3.1. If the conditions a1) − a4), b) and (2.8) hold, then all minimizers of the problem (2.1)–(2.2)

(uε(x) ∈ Mε) converge to the unique minimizer u(x) of the problem (2.9)–(2.10) in the following sense

maxuε∈Mε

∫Gε

|uε(x) − u(x)χε(x)|2dx → 0,

as ε → 0, where the function p(x) is defined in the condition b). This convergence takes place in the followingrange of parameters α, and β :

Λ+ = 1 < α ≤ 3, β ≥ 3 − 2α⋃

α ≥ 3, −∞ < β < ∞


when gε > 0 and

Λ− = 1 < α ≤ 3, β ≥ 3 − 2α⋃

α ≥ 3, β ≥ −α

when gε < 0.The domains Λ+ and Λ− are presented (shaded) in Fig. 3.1, 3.2. Notice that the function p(x) = 0 only

on the bold lines. Also, note that we are able to prove a probabilistic analog of this theorem only for thesubdomains located in Λ± to the right of the vertical line passing through the ponit α = 2.

Fig. 3.1. (g > 0)

Fig. 3.2. (g < 0)

4. Probabilistic analog of the conditions of Theorem 3.1. First, note that the conditions a1) anda2) are satisfied with probability 1 (this follows from conditions 1) and 3) of Section 2).


We now show that condition a3) holds ”in probability”. Let µ be a number such that 2/α < µ < 1 and letT i

rεbe the balls centered at the points xi

ε and of radii rε = εαµ. It is clear that Biε ⊂ T i

rε. Since the support G′

of the function f(x; a) with respect to x is a compact set in G, then (for a sufficiently small ε), the balls T irε

do not intersect the boundary ∂G. Consider the event Aµε from Ωε such that the balls T i

rεdo not have pairwise

intersection, i.e.

Aµε = ωε ∈ Ωε : T i

rε

⋂T j

rε= ∅, i, j = 1, ..., Nε, i = j.

Lemma 4.1. If the conditions 1)–2) of Theorem 2.1 hold, then

limε→0

PεAµε = 1.

Proof. Let Aµε = Ωε \ Aµ

ε be the event that at least one pair of balls T irε

and T jrε

intersect, i.e. Aµε is a

complement to Aµε . Then

PεAµε = 1 − PεAµ

ε (4.1)

and

PεAµε =

Nε∑j,i=1j>i

∫G

∞∫0

∫T i2rε

∞∫0

f ε2(x

i, xj ; ai, aj)dajdxjdaidxi =

=Nε(Nε − 1)

2

∫G

∞∫0

∫T 12rε

∞∫0

f ε2(x

1, x2; a1, a2)da2dx2da1dx1.

Since Nε = ε−3 and meas (T 12rε

) = 32π3 ε3µα), from conditions 2), 1) and this equality we get

PεAµε ≤ ε−6

2

∫G

∞∫0

∫T 12rε

∞∫0

f ε1(x

1; a1)f ε1(x2; a2)da2dx2da1dx1 ≤

≤ ε−6

2

∫G

∞∫0

f ε(x1; a1)

∫

T 12rε

∞∫0

f ε(x2; a2)da2dx2

da1dx1 ≤ Cε3µα−6.

Since µα > 2, the statement of the lemma follows from (4.1).Corollary 4.2. It follows from Lemma 4.1 that, if we choose κ in condition a3) such that 2/α < µ < κ < 1,

then for any i (i = 1, ..., Nε) the inequalities Riε ≥ aκiε hold in probability, i.e.

limε→0

Pεωε ∈ Ωε : Riε ≥ aκiε, i = 1, ..., Nε = 1.

Let us consider now the condition a4). To this end we introduce a random variable

ζεσ(ωε) =

Nε∑i=1

bσiε

R3(σ−1)iε

,

where biε = biε(ωε) and Riε = Riε(ωε) are random variables defined by (3.2), (3.1).Lemma 4.3. If 3/2 < σ < 2, then

limε→0

Pεωε ∈ Ωε : ζεσ(ωε) ≤ N ≥ 1 − C(σ)

N


for any N > 0, where C(σ) is independent of N .Proof. It follows from the definition of Riε that

ζεσ(ωε) ≤ ζε

1σ(ωε) + ζε2σ(ωε),

where ζε1σ(ωε) and ζε

2σ(ωε) are random variables, which are defined as follows

ζε1σ(ωε) =

Nε∑i=1

bσiε max

j =i

1|xj

ε − xiε|3(σ−1)

;

ζε2σ(ωε) =

Nε∑i=1

bσiε

[ρ(xiε, ∂G)]3(σ−1)

.

