ENERGY-MINIMIZING INCOMPRESSIBLE NEMATIC ELASTOMERS

Transcription

1 ENERGY-MINIMIZING INCOMPRESSIBLE NEMATIC ELASTOMERS PATRICIA BAUMAN* DEPARTMENT OF MATHEMATICS PURDUE UNIVERSITY WEST LAFAYETTE, INDIANA, 4796 AND ANDREA C. RUBIANO** FRANKLIN W. OLIN COLLEGE OF ENGINEERING NEEDHAM, MASSACHUSETTS 2492 ABSTRACT. We prove weak lower semi-continuity and existence of energy-minimizers for a free energy describing stable deformations and the corresponding director configuration of an incompressible nematic liquid-crystal elastomer subject to physically realistic boundary conditions. The energy is a sum of the trace formula developed by Warner, Terentjev and Bladon (coupling the deformation gradient and the director field and the Landau-de Gennes energy in terms of the gradient of the director field and the bulk term for the director with coefficients depending on temperature. A key step in our analysis is to prove that the energy density has a convex extension to non-unit length director fields. Our results apply to the setting of physical experiments in which an incompressible elastomer occupying a thin reference domain in R 3 is clamped on its sides and stretched perpendicular to its initial director field, resulting in shape-changes and reorientation of the director in patterns (including stripes that depend on the modulus of the stretching and regions of shear in the minimizing deformation-director state.. INTRODUCTION Incompressible nematic liquid crystal elastomers are incompressible elastic materials formed by networks of polymer chains and mesogens that have intrinsic nematic liquid crystal ordering. Their behavior combines properties of rubber-like solids and nematic liquid crystals, giving them unique stimuli response behaviors. One of their most interesting features is the coupling between the observable shape and the orientation of the liquid crystal nematic rod-like units. Nematic elastomers can deform significantly due to an application of an electric field, which causes liquid-crystal director rotations and in turn changes the shape of the elastomer. Conversely, a deformation of the elastomer results in changes in the director orientations. In this paper, we investigate a free energy in which a minimizer (y, n represents a stable incompressible deformation y = (y (x, x 2, x 3, y 2 (x, x 2, x 3, y 3 (x, x 2, x 3 = y(x 99 Mathematics Subject Classification. Primary 35J66; Secondary 35J2,49J3. Key words and phrases. Nematic elastomers, calculus of variations, existence, lower semi-continuity, liquid crystals. *Research supported by NSF Grants FRG-DMS and DMS **Current address: Princeton Consultants, 2 Research Way, Princeton, NJ 854. Research supported by NSF Grant FRG-DMS

2 of a nematic elastomer occupying a reference configuration Ω in R 3 (with constant initial (reference director field n S 2 and the corresponding current director field n = n (x, x 2, x 3, n 2 (x, x 2, x 3, n 3 (x, x 2, x 3 = n(x. We assume that the reference domain Ω is a bounded, simply connected Lipschitz domain and that (partial displacement boundary conditions on y and n (or weak anchoring conditions on n are imposed. The free energy is a sum of two terms. The first is an integral of the trace formula developed by the physicists Bladon, Terentjev and Warner that couples the deformation gradient y and director field n (see [4], [5], and [6]. The second term is the Landau-de Gennes energy for variations of the director. The BTW trace formula was formulated as an extension of classical molecular-statistical rubber elasticity. The book [24] by Warner and Terentjev is an excellent source for this theory, known as the neo-classical theory, and its developments. The problem we consider is motivated by well-known physical experiments in which an incompressible nematic elastomer occupying a region in R 3 with constant height and rectangular cross section ( a, a ( b, b and with reference director field n = e 2 is (partially clamped on the two sides Ω {x = ±a}, and the clamped regions are then stretched outward in the directions ± e on the right and left sides, respectively. It is observed that the pattern of the director field n in the deformed material depends on the modulus of the stretching; at small stretching, n tends to stay mostly aligned in the bulk close to the direction e 2 = n. As the stretching increases, stripe regions form in portions of the bulk in which the direction of n in each of these regions are roughly aligned with each other in a direction that is not parallel to n and is different in alternating stripes. Finally, when the modulus of the stretch is large, the direction of n tends to align parallel to e and the striping regions are small regions near the clamped edges. It is observed that the deformation y(x exhibits shearing in the stripe regions. (See [5], [6], [2], [23]... The Coupled Bladon-Terentjev-Warner/Frank Energy. First, let us describe the model that we consider and the energy to be minimized, as well as assumptions that we shall use throughout the paper. The Landau-de Gennes energy is given in terms of a 3 x 3 matrix-valued function Q = Q(x that is trace-free and symmetric. Assuming equal elastic constants, it is given by k 2 Q ATr(Q2 4 9 BTr(Q3 + 9 C(Tr(Q2 2 Ω where k >, A, B, C are constants depending on temperature, and Tr denotes the trace. Here we consider uniaxial nematic elastomers in which the average second moments of the mesogenic rod-like units are described by the nematic order parameter Q = 3 ( 2 s n n 3 I. with n = n(x in S 2 and s = s(x R. Here I denotes the 3 x 3 identity matrix. We remark that matrices Q = Q(x of this type have n as an eigenvector of Q and its eigenvalues are functions of the scalar parameter s. In this case, the terms involving trace in the Landau-de Gennes energy can be rewritten as functions of s. Uniaxial phases occur when s = s(x > (in which case the molecules near x mainly align in the direction of n(x and when s(x < (in which case the molecules mainly align perpendicular to n(x. When s(x =, all 2

3 eigenvalues are zero, and all directions of alignment have equal probability; this is known as the isotropic phase. The energy density for an incompressible nematic elastomer defined by the Bladon-Terentjev- Warner trace formula is formulated in terms of the deformation gradient F = y(x and the Landau-de Gennes tensor Q = Q(x through its associated principle director n and order parameter s. Denote by = (s > and l = l (s > the s-dependent effective length of molecular-steps in the directions perpendicular and parallel to n, respectively, and by r = r(s := l the anisotropy parameter at a microscopic level. Define the step length tensor L = L(n; s by ( L = (l n n + I. for n and s as above. The matrix L = L(n; s at a point x in Ω describes the average shape anisotropy of the polymer backbones in the nematic elastomer. The behavior of a nematic elastomer under a deformation is influenced by the molecular alignment conditions of the material when it was initially prepared. We assume that n S 2 is a fixed (but arbitrary constant initial (reference director field defined in the reference configuration Ω (before a deformation is imposed corresponding to a constant state Q = 3 2 s ( n n 3 I where s. Define L := L(n ; s. The matrix L = L(n; s defined in ( is a symmetric, positive definite matrix with eigenvalues l ( multiplicity and ( multiplicity 2. Also, the eigenspace corresponding to l is spanned by n, and the eigenspace corresponding to is the 2-dimensional subspace of vectors in R 3 that are orthogonal to n. The tensor L /2 = L(n; s /2 is well defined for every unit-length vector n, and simple calculations show that (2 L = [ I + ( r ] n n, L 2 = l [I ( r ] n n. Using the above definitions the energy density known as the Bladon-Terentjev-Warner trace formula for an incompressible nematic elastomer is defined by: F BTW (n, y = µ { ( Tr L ( y T L y + ln ( } det LL, 2 where µ > is the constant linear shear modulus of the material and F T denotes the transpose of the matrix F. (See [24], Chapter 6. We assume throughout this paper that s > and s are fixed uniform constants in the above model (independent of n(x and x in Ω such that the following conditions hold: l,, l, and l in the BTW trace formula are all positive constants r l > and r l l L = L(n and L = L(n are given by equation ( ln(det LL is constant throughout the domain (since the eigenvalues of L and L are constant in the domain For example, the above conditions hold for the model of freely-jointed main-chain elastomers, which have l (s = a( + 2s and (s = a( s where a >, provided that s (, and s [, are fixed. (See [24], Chapters 3 and 6. These conditions were also assumed in previous studies related to the construction of an effective energy by Conti, 3

4 DeSimone and Dolzmann for a modified BTW trace formula. (See [], [], [2], [3], and [4]. Since s is a positive constant, the terms involving Tr(Q in the Landau-de Gennes energy density are all constant. Dropping these constant terms without loss of generality, the Landau-de Gennes energy density becomes the equal constant version of the Frank energy, namely F Fr ( n = κ n 2. where κ = 9s 2 k/4. As described in [9] and [24], a Frank energy (or Landau-de Gennes energy is to be added to the BTW energy to give a continuum theory of nematic elastomers. Dropping the constant log term in the original trace formula F BTW without loss of generality (since here we are concerned with energy minimizers, we define (3 F BTW (n, y = µ 2 { Tr ( L ( y T [L(n] y }. The coupled BTW/Frank energy that we consider here is then: (4 I (y, n, y, n = Ω Ω F BTW (n, y + F Fr ( n µ { ( Tr L ( y T [L(n] y } + κ n 2 2 where n W,2 (Ω; S 2, y W,2 (Ω; R 3, and det y = almost everywhere in Ω. According to experiments, the coefficient µ in F BTW is much greater than the Frank coefficient κ in I. This means that in a minimizer, large gradients of n cost little energy compared to large gradients of y. We remark that the nonconvex effective energies developed in the papers cited above by Conti, DeSimone and Dolzmann assumed that variations in n cost no energy. Starting with a modified BTW trace formula depending on n and F = y(x (with no penalty on n, they then minimized the energy density with F fixed among all n in S 2 to get an energy density depending only on F; they computed the quasiconvexication of the resulting energy to obtain an effective energy. (See [], [2] and [3]. Using this effective energy, they have computed numerical solutions that exhibit the stripe pattern formations observed in the physical experiments and the stress-strain curve associated with soft elasticity. (See [] and [4]. On the other hand, the inclusion of a Frank or Landau-de Gennes energy allows one to prescribe anchoring conditions on y and n at the boundary as in experiments. It may also be useful for analyzing the difference in the scale of oscillations in n and y in stripe formations observed in stable deformations of incompressible elastomers. We note that in a more realistic model, the Frank energy density in the last term above would be integrated over the current configuration y(ω and derivatives of the director n would be taken with respect to the current variables y instead of x in this term. This point is generally ignored at the present time, as techniques for the set-up and analysis of the resulting problem are not yet sufficiently developed (for example, it requires y(ω to be 4

5 an open set (up to a set of measure zero in which n(y = n(y(x is in W,2 (y(ω, as well as invertibility properties on y to describe the problem rigorously; it is considered an interesting first step to analyze minimizers of the coupled energy I as defined above. (See [9]. Our goal in this paper is to show that minimizers of I exist among a class of test functions in (5 A := {u = (y, n W,2 (Ω; R 3 R 3 : det y = and n = a.e. in Ω}, suitable for modeling the stretching experiments for thin incompressible nematic elastomers with physically realistic boundary conditions imposed on all or part of Ω for y and n. We will also investigate invertibility properties of the deformation y. Challenges in this study arise from the restrictions imposed in our admissible set A and from the nonconvex coupling between the deformation gradient y and the director n in the trace formula..2. Main Results. In Lemma 2. we compute that F BTW (n, A = µ ( l { A 2 r A T n 2 + r 2 An 2 r r 2 (A T 2 } n, n 2 for all (n, A in S 2 M 3, where M 3 is the class of 3 x 3 real-valued matrices, r ( r and r 2 r. From our assumptions in Section., we have < r < and r 2. Thus ( µ l I (y, n, y, n = { y 2 r ( y T n for all (y, n W,2 (Ω; S 2 R 3. Our main results can be described as follows: In Section 3, we prove by construction: Ω r 2 [ ( yn 2 r ( yn, n 2]} + κ n 2 Theorem. The function F BWT (n, A has a continuous real-valued extension to a function F e (m, A defined on R 3 M 3 R such that: (i F e (m, A is convex with respect to A, ( for ( every fixed m ( R 3. ( (ii There exist positive constants c = c r, µ l 2 and c = c r, r, µ l 2 independent of m and A, such that c A 2 F e (m, A c A 2 for every m R 3 and A M 3. (See Lemma 3. and Theorem (3.2. Thus we can define an extended coupled energy: I e (y, m = F e (m, y + κ m 2 Ω for all (y, m W,2 (Ω; R 3 M 3 which is coercive and convex in ( y, n, sequentially weakly lower semi-continuous, and agrees with I (y, m for (y, m as above when m = 5

6 almost everywhere in Ω. (See Corollary 3.3 and Theorem 4.. It follows that minimizing sequences of I among test functions in A : {u = (y, n W,2 (Ω; R 3 R 3 : n = a.e. in Ω}, subject to appropriate anchoring or partial anchoring conditions on (y, n, have subsequences that converge to a minimizing test function in this set. (See Theorems 4.2 and 4.3. The same result need not hold in A (defined in (5, since counterexamples due to Ball and Murat (see [3] show that the map y W,2 (Ω; R 3 det y is not weakly sequentially continuous into L (Ω. Motivated by the stretching experiments on incompressible nematic elastomers occupying rectangles in R 3 that are clamped on the sides with reference director field n = e 2, we define the reference film framework to be the setting in which: H. The reference domain Ω is given by Ω = ˆΩ ( ɛ, ɛ, where ˆΩ R 2 is a bounded Lipschitz, simply connected domain in R 2, and ɛ is a fixed positive number. H2. Γ = ˆΓ ( ɛ, ɛ and Γ 2 = ˆΓ 2 ( ɛ, ɛ, where ˆΓ and ˆΓ 2 are each a finite union of relatively open connected subsets of ˆΩ whose closures are disjoint. This allows examples in which the cross section ˆΩ of Ω is a rectangle, Γ is the union of two opposite sides (or all of ˆΩ ( ɛ, ɛ and Γ 2 = or Γ 2 = Γ. For Ω as above, define the set of incompressible generalized shear deformations by: GS {y W,2 (Ω; R 3 : y(x = (ŷ (x, x 2, ŷ 2 (x, x 2, ξ(x, x 2 + x 3 γ (x, x 2 (ŷ ( ˆx, ŷ 2 ( ˆx, ξ( ˆx + x 3 γ( ˆx (ŷ( ˆx, ξ( ˆx + x 3 γ( ˆx with γ( ˆx > a.e. in ˆΩ and det y = a.e. in Ω. Deformations in this class include simple or anti-plane shears, homogeneous diagonal extensions, and shears or invertible diffeomorphisms in planes parallel to the x, x 2 plane composed with nonhomogeneous translations and dilations in the x 3 direction. Let Given (y, n A GS, define A GS, (Ω = A GS, := {(y, n A : y GS }. A GS, A GS, (y, n = { (y, n A GS, : y = y in Γ and n = n in Γ 2 in the sense of trace }. The above class of functions defines an admissible class of incompressible deformations and director fields with prescribed boundary conditions in which we investigate the existence of minimizers. We prove the following results in the thin film framework: Lemma 2. Given R >, let B = B(R = {y W,2 (Ω; R 3 : y W,2 (Ω R}. The set GS B is sequentially weakly compact (and therefore sequentially weakly closed in W,2 (Ω; R 3. (See Lemma 5.2. The proof of Lemma 2 uses the fact that for y (ŷ, ξ + x 3 γ in GS, we have det ŷ L (Ω, γ(x, x 2 = det ŷ( ˆx > a.e. in ˆΩ, and γ W,2 ( ˆΩ. 6

7 The following result provides existence of incompressible generalized shear energy-minimizers (y, n subject to boundary conditions that are relevant to the stretching experiments on incompressible elastomers: Theorem 3. Assume S is a nonempty subset of W,2 (Ω; R 3 R 3 such that for each R >, S {(y, n W,2 (Ω; R 3 R 3 : (y, n R} is sequentially weakly closed. If (y, n S A GS,, there exists a function (ỹ, ñ in S A GS, (Ω such that I (ỹ, ñ = inf (y,n S A GS, I (y, n. In particular, if (y, n A GS, (Ω, there exists (ỹ, ñ in A (Ω such that GS, I (ỹ, ñ = inf I (ỹ, ñ. (y,n A GS, (See Theorem 5.3. We also prove existence of minimizers with strong anchoring of y in Γ but weak anchoring of n in Γ 2. If ˆΓ = ˆΩ and the trace of (y, y 2 on ˆΩ satisfies certain regularity and topological properties, then the minimizer (ỹ, ñ above satisfies ỹ (ỹ, ỹ 2 is continuous in ˆΩ and one-to-one almost everywhere in ỹ ( ˆΩ. This is physically relevant, since physically realistic deformations of an elastic material should not allow interpenetration of matter. More precisely, we have: Corollary 4. Assume ˆΩ is a bounded C 3 domain, Γ = ