Singular Limits in Polymer Stabilized Liquid Crystals P. Bauman * , D. Phillips † and Q. Shen Department of Mathematics Purdue University West Lafayette, IN 47907 Synopsis We investigate equilibrium configurations for a polymer stabilized liquid crys- tal material subject to an applied magnetic field. The configurations are deter- mined by energy minimization where the energies of the system include those of bulk, surface, and external field. The Euler-Lagrange equation is a nonlinear PDE with nonlinear boundary conditions defined on a perforated domain mod- eling the cross section of the liquid crystal-polymer fiber composite. We analyze the critical values for the external magnetic field representing Fredericks tran- sitions and describe the equilibrium configurations under any magnitude of the external field. We also discuss the limit of the critical values and configurations as the number of polymer fibers approaches infinity. In the case where away from the boundary of the composite, the fibers are part of a periodic array, we prove that non-constant configurations develop order-one oscillations on the scale of the array’s period. Further more we determine the small-scale structure of the configurations as the period tends to zero. 1 Introduction 1.1 Polymer-liquid crystal composites A nematic liquid crystal is a material that exists in an intermediate phase between liquid and solid . Its molecules are long thin rods that tend to align with one another. * Partially supported by the National Science Foundation under Grant No.9971974 † Partially supported by the National Science Foundation under Grant No.9971713 1
Transcript
Singular Limits in Polymer Stabilized LiquidCrystals
P. Bauman∗, D. Phillips†and Q. Shen
Department of MathematicsPurdue University
West Lafayette, IN 47907
Synopsis
We investigate equilibrium configurations for a polymer stabilized liquid crys-tal material subject to an applied magnetic field. The configurations are deter-mined by energy minimization where the energies of the system include thoseof bulk, surface, and external field. The Euler-Lagrange equation is a nonlinearPDE with nonlinear boundary conditions defined on a perforated domain mod-eling the cross section of the liquid crystal-polymer fiber composite. We analyzethe critical values for the external magnetic field representing Fredericks tran-sitions and describe the equilibrium configurations under any magnitude of theexternal field. We also discuss the limit of the critical values and configurationsas the number of polymer fibers approaches infinity. In the case where away fromthe boundary of the composite, the fibers are part of a periodic array, we provethat non-constant configurations develop order-one oscillations on the scale ofthe array’s period. Further more we determine the small-scale structure of theconfigurations as the period tends to zero.
1 Introduction
1.1 Polymer-liquid crystal composites
A nematic liquid crystal is a material that exists in an intermediate phase betweenliquid and solid . Its molecules are long thin rods that tend to align with one another.
∗Partially supported by the National Science Foundation under Grant No.9971974†Partially supported by the National Science Foundation under Grant No.9971713
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The local average of the molecules’ principal axes near a point is identified with a unitvector n called the director. (See [5].) The resulting director field, n(x) for x rangingover the material body, defines the liquid crystal’s configuration. The configurationdetermines the optical characteristics of the liquid crystal and is very sensitive to anapplied electric or magnetic field.
A Fredericks transition describes the onset of change, away from a uniform state,for a director configuration of a liquid crystal that is subjected to an applied field. Themagnitude of the applied field for which a Fredericks transition occurs is called thecritical value or threshold of the transition. Both the thresholds of the transitions andthe configurations of the director fields are important for applications of liquid crystalsin information display and processing.
New composite materials have been developed so as to stabilize particular configu-rations and to influence where critical values of applied fields occur. These are calledpolymer stabilized liquid crystals (PSLC). (See [3, 4, 6, 7, 8, 9, 13, 15] as well as ref-erences therein.) A composite is formed by embedding a polymer network in a liquidcrystal matrix. In the absence of an applied field, the director field for a liquid crystaltends to align itself with the polymer network resulting in a preferred configuration. Ifan external field is applied to PSLC, it competes with the polymer network to determinethe configuration in the liquid crystal. In this paper, assuming that a fixed polymernetwork has been assembled we examine a mathematical model obtaining qualitativedescriptions for director configurations and their transitions under various magnitudesof the external magnetic field.
1.2 The model
We consider a polymer-liquid crystal composite in the shape of a long cylinder withits cross section in the x1x2 plane. The polymer network is modeled as a collection offibers (rods) that extend through the cylinder in the x3-direction. Here we ignore any
cross-links in the network. Let T =N⋃i=1
Ti denote the cross section of the fibers, where
we assume the T i are mutually disjoint and that the Ti are simply connected opensets with ∂Ti of class C2,α for some 0 < α < 1. Let Ω be a bounded domain in the x1x2
plane with a C2,α boundary. Set D := Ω \ T . This set will denote the cross section ofthe liquid crystal matrix. Throughout the paper we assume that D is connected andthat Ω ∩ Ti 6= ∅ for each i. See Figure 1.
2
D
SST
S
ei
The cross section of a polymer-liquid crystal composite.Figure 1
Consider a cylinder whose height, L, is large relative to the diameter of its crosssection. Set diam(Ω) = W . We assume that the cylinder occupies (Ω ∪ T ) × (0, L)with L >> W . Set D = D × (0, L). This body denotes the perforated cylindricalregion occupied by the liquid crystal. Furthermore let S = ∂D \ ∂Ω, Se = ∂D \ S,S = S × (0, L) and Se = Se × (0, L). Then S denotes the interface between the liquidcrystal and the polymer fibers and Se represents the lateral surface between the liquidcrystal and the exterior of the composite. See Figure 2.
D
S
S e
Polymer-liquid crystal composite.Figure 2
We use energy minimization to analyze configurations for the liquid crystal andits Fredericks transitions under an external magnetic field. There are three types ofrelevant energies to take into account. These are the bulk energy of the liquid crystal,the energy due to the application of the external magnetic field and the surface energieson the interface between liquid crystal and polymer S, and on the outer surface Se .
Let n = (n1, n2, n3) be the unit vector field denoting the director of the liquidcrystal. The Frank-Oseen theory (see [5]) defines the elastic bulk energy density of anematic liquid crystal as
where K1, K2, K3, and K4 are material constants for the liquid crystal depending onthe temperature. If K1 = K2 = K3 = K > 0 and K4 = 0, this gives the one constantapproximation case where the energy density becomes
K|∇n|2.
An external magnetic field can change the configuration of the liquid crystal. Undera uniform magnetic field the energy density is
m(n) = −χH2(h · n)2
where χ > 0 is a material constant, H is the magnitude of the field and h is a unitvector representing the direction of the field. In this paper we take h perpendicular tothe fibers. We shall assume that
h = e2
so that the field tends to make the directors turn in the x2-direction.The interaction between the nematic liquid crystal and the polymer network is
described by a weak anchoring condition on the interface expressed as a surface energy.We are interested in the effect of having a large contact area between the liquid crystaland the polymer on the qualitative features of the Fredericks transition.
On the interface between the liquid crystal and the polymer, S, we consider a weakanchoring condition with the Rapini-Papoular surface energy density
εw(1− (n · q)2)(1.1)
where w > 0 is a material constant representing the strength of the surface energyand ε is a dimensionless scaling factor for which ε−1 is proportional to the normalizedinter-facial surface area H2(S)/WL = H1(S)/W . (Here Hn(S) is the n-dimensionalHausdorff measure of S.) The unit vector q is parallel to the easy axis of the liquidcrystal defined on the surface of the polymer. In this paper we shall assume q = e3,i.e., in the direction of the polymer fibers.
Remark. The scaling factor ε is distinguished so that the surface energy remainsbounded when considering a family of networks having larger and larger surface area,ε−1 → ∞. Without this factor the weak anchoring condition, (1.1), becomes a stronganchoring condition (n = ±q on S) as ε→ 0.
On the outside boundary, Se, we impose another weak anchoring condition withsurface energy density
βw(1− (n · q)2),
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where β > 0 is another dimensionless scaling factor.Since L >> W the surface areas of the top and bottom of the cylinder segment are
small relative to that of Se . As a result, we ignore contributions to the surface energydue to them.
For the one-constant approximation model then, we consider configurations thatminimize the following energy
E(3)(n) =∫DK|∇n|2dx1dx2dx3 +
∫Sεw(1− (n · e3)2)dH1dx3(1.2)
+∫Seβw(1− (n · e3)2)dH1dx3 −
∫DχH2(e2 · n)2dx1dx2dx3
where we seek solutions in H1(D; S2).Our energy is based on a fibril model given in[19]. The specific structure we consider is an idealization of that observed in [10] .
1.3 Statement of main results
In Section 2 the problem is made dimensionless by introducing the tansformations
(x1, x2, x3) =
(x1
W,x2
W,x3
W
)and n(x1, x2, x3) = n(x1, x2, x3). Using these (1.2) be-
comes
W
∫D
K|∇n|2dx1dx2dx3 +W 2
∫S
εw(1− (n · e3)2)dH1dx3
+W 2
∫Se
βw(1− (n · e3)2)dH1dx3 −W 3
∫D
χH2(e2 · n)2dx1dx2dx3
where diam(Ω) = 1. Suppressing the tilde, we show that minimizers take the form
n = n(x1, x2) = (0, sin θ, cos θ)
where θ(x1, x2) ∈ H1(D) is the angle between e3 and n(x1, x2).We will take K/w = ε2W and focus on the case of ε small, physically indicating
a stronger surface energy relative to the bulk liquid crystal energy. Setting M =WχH2/w we arrive at an unconstrained, scalar minimum problem on the cross section.Minimize
E(θ) = ε2∫D| ∇θ |2 dx1dx2 + ε
∫S
sin2 θdH1(1.3)
+β
∫Se
sin2 θdH1 −M∫D
sin2 θdx1dx2.
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for θ in H1(D). The associated equilibrium problem is
ε2∆θ +M sin θ cos θ = 0 in D,
ε∂θ
∂ν+ sin θ cos θ = 0 on S,(1.4)
ε2∂θ
∂ν+ β sin θ cos θ = 0 on Se.
We show that it suffices to consider solutions valued in [0, π/2], and that these weaksolutions are classical away from S ∩Se. Note that θ ≡ 0 and θ ≡ π/2 are solutions to(1.4). In Section 3, for the case where S ∩ Se = ∅, we establish a comparison principlefor solutions to (1.4). Using this we prove that θ ≡ 0 is the unique minimizer if thestrength of the magnetic field, M , is small and θ ≡ π/2 is the unique minimizer if Mis large. In particular, we prove that there exist two values for M , µ < µ, at whichthe second variation of E becomes degenerately stable for θ ≡ 0 (with M = µ) andθ ≡ π/2 (with M = µ) . These are the critical values of the two Fredricks transitionsfor this problem. We show that θ ≡ 0 is the unique minimizer for (1.3) if M ≤ µ andthat θ ≡ π/2 is the unique minimizer for (1.3) if µ ≤ M . If µ < M < µ we provethat there exists a unique minimizer and that it is increasing with M . These solutionsthen describe the stable, quasi-static transition from the states θ ≡ 0 to θ ≡ π/2 as Mincreases from µ to µ.
A principal objective in this work is to analyze the limit of critical values andconfigurations of minimizers as the number of polymer fibers goes to infinity, whichcorresponds to finer and finer distributions in the polymer networks. If the fibers aredistributed uniformly, as part of a periodic array, we compare minimizers for (1.3) tothose for a corresponding cell problem. To describe this we introduce a fundamentalunit cell
Y := (x1, x2) | −1
2< x1, x2 <
1
2.
We let
T ⊂⊂ Y
represent the cross section of a fundamental polymer rod where T is a simply connectedopen set with a C2,α boundary. Define Y ∗ := Y \ T as the region in Y occupied by theliquid crystal. For each ε > 0, we distinguish a cross section homothetic to T,
T (ε) := εx | x ∈ T.
Then
T (ε) + εz | z ∈ Z × Z
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where Z = . . . ,−1, 0, 1, . . . is an ε-periodic array of cross sections, and
Y(ε) := R2 \⋃
z∈Z×Z(T (ε) + εz)
is the cross section of the ε-periodic liquid crystal matrix in R2.
Let
P(Y ∗) = w ∈ H1(Y ∗) :
w(−1
2, x2) = w(
1
2, x2) for − 1
2≤ x2 ≤
1
2and
w(x1,−1
2) = w(x1,
1
2) for − 1
2≤ x1 ≤
1
2
and define
Ep(w) =
∫Y ∗
(| ∇w |2 −M sin2(w))dx1dx2 +
∫∂T
sin2(w)dH1.
In Section 4.2 we prove that Ep has a unique minimizer, valued in [0, π/2], in theclass of periodic functions P(Y ∗), which we denote as ϕ(x,M). Furthermore we showthat there exist two critical values, λ and λ, so that if M ≤ λ then ϕ(x,M) ≡ 0, ifλ ≤ M then ϕ(x,M) ≡ π/2, and such that ϕ(x,M) is increasing in M if λ < M < λ.Let ϕ(x,M) be the 1-periodic extension of ϕ(x,M) to Y(1) and define
ϕε(x,M) = ϕ(x/ε,M)
for x in the cross section of the ε-periodic matrix Y(ε).Set D = D(ε) and consider the case where D(ε) = Ω ∩ Y(ε). We prove that
limε→0
µ(ε) = λ. (See Theorem 4.6.) Thus the lower critical value for the cell problem is a
good approximation to µ(ε) if fine periodic networks are considered. However though,the upper threshold is influenced by a boundary layer near the exterior boundary Seand tends to infinity, lim
ε→0µ(ε) = ∞. (See Theorem 4.7.) Thus the cell problem does
not give an accurate estimate for the transition’s critical value at θ ≡ π/2.In Section 4.4 we prove our main result for minimizers, θε(x,M), to (1.3). Assume
for each compact set K ⊂ Ω there is an ε0(K) so that D(ε) ∩ K = Y(ε) ∩ K for allε < ε0. We show that
θε(x,M)− ϕε(x,M)→ 0 as ε→ 0,(1.5)
uniformly for x ∈ K ∩ D(ε) . Thus the minimizer of the corresponding cell problemprovides a uniform approximation to minimizers for the composite system, away from
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the composite’s exterior. (See Theorem 4.10.) Notice that the minimizers θε(x,M) donot converge pointwise but develop oscillations as ε→ 0. Also notice that even thougha complete transition to θ ≡ π/2 can not occur until M ≥ µ(ε) >> λ, in fact θε(x,M)is uniformly close to π/2 away from Se for all M ≥ λ and ε sufficiently small. This isdue to ϕε(x,M) ≡ π/2 if M ≥ λ and (1.5).
Lastly, we comment on previous and related work. Prior studies for the analysis ofequilibrium configurations focused on pure liquid crystal in special settings. These ledto one-dimensional mathematical problems (e.g., nonlinear ODE with various boundaryconditions). See Virga [17] for discussion in one-dimensional settings. However though,for composite materials, a one-dimensional model is not suitable. In the area of two-dimensional settings, Wang [18] analyzed the existence of Fredericks transitions for a“light-nematic” system in a rectangle domain without any holes. The critical valueproblems in composite systems that we are considering also have connections witheigenvalue problems in perforated domains (See Vanninathan [16] and Kaizu [12]).
2 Existence results
Consider the minimum problem for the one-constant approximation model given in(1.2). The problem is made dimensionless by writing
E(3)(n)
KW=
1
W
∫D
|∇n|2dx1dx2dx3 +ε
Wb
∫S
(1− (n · e3)2)dH1dx3
+β
Wb
∫Sε
(1− (n · e3)2)dH1dx3 −1
Wξ2
∫D
(e2 · n)2dx1dx2dx3
where b = K/w is the extrapolation length and ξ =
√K
χH2is the magnetic coherence
length (see [5]). We let E(3) = E(3)/KW , (x1, x2, x3) =
(x1
W,x2
W,x3
W
), n(x1, x2, x3) =
n(x1, x2, x3), b′ = b/W , ξ′ = ξ/W and suppress the tilde. The non dimensional versionof the total energy becomes
E(3)(n) =
∫D
|∇n|2dx1dx2dx3 +ε
b′
∫S
(1− (n · e3)2)dH1dx3
+β
b′
∫Se
(1− (n · e3)2)dH1dx3 −1
ξ′2
∫D
(e2 · n)2dx1dx2dx3
where the regions and boundaries have been scaled accordingly. In particular, nowD = Ω \ T × (0, L
W), diam (Ω) = 1 and L
W>> 1.
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We seek minimizers of E(3) in F (3) = n(x1, x2, x3) ∈ H1(D; S2). In the followingtheorem we use the same ideas as in [2] to show that a minimizer does not depend onx3.
Let
F = n(x1, x2) ∈ H1(D; S2),
E(n) =
∫D|∇x1x2n|2dx1dx2 +
ε
b′
∫S(1− n2
3)dH1
+β
b′
∫Se
(1− n23)dH1 − 1
ξ′2
∫Dn2
2dx1dx2.
Theorem 2.1 A function n minimizes E(3) in F (3) if and only if n = n(x1, x2) ∈ Fand minimizes E in F . Moreover minimizers for E in F exist and are analytic in D.
Proof: Minimizers for E in F exist and are classical (analytic) in D (See [11]). Letn0 ∈ F such that E(n0) = min
n∈FE(n) and n any element in F (3). Then
E(3)(n) =
∫ LW
0
E(n)(x3)dx3 +
∫ LW
0
∫D| ∂n
∂x3|2 dx1dx2dx3
≥ E(n0)L
W.
Thus n0 minimizes E(3) in F (3) and any minimizer of E(3) in F (3) is independent of x3.
Theorem 2.1 asserts that we only need to discuss the two dimensional problem offinding minimizers for E in F .
If there is no external magnetic field, i.e., ξ =∞, then it is obvious that E(n) ≥ 0and E(n) = 0 if and only if n = ± e3. These are trivial minimizers. Our next lemmashows that the components of a minimizer are either identically zero or strictly of onesign.
Lemma 2.2 If n is a minimizer for E in F and ni is a component of n then eitherni ≡ 0 in D or ni(x) 6= 0 for all x ∈ D.
Proof: Consider n ∈ F where n is obtained from n by replacing ni with |ni|. Thenwe see n is also a minimizer. Thus it is without loss of generality to assume ni ≥ 0.Since n is regular it satisfies an equilibrium equation in D of the form ∆nj = λjnj inD for 1 ≤ j ≤ 3 where λj = λj(x) is a smooth function (see [17]). From the maximumprinciple however nonnegative solutions are either identically zero or never vanish.
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We see that it is without loss of generality to only discuss minimizers with non-negative components. The next theorem however shows that the n1 component of aminimizer is always zero. Recall e1 is perpendicular to both the easy axis of the weakanchoring condition, e3, and the direction of the applied field, e2.
Theorem 2.3 If n = (n1, n2, n3) is a minimizer of E in F then n1 ≡ 0 in D.
Proof: We argue by contradiction. In light of the previous lemma then we can assumethat n1 > 0 in D. Consider the unit vector field
This implies that n is not a minimizer for E, which is a contradiction.
Lemma 2.2 and Theorem 2.3 combine to give the following corollary.
Corollary 2.4 Let n = (n1, n2, n3) be a minimizer of E in F with nonnegative com-ponents. Then either n = (0, 1, 0), n = (0, 0, 1) or n = (0, n2, n3) with 0 < n2, n3 < 1in D.
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Proof: From the theorem we have n1 ≡ 0. If n2(x) = 0 for some x ∈ D then thelemma implies that n2 ≡ 0. Since n2
2 + n23 = 1 we see that n = (0, 0, 1). Similarly if
n3(x) = 0 for some x ∈ D it follows that n = (0, 1, 0). The remaining possibility is0 < n2, n3 < 1 in D.
The corollary allows us to introduce a new scalar variable, θ, such that 0 ≤θ(x1, x2) ≤ π/2 and for which
n(x1, x2) = (0, sin θ, cos θ)(2.1)
Note θ(x1, x2) is the angle between e3 and n(x1, x2).We see that θ is analytic in D, θ ∈ H1(D), and that
E(n) = E(θ)(2.2)
where
E(θ) =
∫D
(| ∇θ |2 − 1
ξ′2sin2 θ)dx1dx2 +
ε
b′
∫S
sin2 θdH1 +β
b′
∫Se
sin2 θdH1.
Moreover θ is a solution to the following boundary value problem in the sense of H1
∆θ +
(1
ξ′2
)sin θ cos θ = 0 in D,
∂θ
∂ν+( εb′
)sin θ cos θ = 0 on S,(2.3)
∂θ
∂ν+
(β
b′
)sin θ cos θ = 0 on Se,
where ν is the exterior normal to D.
Theorem 2.5 A function θ is a minimizer for E in H1(D) if and only if there is ann ∈ F minimizing E where θ and n are related by (2.2). Moreover every minimizer θsatisfies
kπ
2≤ θ ≤
(k + 1
2
)π for some integer k
and each bounded equilibrium θ ∈ C2,α(D \ (S ∩ Se)).
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Remark. S ∩ Se is the subset of the cross section where the edges of the polymerfibers intersect the exterior surface of the liquid crystal. In particular, if the fibers areall located in the interior of the composite then S ∩ Se = ∅.
Proof: We only need to verify the regularity of θ near ∂D \ (S ∩ Se). First we recallbounded weak solutions to (2.3) are of class Cγ(D\ (S ∩Se)) for some γ > 0 (see [14]).Second (using linear elliptic theory), weak solutions for which ∆θ ∈ Cγ(D \ (S ∩ Se))and for which
∂θ
∂ν∈ Cγ(∂D \ (S ∩ Se)) are such that θ ∈ C1,γ(D \ (S ∩ Se)). (See [1].)
Appealing to linear theory once more we have θ ∈ C2,α(D \ (S ∩ Se)).Since we are assuming n has nonnegative components we restrict our attention.
Corollary 2.6 If θ is a solution to (2.3) such that 0 ≤ θ ≤ π2
then either θ ≡ 0, π2
or
0 < θ < π2
for x ∈ D \ (S ∩ Se).
Proof: By the previous theorem θ is a classical solution to (2.3). The equation in(2.3) can be written as ∆θ+λ(x)θ = 0. By the maximum principle if θ ≥ 0 then either
θ > 0 in D or θ ≡ 0 in D. At the boundary we see from (2.3) that∂θ
∂ν= 0 if θ = 0 at
some point in ∂D \ (S ∩Se), contradicting the Hopf maximum principle. We concludethat either θ > 0 in D\ (S ∩Se) or θ ≡ 0 in D. Setting ϕ := π
2−θ an identical analysis
for ϕ implies either θ < π2
in D \ (S ∩ Se) or θ ≡ π2
in D.
The normalized extrapolation length b′ = bW
indicates locally the strength of thebulk energy with respect to the surface energy of the polymer fibers. If the surfaceenergy is relatively strong then b′ is small. In this paper we are interested in thecase of small extrapolation length. More specifically we consider b′ = ε2 where ε−1 iscomparable to the total surface area of the polymer network H1(S). This will allowsolutions to develop order one oscillations on an ε− lengthscale.
Remark. From this point forward we assume that β is fixed and KwW
= bW
= b′ = ε2.
Thus we have M = WχH2/w = ε2/ξ′2. Setting
E(θ) := ε2E(θ)
we arrive at the energy (1.3) and the associated equilibrium problem (1.4).
In Section 3 we will investigate the critical values for transition as well as the cor-responding configurations of the liquid crystal in the composite under various appliedexternal magnetic fields. This will be done by examining the solutions to (1.4) takingvalues in [0, π/2]. In Section 4 we will assume the polymer network has a periodic struc-ture (scaled by ε). We will examine the limit of the critical values and configurationsas the number of polymer rods goes to infinity, or equivalently ε→ 0.
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3 Analysis of equilibriums for E in H1(D)
In this section we assume that the parameter ε and the domain D are fixed with ε > 0.Moreover we assume that the polymer fibers are located in the interior of the composite.As such we assume that
∂D = S ∪ Se with S ∩ Se = ∅.
We will find two transition thresholds for M , µ and µ, such that if 0 ≤ M ≤ µ thenthe director field (2.1) with minimal energy has θ ≡ 0. A heterogeneous intermediatestate is the minimizer if µ < M < µ and if µ ≤ M the minimal state has θ ≡ π/2.
We first prove a comparison lemma relating equilibrium configurations to fieldstrengths.
Lemma 3.1 Assume S ∩ Se = ∅. Consider solutions to (1.4) with θ = θi, M = Mi
for i = 1, 2 and 0 < θ1, θ2 < π/2 in D. If M1 < M2 then θ1 < θ2 in D. If M1 = M2
then θ1 = θ2.
Proof: Let F (θ) be such that F ′(θ) = 1/ sin θ cos θ. Set v1 = F (θ1) and v2 = F (θ2).Note that since 0 < θ1, θ2 < π/2 then −∞ < v1, v2 < ∞ in D, and that since F ′ > 0we have θ1 < θ2 if and only if v1 < v2. From the proof of Theorem 2.5 we havev1, v2 ∈ C2(D). From (1.4) then they each satisfy
∆vi −F ′′(θi)
F ′(θi)2| ∇vi |2= −Mi
ε2in D,
∂vi∂ν
= −1
εon S,
∂vi
∂ν= − β
ε2on Se
where θi = θ(vi) = F−1(vi). Set
g(v) = −F ′′(θ(v))/F ′(θ(v))2
= cos(2θ(v)).
Note
g′(v) = − sin2(2θ(v)) < 0.(3.1)
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Thus v1 and v2 each satisfy
∆vi + g(vi) | ∇vi |2 = −Mi
ε2in D,
∂vi
∂ν= −1
εon S,(3.2)
∂vi
∂ν= − β
ε2on Se.
Set z = v1 − v2 and take the differences of the equations for i = 1 and 2. This can beviewed as
∆z + a · ∇z + dz =(M2 −M1)
ε2in D,
∂z
∂ν= 0 on ∂D
where a = g(v2)∇(v1 + v2) and
d = |∇v1|2(g(v1)− g(v2))
(v1 − v2).
Note from (3.1) that d ≤ 0.We apply the strong maximum principle to z. If M1 < M2 the principle asserts
that either z < 0 in D or z equals a nonnegative constant. Thus either
v1 < v2 on D or v1 − v2 ≡ Const. ≥ 0 in D.(3.3)
In the latter case we get
(g(v1)− g(v2)) | ∇v1 |2= (M2 −M1)/ε2 > 0 in D.
In view of (3.1) this would be impossible.If M1 = M2 we again conclude that (3.3) holds. The second case implies that
(g(v1)− g(v2)) | ∇v1 |2= 0 in D.
Thus either v1 = v2 in D or ∇v1 = 0 in D. Clearly the boundary conditions in (3.2)prohibit a constant solution. Thus either v1 < v2 in D or v1 = v2 in D. Reversing theroles of v1 and v2 we see that v1 = v2.
In order to distinguish stable equilibrium we consider the second variation of E . Let
D2E(θ;w) =d2
dt2E(θ + tw) |t=0 for θ, w ∈ H1(D).
14
Define µ and µ such that
µ := infw∈H1(D)
‖w‖2=1
(ε2∫D| ∇w |2 dx1dx2 + ε
∫Sw2dH1 + β
∫Sew2dH1)
= infw∈H1(D)
‖w‖2=1
1
2D2E(0;w) +M(3.4)
and
µ := − infw∈H1(D)
‖w‖2=1
(ε2∫D
| ∇w |2 dx1dx2 − ε∫Sw2dH1 − β
∫Sew2dH1)
= M − infw∈H1(D)
‖w‖2=1
1
2D2E(π/2;w)(3.5)
Note θ ≡ 0 is a stable equilibrium if and only if M < µ and θ ≡ π2
is stable if and onlyif M > µ.
Lemma 3.2 The critical values satisfy µ < µ.
Proof: Set w = (H2(D))−1/2 in (3.4) and (3.5). Then
µ ≤ (εH1(S) + βH1(Se))/H2(D) ≤ µ.
Moreover equality occurs in either statement if and only if the constant function is anextremal for the corresponding eigenvalue problem. Constant functions do not satisfythe natural boundary conditions for extremal solutions thus both inequalities are strict.
The next theorem describes solutions to (1.4) for different values of M .
Theorem 3.3 Assume D is such that S ∩ Se = ∅. Consider solutions to (1.4) valuedin [0, π/2]. If 0 ≤ M ≤ µ or µ ≤ M then there are exactly two solutions, θ ≡ 0 andθ ≡ π/2. If µ < M < µ there is exactly one more solution, θ(x,M). This solutionsatisfies
0 < θ(x,M) < π/2 for x ∈ D.
and is strictly increasing in M for each x.
15
Extend θ(·,M) by defining
θ(·,M) ≡ 0 if M ≤ µ,
≡ π/2 if µ ≤M.
Then for each M ≥ 0, θ(·,M) is the unique minimizer for E among functions valuedin [0, π/2] moreover θ ∈ C([0,∞);C2(D)).
Proof: First consider µ < M < µ. Then from (3.4) and (3.5) we have
infw∈H1(D)
‖w‖2=1
D2E(0;w) = 2(µ−M) < 0(3.6)
and
infw∈H1(D)
‖w‖2=1
D2E(π/2;w) = 2(M − µ) < 0.(3.7)
Thus neither θ ≡ 0 or θ ≡ π/2 are minimizers for E for these values of M . On theother hand from Theorems 2.1 and 2.5 a minimizer exists valued in [0, π/2]. FromCorollary 2.6 a third solution must take its values in (0, π/2) and from Lemma 3.1 itmust be unique. Thus θ(x,M) is uniquely determined for µ < M < µ as claimed.From Theorem 3.3, θ(x,M) is strictly increasing in M for µ < M < µ. By ellipticestimates and uniqueness we have
θ(x, ·) ∈ C((µ, µ);C2(D)).
Next due to the monotonicity of θ(x,M) we see
limM→µ+
θ(x,M) =: w(x) and limM→µ−
θ(x,M) =: w(x)
exist where the convergence is in C2(D). Moreover 0 ≤ w < π/2 and 0 < w ≤ π/2in D. We claim that w ≡ 0 and w ≡ π/2. To prove this for w we assume that it isnot the case. Then by Corollary 2.6 we see that 0 < w < π/2 in D. Since θ(·,M) isminimizer if µ < M < µ it follows that the limit, w, is also a minimizer if M = µ.Thus E(w) ≤ E(0) = 0.
Set z = sin(w) 6≡ 0. Then∫D| ∇z |2 dx1dx2 <
∫D| ∇w |2 dx1dx2.
16
Thus∫D
(ε2 | ∇z |2 −µz2)dx1dx2 + ε
∫Sz2dH1 + β
∫Sez2dH1
<
∫D
(ε2 | ∇w |2 −µz2)dx1dx2 + ε
∫Sz2dH1 + β
∫Sez2dH1
= E(w) ≤ 0.
This contradicts the definition of µ in (3.4) and establishes that w ≡ 0. The argumentfor showing that w ≡ π/2 is similar. Thus we have proved all of the assertions forµ ≤ M ≤ µ.
We next consider 0 ≤ M ≤ µ. Let us assume that there is a solution, θ, to (1.4)
with 0 < θ < π/2. Given η > 0 choose δ > 0 so that 0 < θ(x, µ + δ) < η in D. Thenusing Lemma 3.1 we see
0 < θ(x) < θ(x, µ+ δ) < η in D.
Since η is arbitrary we see that such a θ cannot exist and that θ ≡ 0 and θ ≡ π/2 arethe only solutions for M ≤ µ. Finally from (3.7) we see that θ ≡ π/2 is unstable forM < µ. Thus θ ≡ π/2 cannot be a minimizer for E if M ≤ µ. It follows that θ ≡ 0 isthe unique minimizer in this case. The argument for M ≥ µ is identical.
4 Periodic Networks
4.1 Fundamental Cells.
By assuming more structure on the polymer network away from the boundary of thecomposite we are able to obtain further details concerning the transition fields andconfigurations. In order to describe these networks we denote
Y := (x1, x2) | −1
2< x1, x2 <
1
2
as the fundamental unit cell,
T ⊂⊂ Y(4.1)
representing the cross section of a fundamental polymer rod where T is a simply con-nected open set with a C2,α boundary. Define Y ∗ := Y \T as the region in Y occupiedby the liquid crystal. We set
Y (ε) := εx | x ∈ Y , T (ε) := εx | x ∈ T and Y ∗(ε) := εx | x ∈ Y ∗.(4.2)
17
For each ε > 0 we tile R2 with translates of the ε-cell Y (ε). Set
Y(ε) := R2 \⋃
z∈Z×Z(T (ε) + εz)
where Z = . . . ,−1, 0, 1, . . ..
4.2 The Cell Problem
In this section we introduce an ε-periodic variational problem defined on Y(ε) andcollect the characteristic features of its solution. These characteristics will correspondto the feature found for solutions of (1.4) such that S ∩ Se = ∅ and their proofs followin the same manner. Let
P(Y ∗) = w ∈ H1(Y ∗) :
w(−1
2, x2) = w(
1
2, x2) for − 1
2≤ x2 ≤
1
2and
w(x1,−1
2) = w(x1,
1
2) for − 1
2≤ x1 ≤
1
2
Set
Ep(w) =
∫Y ∗
(| ∇w |2 −M sin2(w))dx1dx2 +
∫∂T
sin2(w)dH1.(4.3)
Theorem 4.1 There exist minimizers for Ep in P(Y ∗) taking values in [0, π/2].
Theorem 4.2 Let ϕ be a bounded equilibrium for Ep in P and ϕ its 1-periodic extension
to Y(1). Then ϕ ∈ C2,α(Y(1)) and it satisfies
∆ϕ+M sin ϕ cos ϕ = 0 in Y(1),(4.4)
∂ϕ
∂ν+ sin ϕ cos ϕ = 0 on ∂Y(1).
Theorem 4.3 Let ϕ1 and ϕ2 be solutions to (4.3) with 0 < ϕ1, ϕ2 < π/2 and where0 ≤M1 ≤M2. If M1 < M2 then ϕ1 < ϕ2 in Y(1). If M1 = M2 then ϕ1 = ϕ2.
Define
λ = infw∈P(Y ∗)
‖w‖2=1
(∫Y ∗| ∇w |2 dx1dx2 +
∫∂T
w2dH1
).(4.5)
λ = − infw∈P(Y ∗)
‖w‖2=1
(∫Y ∗| ∇w |2 dx1dx2 −
∫∂T
w2dH1
).(4.6)
18
Theorem 4.4 The infima are achieved in (4.5) and (4.6) by w, w ∈ C2,α(Y ∗) respec-tively such that 0 < w,w in Y ∗. Moreover the 1-periodic extensions of w and w are inC2,α(Y(1)) and 0 < λ < λ.
Theorem 4.5 There exists ϕ ∈ C([0,∞);C2,α(Y ∗)) such that1) for each M ≥ 0, ϕ(·,M) is the unique minimizer for Ep in P(Y ∗) for which
0 ≤ ϕ ≤ π/2.2) ϕ(x,M) ≡ 0 if M ≤ λ andϕ(x,M) ≡ π/2 if λ ≤M ,
3) ϕ(x,M) is a strictly increasing in M for λ ≤M ≤ λ, for each x ∈ Y ∗.
4.3 Asymptotic Analysis of Critical Values
In this section we consider a family of polymer networks determined by T = T (ε) forε > 0 and corresponding solutions to (1.4). For each ε > 0, T (ε) is a finite union ofdisjoint open sets with regular boundaries as described in Section 2. We investigateµ(ε) and µ(ε) as ε → 0. These quantities are defined by (3.4) and (3.5), the criticalvalues at which the solutions θ ≡ 0 and θ ≡ π/2 respectively lose stability. Let D(ε)be the domain of liquid crystal and S(ε) be the interface.
For the analysis of µ(ε) we assume that T (ε) have a given ε-periodic structure.In this instance we show that µ(ε)→ λ as ε→ 0 where λ is the lower critical value forthe cell problem defined in Section 4.2.
Theorem 4.6 Let T be the cross section for a fundamental fiber as in (4.1). Supposethat T (ε) = Ω ∩
⋃z∈Z×Z
(T (ε) + εz). Then limε→0
µ(ε) = λ.
Proof: We derive an upper and a lower estimate for µ(ε).
Step 1. We construct a test function for (3.4) that vanishes near Se. Consider w(x),the positive extremal to (4.5) defined for x ∈ Y ∗. Extend w as a periodic function toY(1). It follows from Theorem 4.4 that there are constants 0 < c1 < c2 so that
c1 ≤ w ≤ c2,(4.7)
and
w ∈ C2,α(Y(1)).(4.8)
Define
wε(x) = w(x/ε) for x ∈ Y(ε).
19
Let φε ∈ C1c (Ω) such that 0 ≤ φε ≤ 1, φε = 1 if x ∈ Ω and dist(x, ∂Ω) ≥ 2ε, φε = 0 if
x ∈ Ω and dist(x, ∂Ω) ≤ ε, and | ∇φε |≤ c3/ε.Set ζε := φεwε in D(ε). We insert ζε into (3.4). It follows that
µ(ε) ≤(ε2∫D(ε)
| ∇wε |2 dx1dx2 + ε
∫S(ε)
w2εdH1 + c4ε
)/∫D(ε)
φ2εw
2εdx1dx2.
Let M(ε) be the union of the tiles, Y ∗(ε) + εz for z ∈ Z × Z contained in D(ε). Itfollows that D(ε) \M(ε) is contained in a 2ε neighborhood of Se(ε). Using (4.7) and(4.8) we see
µ(ε) ≤ε2∫M(ε)
| ∇wε |2 dx1dx2 + ε∫S(ε)∩∂M(ε)
w2εdH1∫
M(ε)w2εdx1dx2
+ c5ε.
Let N(ε) be the number of tiles making upM(ε). Then
ε2∫M(ε)|∇wε|2dx1dx2 + ε
∫S(ε)∩∂M(ε)
w2εdH1)∫
M(ε)w2εdx1dx2
=N(ε)(ε2
∫Y ∗(ε) |∇wε|2dx1dx2 + ε
∫T (ε)
w2εdH1)
N(ε)∫Y ∗(ε)w
2εdx1dx2
=(∫Y ∗ |∇w|2dx1dx2 +
∫Tw2dH1)∫
Y ∗ w2dx1dx2
= λ
where we used (4.5).Thus we see
µ(ε) ≤ λ+ c5ε.(4.9)
Step 2. Let uε be the positive extremal to (3.4). Set vε := uε/wε. Using (4.7) and(4.8) we have vε ∈ H1(D(ε)). Thus inserting uε into (3.4) we have
µ(ε) = ε2∫D(ε)
∇wε · ∇(wεv2ε )dx1dx2
+ ε2∫D(ε)
w2ε | ∇vε |2 dx1dx2
+ ε
∫S(ε)
w2εv
2εdH1 + β
∫Se(ε)
w2εv
2εdH1.
20
Integrating by parts in the first term on the right and using the fact that wε satisfies
ε2∆wε = −λwε in Y(ε),
ε∂wε∂ν
= −wε on ∂Y(ε)
we get
µ(ε) ≥ λ+
∫Se(ε)
(ε2∂wε∂ν
+ βwε)wεv2εdH1.
Since | ∇wε |≤ c3/ε and wε ≥ c1 > 0 we see (ε2∂wε
∂ν+ βwε) > 0 for ε sufficiently small.
Thus µ(ε) ≥ λ for ε small enough.Combining this inequality with (4.9) proves the theorem.
We now show that as long as Se(ε) is at least a fixed portion of the exterior boundaryof D(ε) then µ(ε)→∞ as ε→ 0. This is independent of the nature of T (ε) inside thecomposite and is markedly different from what occurs for the cell problem where theupper transition threshold, λ, is finite.
Theorem 4.7 Assume there is a δ > 0 so that H1(Se(ε)) ≥ δ for all ε sufficientlysmall. Then
limε→0
ε · µ(ε) > 0.
Proof: Let ηε = 1−φε where φε is the cut-off function defined in the previous theorem.Inserting ηε as a test function in (3.5) we see
µ(ε)
∫Ω
η2εdx1dx2 ≥ −ε2
∫Ω
| ∇ηε |2 dx1dx2 + β
∫Se(ε)
η2εdH1.
Thus εµ(ε) ≥ c1(βδ − c2ε) for some constants c1, c2 > 0, for all ε > 0 and sufficientlysmall.
4.4 Asymptotic Analysis of minimizers
In the previous section we saw that the critical fields are sensitive to the structure ofT (ε) near the exterior boundary Se even if T (ε) is assumed to be ε-periodic. In thissection we show that if the T (ε) are ε-periodic in the bulk of the composite, away
21
from Se, then the solutions to the cell problem found in Section 4.2 give an accuratedescription of minimizers for (1.3) away from Se.
Given a fundamental cross section T = T (1), as in (4.1), we introduce a bulk energydefined on open bounded subsets of Y(ε). Fix ε > 0 and M ≥ 0.
Let O be a bounded open subset of Y(ε) with a Lipschitz continuous boundary. Set
B(w; ε,M,O) =
∫O
(ε2 | ∇w |2 −M sin2w) dx1dx2 + ε
∫O∩∂Y(ε)
sin2w dH1
for w ∈ H1(O).
Definition. For ε, M , and O fixed we say that w ∈ H1(O) is a local minimizerfor B if
B(w; ε,M,O) ≤ B(v; ε,M,O) for all v ∈ H1(O)
such that w = v on ∂O \ ∂Y(ε).
For n ∈ N and ε > 0. Let C(n, ε) be the perforated cell
C(n, ε) = (x1, x2) : (−n +1
2)ε < x1, x2 < (n− 1
2)ε ∩ Y(ε).
Note that C(n, ε) is tiled by (2n− 1)2 translations of Y ∗(ε).We first prove several estimates for local minimizers. Denote by ∂eC(n, ε) the exte-
rior boundary for C(n, ε),
∂eC(n, ε) = (x1, x2) : x1 or x2 ∈ (−n+1
2)ε, (n− 1
2)ε.
We distinguish several functions used for comparison and describe their properties.Consider
u ∈ C2(C(n, ε) \ ∂eC(n, ε)) ∩ C(C(n, ε))
satisfying
ε2∆u+M sin u cosu = 0 in C(n, ε),(4.10)
ε∂u
∂ν+ sin u cosu = 0 on ∂C(n, ε) \ ∂eC(n, ε),(4.11)
22
u = u0 on ∂eC(n, ε).(4.12)
If u0 ∈ H1/2(∂eC(n, ε)) ∩ C(∂eC(n, ε)) and 0 ≤ u0 ≤ π/2 the methods from Section2 can be applied to show that a solution, u, to (4.10)-(4.12) exists and that 0 ≤ u ≤ π
2.
Moreover if 0 < u0 < π2
then the solution is unique, 0 < u < π2
in C(n, ε) and using theproof of Lemma 3.1 we can compare solutions. Specifically if u solves (4.10), (4.11)
with M replaced by M such that M ≤ M , assuming 0 < u, and 0 ≤ u ≤ u < π2
on∂eC(n, ε) then it follows that u ≤ u in C(n, ε).
To go further, let ua solve (4.10)–(4.12) with u0 ≡ a where a is a constant such that0 < a < π
2. Then u is increasing in a. Set u(x, ε,M, n) := lim
a↑π2
ua(x, ε,M, n). Here the
convergence is in C(C(n, ε)) ∩ H1(C(n, ε)). We see the function u(x, ε,M, n) has theproperties
u(x, ε, M , n) ≤ u(x, ε,M, n) if M < M
and
u(x) ≤ u(x, ε,M, n)
for any solution , u, to (4.10)–(4.12) with 0 ≤ u0 < π2.
Next we note (arguing just as in Section 2) that there is a local minimizer, w, forB(·; ε,M, C(n, ε)) with w = a on ∂eC(n, ε) taking values in (0, π
2). Since ua is the unique
solution to (4.10)–(4.12) with values in (0, π2) and u0 = a it must be a local minimizer.
As the limit of local minimizers is a local minimizer we have that u is a local minimizerwith u0 = π
2.
Similarly we can construct the local minimizer u(x, ε,M, n) := lima↓0
ua(x, ε,M, n)
having the properties
u(x, ε, M , n) ≤ u(x, ε,M, n) if M < M
and
u(x, ε,M, n) ≤ u(x)
for any solution, u, to (4.10)–(4.12) with 0 < u0 ≤ π2.
We are now prepared to systematically derive estimates for local minimizers.
23
Lemma 4.8 Given constants M and M such that λ < M ≤M < λ there is an integern0 and a constant ρ > 0 so that if 0 ≤M ≤M then
u(x, ε,M, n0) ≤ π
2− ρ for x ∈ C(1, ε).
If M ≤M then we have
ρ ≤ u(x, ε,M, n0) for x ∈ C(1, ε).
Proof: We first rescale the problem by setting ε = 1. Indeed if wε(x) is defined onC(n, ε) then by setting εy = x for y ∈ C(n, 1) and w(y) := wε(x) we see
ε2B(w; 1,M, C(n, 1)) = B(wε; ε,M, C(n, ε)).
Thus we can rescale without loss of generality.We begin by finding an integer n1 > 4 and a function w ∈ H1(C(n1, 1)) with w = 0
on ∂eC(n1, 1) so that
B(w; 1,M, C(n1, 1)) < 0.(4.13)
Let ϕ = ϕ(x,M) be as in Theorem 4.5 and let ϕ be its 1-periodic extension to Y(1).Let ζ ∈ C2
Since λ < M we have Ep(ϕ,M) < 0 and (4.13) follows.Thus u ≡ 0 is not a local minimizer on C(n1, 1). Now consider u(x, 1,M, n1), the
local minimizer constructed above. As a nonnegative solution to (4.10), (4.11) it mustbe that either u ≡ 0 or u > 0 in C(n1, 1) \ ∂eC(n1, 1). We have just shown that u 6≡ 0.Thus
infx∈C(1,1)
u(x, 1,M, n1) =: ρ1(n1) > 0.
Moreover since u(x, 1,M, n1) ≥ u(x, 1,M, n1) for M > M we have
u(x, 1,M, n1) ≥ ρ1(n1) on C(1, 1) for all M ≥ M.
In the same manner, by showing that u ≡ π2
is not a local minimum on C(n2, 1) forn2 sufficiently large,
24
It follows that
u(x, 1,M, n2) ≤ u(x, 1,M, n2) ≤ supx∈C(1,1)
u(x, 1,M, n2)
=π
2− ρ2(n2) for x ∈ C(1, 1) and M ≤M
where ρ2(n2) > 0.The assertions follow by taking n0 = max(n1, n2) and ρ = min(ρ1, ρ2).
Let ϕε(x,M) = ϕ(x/ε,M) where ϕ is the 1-periodic solution to the cell problem(4.4).
Lemma 4.9 Given η > 0 there is an integer n1(η) so that for all ε > 0 and M ≥ 0we have
| u(x, ε,M, n1)− ϕε(x,M) |< η, | u(x, ε,M, n1)− ϕε(x,M) |< η for x ∈ C(1, ε).(4.14)
Proof: We rescale as before setting ε = 1 and carry out the proof for u. The argumentfor u is similar.
Choose λ < M < M < λ so that
| ϕ(x,M) |< η/2 and | ϕ(x,M)− π/2 |< η/2 for all x.(4.15)
By Theorem 4.5 this is always possible.Next we consider n ≥ n0 where n0 is determined in Lemma 4.8 and M such that
M ≤M ≤M . Now either
u(x, 1,M, n) ≡ 0
or
π
2> u(x, 1,M, n) > 0 on C(n, 1) \ ∂eC(n, 1).
Since u is a local minimizer the former is ruled out by the proof of Lemma 4.8. We nowcompare u(x, 1,M, n) with u(x−z, 1,M, n0) and u(x−z, 1,M, n0) for x ∈ C(n0, 1)+z,for those z ∈ Z × Z such that C(n0, 1) + z ⊂ C(n, 1). We get
for such u, with x ∈ C(n0, 1) + z. From Lemma 4.8 then we see that for any compactK ⊂ R2 and n large enough
0 < ρ ≤ u(x, 1,M, n) ≤ π
2− ρ for x ∈ K ∩ Y(1)
25
provided M ≤M ≤M .Using this and elliptic estimates we can extract a subsequence u(x, 1,Mi, ni)
such that ni → ∞ and Mi → M for some M , with M ≤ M ≤ M that converges inC2(K ∩ Y(1)), for each compact set K to a solution, u, of
∆u+ M sin u cos u = 0 in Y(1),(4.16)
∂u
∂ν+ sin u cos u = 0 on ∂Y(1),(4.17)
and such that
0 < ρ < u <π
2− ρ on Y(1).
We now show that u = ϕ(x, M). Note ϕ satisfies (4.16) and (4.17) and is uniformlybounded away from 0 and π
2as well.
We argue by contradiction. Assume for definiteness that u(x0)− ϕ(x0, M) > 0 forsome x0 ∈ Y(1). Set v1 = F (u) and v2 = F (ϕ) where F is defined in Lemma 3.1. Notev1 and v2 are uniformly bounded in Y(1) and v1(x0) − v2(x0) > 0. The argument inthe proof of Lemma 3.1 asserts that v1−v2 cannot attain a positive maximum in Y(1).It follows that there must be a sequence xn ⊂ Y(1) such that | xn |→ ∞ as n→∞and
limn→∞
(v1(xn)− v2(xn)) = supY(1)
(v1 − v2) > 0.
Choose zn ∈ Z × Z so that xn ∈ C(1, 1) + zn. Set
w1,n(x) := v1(x + zn), w2,n(x) := v2(x + zn).
There exist subsequences, w1,ni and w2,ni converging in C2(K∩Y(1)) for each compactset K to w1 and w2 respectively where these limits satisfy (4.16) and (4.17). Howeverw1−w2 achieves a positive maximum at some point in C(1, 1) and this is a contradiction.
Thus u = ϕ on Y(1). This implies that the full sequence, u(x, 1,M, n) convergesuniformly to ϕ(x,M) on compact subsets of Y(1) for each M ≤ M ≤ M . Moreoverthe convergence is uniform in M for M ≤M ≤ M as well.
In particular given η > 0 there is an n1(η) ≥ n0 so that with K = C(1, 1)
where we have used the monotonicity of u in M , (4.15), and (4.18). Similarly if M > M
we have
0 ≤ π
2− u(x, 1,M, n1) ≤ π
2− u(x, 1,M, n1)
=π
2− ϕ(x,M) + (ϕ(x,M)− u(x, 1,M, n1))
<η
2+η
2.
Combining this, (4.18) and (4.19) completes the proof of (4.14).
We next introduce a family of cross sections T (ε) that are eventually ε-periodicon every compact subset of Ω as ε→ 0.
Definition. We say that the family T (ε) : ε > 0 is locally periodic if there is afundamental cross section T , as in (4.1), and a function r(ε) ≥ 2ε for ε > 0 such thatlimε→0
r(ε) = 0, for which
T (ε) ∩ x ∈ Ω : dist(x, ∂Ω) > r(ε)=
⋃z∈Z×Z
(T (ε) + εz) ∩ x ∈ Ω : dist(x, ∂Ω) > r(ε) for each ε > 0.
We can now state the main result for this section.
Theorem 4.10 Let T (ε) : ε > 0 be a family that is locally periodic and let θε bea family of minimizers for (1.3) in H1(D(ε)) such that 0 ≤ θε ≤ π
2. Then if K is a
compact subset of Ω we have
limε→0‖θε(x,M)− ϕε(x,M)‖C(K∩D(ε)) = 0.
Moreover the convergence is uniform in M for M ≥ 0.
Proof: Let K be a compact subset of Ω. Recall D(ε) = Ω\T (ε). Given η > 0, chooseλ < M < M < λ so that ϕ(x,M) < η/2 and π
2− ϕ(x,M) < η/2 for all x. Next set
27
n := max(n0(M(η),M(η)), n1(η)) where n0 was introduced in Lemma 4.8 and n1 inLemma 4.9. Finally we take ε0 sufficiently small so that for ε < ε0 the family
N := C(1, ε) + εz : where z ∈ Z × Z such that
C(n, ε) + εz ⊂ x ∈ Ω : dist(x, ∂Ω) > r(ε),
forms a cover for K∩D(ε). Thus we are requiring that the sets C(n, ε)+εz are containedin the region of D(ε) for which T (ε) is ε-periodic.
Let M ≤ M ≤ M and let θε(x,M) be a minimizer to (1.3) in H1(D(ε)) with0 ≤ θε ≤ π
2. According to Corollary 2.6 either θε ≡ 0, θε ≡ π
2or 0 < θε <
π2
in D(ε)\Se.By the choice of ε and n, the set D(ε) contains a cell, C(n, ε) + εz for some z ∈ Z ×Z.Now θε is a local minimizer on C(n, ε) + εz and the proof of Lemma 4.8 asserts that θεcan be neither 0 or π
2. Thus 0 < θε <
π2
in D(ε) \ Se.Next for x′ ∈ K ∩ D(ε) select an element in N containing x′, C(1, ε) + εz′. We can
compare u(x− εz′, ε,M, n), θε(x,M) and u(x− εz′, ε,M, n) on C(n, ε) + εz′ and applyLemma 4.9. We obtain
ϕε(x′,M)− η < u ≤ θε(x
′,M) ≤ u < ϕε(x′,M) + η
Thus we have
| θε(x,M)− ϕε(x,M) |< η for all x ∈ K ∩ D(ε), M ≤M ≤M and ε < ε0.
The cases of M < M and M ≤M are treated just as in the proof of Lemma 4.9. Thusthe proof is complete.
Remark. Since θε and ϕε are bounded we see that under the hypotheses of Theorem4.10 that
limε→0‖θε(x,M)− ϕε(x,M)‖L2(D(ε)) = 0.
5 Discussion
In this paper, using a fibril model with a fixed polymer network, we describe the equi-librium configurations for a polymer stabilized liquid crystal. In a two-dimensionalsetting, we show the existence of lower and upper critical values for an external mag-netic field such that the minimizer is non constant when the field is between these twothresholds (i.e., the liquid crystal configuration is non-uniform). When the field is be-low the lower critical value, the configuration is uniform in the direction determined bythe polymer network and boundary conditions. If the field is above the upper critical
28
value, the configuration is uniform in the direction of the external magnetic field. Wealso study the properties of these configurations under various magnitudes of externalfields.
In the case of a uniform polymer network with a periodic structure, as the numberof polymer fibers goes to infinity, we show away from Se that minimizers becomeuniformly close to the oscillating solutions of the scaled cell problem. Moreover thelower critical values of the external field for the composite system approach those ofthe corresponding cell problem. Thus, as ε→ 0, minimizing configurations are parallelto the easy axis, e3, for nearly the same values of magnetic intensity as for the cellproblem. However the situation is different for the upper critical values. The exteriorboundary condition forces the upper critical values of the external field to tend toinfinity as the number of fibers increases. This result also holds if the polymer networkis not uniformly distributed. Thus as ε → 0, we have the phenomenon of minimizersbeing close but not identically parallel to the direction of the applied field, h, forM = WχH2/w in a large interval to the right of the upper critical value for the cellproblem, λ. This seems to be the most significant effect of the exterior boundary.
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