Therefore, it follows from the Chebyshev’s inequality that

Pεωε ∈ Ωε : ζεσ(ωε) ≤ N ≥ Pεωε ∈ Ωε :

2∑i=1

ζεiσ(ωε) ≤ N ≥

≥ 1 −M(

2∑i=1

ζεiσ

)N

= 1 − M (ζε1σ)

N− M (ζε

2σ)N

(4.2)

for any N > 0. Here M(·) is the expectation and in the second inequality it is taken into account that the

random variable2∑

i=1

ζεiσ(ωε) is positive.

It follows from the properties of the distribution functions (see conditions of Theorem 2.1) that

M (ζε2σ(ωε)) = Nε

∫G

∞∫0

bσε (a)f ε

1(x; a)[ρ(x, ∂G)]3(σ−1)

dadx ≤ Cbσε (εα)Nε

∫G′

dx

[ρ(x, ∂G)]3(σ−1),(4.3)

where G′ is the support of the function f(x; a) with respect to the variable x ∈ G and the functions bε(a) aredefined as follows

bε(a) = |gε| a2 ≡ |g|εβa2 for 2 < α ≤ 3, β ≥ 3 − 2α and α ≥ 3, β ≥ −α;

a for α ≥ 3, β < −α.

Notice that bε(εα) ≤ Cεα for any α, β ∈ Λ+. Since G′ ⊂ G, then ρ(x, ∂G) ≥ ρ0 > 0 for x ∈ G′. Therefore, itfollows from (4.3) that

M (ζε2σ) ≤ Cεσα−3,

where C is a constant independent of ε.Since σ > 3/2 and α > 2, from this inequality we have

limε→0

M (ζε2σ) = 0.(4.4)

Next we estimate the expectation of the random variable ζε1σ(ωε). It is clear that

maxj =i

1|xj − xi|3(σ−1)

≤ L3(σ−1)ε +

∑j =i

χLε(|xj − xi|)|xj − xi|3(σ−1)

for any Lε > 0. Here χLε(t) is the indicator of the segment [0, L−1ε ]. Therefore, it follows from the properties of

the distribution functions (see the conditions of Theorem 2.1) that

M (ζε1σ) =

∞∫0

∫G

. . .

∞∫0

∫G

Nε∑i=1

bσε (ai)max

j =i

1|xj − xi|3(σ−1)

×


×f εNε

(x1, ...xNε ; a1, ...aNε)dx1da1...dxNεdaNε ≤ C1L3(σ−1)ε Nεb

σε (εα)+

+Nε∑i=1

∑j =i

∞∫0

∫G

. . .

∞∫0

∫G

bσε (ai)χLε(|xj − xi|)|xj − xi|3(σ−1)

×

×f εNε

(x1, ...xNε ; a1, ...aNε)dx1da1...dxNεdaNε ≤ C1L3(σ−1)ε ε−3bσ

ε (εα)+

+Nε(Nε − 1)∫G

∞∫0

∫G

∞∫0

bσε (a1)χLε(|x2 − x1|)|x2 − x1|3(σ−1)

f ε1(x

1; a1)f ε1(x2; a2)da1dx1da2dx2 ≤

≤ C1L3(σ−1)ε εση−3 + C2ε

ση−6

L−1ε∫

0

r2dr

r3(σ−1)≤ C1(σ)

[L3(σ−1)

ε εση−3 + L3σ−6ε εση−6

],(4.5)

where C1(σ) is a constant independent of ε and η = η(α, β) is defined as follows :

η(α, β) =

β + 2α for 2 < α ≤ 3, β ≥ 3 − 2α and α ≥ 3, β ≥ −α;α for α ≥ 3, β < −α.

Notice, that when we calculated the integral with respect to r from 0 to L−1ε we used the condition σ < 2 for

the first time.Now we choose Lε as follows :

Lε =

ε−1 for 2 < α ≤ 3;ε−α/3 for 3 ≤ α ≤ ∞.

Since η(α, β) ≥ 3 for 2 < α ≤ 3 and η(α, β) ≥ α for 3 ≤ α ≤ ∞, it follows from (4.5) that

limε→0

M (ζε1σ) ≤ 2C1(σ).(4.6)

The statement of the Lemma follows now from (4.2), (4.4), and (4.6).We next show that condition b) also holds in probability.Let ϕ(x) be an arbitrary function from C(G). Consider a random variable

ζεϕ(ωε) =

∫G

pαβ(x, ωε)ϕ(x)dx,

where pαβ(x, ωε) is defined by (3.3), (3.4), and (3.5).Lemma 4.4. Suppose that the conditions 1)–3) of Theorem 2.1 hold, then

limε→0

Pε

ωε ∈ Ωε :

∣∣∣∣∣∣ζεϕ(ωε) −

∫G

p(x)ϕ(x)dx

∣∣∣∣∣∣ < δ

= 1,

where p(x) is the function defined by (2.6), (2.7).Proof. It follows from the Chebyshev’s inquality [23] that

Pεωε ∈ Ωε :∣∣ζε

ϕ(ωε) − M(ζεϕ)∣∣ < δ ≥ 1 − D(ζε

ϕ)δ2

,(4.7)

where M(ζεϕ) and D(ζε

ϕ) are the expectation and the variance of the random variable ζεϕ, respectively.


Using (3.3) we can represent ζεϕ as follows :

ζεϕ(ωε) =

Nε∑i=1

pεiϕ(xi

ε),

where xiε = xi

ε(ωε) are the random centers of the balls Biε and pε

i = pεsi(ωε) + pε

vi(ωε) are the random variableswhich are defined by (3.4), (3.5). Using this representation, the properties of the distribution functions, andcondition 1) we have

M(ζεϕ

)=

∞∫0

∫G

. . .

∞∫0

∫G

Nε∑i=1

pε(ai)ϕ(xi)f εNε

(x1, ...xNε ; a1, ...aNε)dx1da1...dxNεdaNε =

= Nε

∫G

∞∫0

pε(a)ϕ(x)f ε1(x; a)dxda = Nε

∫G

∞∫0

pε(a)ϕ(x)ε−αf(x; ε−αa)dxda,

where pε(a) = pεs(a) + pε

v(a) and pεs(a), pε

v(a) are defined by (3.4), (3.5) with a instead of aiε.Since Nε = ε−3, from this equality, condition 3), and (3.4), (3.5) we get

M(ζεϕ

)=∫G

ϕ(x)

∞∫0

pαβ(a)f(x; a)dadx =∫G

p(x)ϕ(x)dx ≡ pϕ,(4.8)

where pαβ(x) is defined by (2.7) and p(x) is defined by (2.6).Similarly, taking into account (4.8) and condition 2), we estimate the variance

D(ζεϕ) = M[(ζε

ϕ − M(ζεϕ))2] =

=

∞∫0

∫G

. . .

∞∫0

∫G

(Nε∑i=1

pε(ai)ϕ(xi) − pϕ

)2

f εNε

(x1, ...xNε ; a1, ...aNε)dx1da1...dxNεdaNε =

= ε6∞∫0

∫G

. . .

∞∫0

∫G

Nε∑i=1

(ε−3pε(ai)ϕ(xi) − pϕ

)2f ε

Nε(x1, ...xNε ; a1, ...aNε)dx1da1...dxNεdaNε+

+ε6∞∫0

∫G

. . .

∞∫0

∫G

Nε∑i,j=1i=j

(ε−3pε(ai)ϕ(xi) − pϕ

) (ε−3pε(aj)ϕ(xj) − pϕ

)×

×f εNε

(x1, ...xNε ; a1, ...aNε)dx1da1...dxNεdaNε = Nεε6

∞∫0

∫G

(ε−3pε(a)ϕ(x) − pϕ)2f ε1(x; a)dxda+

+Nε(Nε − 1)ε6∞∫0

∫G

∞∫0

∫G

(ε−3pε(a1)ϕ(x1) − pϕ)(ε−3pε(a2)ϕ(x2) − pϕ)×

×f ε1(x

1; a1)f ε1(x2; a2)dx1da1dx2da2 = ε3

∞∫0

∫G

(pαβ(a)ϕ(x) − pϕ)2f(x; a)dxda+


+(1 − ε3)

∞∫0

∫G

∞∫0

∫G

(pαβ(a1)ϕ(x1) − pϕ)(pαβ(a2)ϕ(x2) − pϕ)f(x1; a1)f(x2; a2)dx1da1dx2da2.

It follows from (4.8) that the second term in the right hand side of the last equality is equal to zero. Therefore,we get

D(ζεϕ) ≤ Cε3.(4.9)

The statement of the lemma follows now from (4.7), (4.8), (4.9).

5. End of the proof of Theorem 2.1. Consider the following events in the probability space Ωε :

A1ε = ωε ∈ Ωε : a0εα ≤ aiε(ωε) ≤ A0ε

α; (α > 2), i = 1, ..., Nε;

A2ε = ωε ∈ Ωε : Riε ≥ aκiε, (2/α < κ < 1), i = 1, ..., Nε;

A3ε(N) =

ωε ∈ Ωε :

Nε∑i=1

bσiε

R3(σ−1)iε

< N, (3/2 < σ < 2)

;

A4ε(j, m) =

ωε ∈ Ωε :

∣∣∣∣∣∣∫G

pε(x, ωε)ϕj(x)dx −∫G

p(x)ϕj(x)dx

∣∣∣∣∣∣ <1m

;

A5ε(ν) = ωε ∈ Ωε : ρ(ωε) > ν,

where ϕj(x) (j = 1, 2, ...) is a sequence of functions which is dense C(G), ρ(ωε) is a random value defined by(2.3), ν > 0, and m ∈ N.

It follows from condition 1) of Theorem 2.1 that

PεA1ε = 1(5.1)

for any ε > 0.Assume that the conclusion of Theorem 2.1 does not hold. Then there exist δ > 0, ν > 0, and a sequence

εk → 0, k = 1, 2, ... such that

limε=εk→0PεA5ε(ν) > δ.(5.2)

On the other hand, it follows from the Corollary 4.2 and Lemmas 4.3 and 4.4 that for any j, m ∈ N andN > 0 there exist C(σ) and ε = ε(δ, N, j, m) such that

PεA2ε ≥ 1 − δ4 ;

PεA3ε(N) ≥ 1 − C(δ)N ;

PεA4ε(j, m) ≥ 1 − δ2j+m+2

(5.3)

for any ε < ε.Set N = 4C(δ)

δ and consider the event

Aεk= A1εk

⋂A2εk

⋂A3εk

(N)⋂ mk⋂

m=1

jk⋂j=1

A4εk(j, m)

⋂A5εk

(ν),(5.4)


where we choose the numbers mk and jk to be such that the inequalities (5.3) hold for ε = εk and m ≤ mk,j ≤ jk, N = 4C(δ)

δ . Since εk → 0 as k → ∞, the numbers mk and jk tend to infinity as k → ∞.Now we show that there is no k for which the event Aεk

⊂ Ωεkis not empty. It follows from our choice of

N and inequalities (5.3) that

Pεk

A2εk

⋃A3εk

(N)⋃ mk⋃

m=1

jk⋃j=1

A4εk(j, m)

≤ δ

4+

δ

4+ δ

∞∑j,m=1

12j+m+2

=34δ.

Therefore,

Pεk

A2εk

⋂A3εk

(N)⋂ mk⋂

m=1

jk⋂j=1

A4εk(j, m)

≥ 1 − 3

4δ.(5.5)

Here the bar denotes the complementary event.It follows from (5.4), (5.1), (5.2), and (5.5) that the events Aεk

(k = 1, 2, ...) are not empty. For any Aεk

we choose a point ω(εk) = ωk from this set and consider the corresponding realization of the balls Biεk

(ωk)(i = 1, ..., Nεk

) in G and the values ρ(ωk) constructed by using the minimizers uεk(x, ωk) ∈ M(ωk) of problem

(2.1)–(2.2) in the domain Gεk(ωk) = G \⋃i Bi

εk(ωk).

It follows from the definition of Aεkthat all the conditions of Theorem 3.1 are satisfied as ε → 0 but ρ(ωk)

does not tend to zero. This contradiction proves Theorem 2.1.

Acknowledgments. The first author was supported in part by NSF grant DMS-0204637. This workstarted when both authors were visiting the Department of Mathematics and Computer Science, Universityof Akron. They are grateful for the support and hospitality received during this visit. We wish to thank D.Golovaty for careful reading of the manuscript and very useful suggestions.

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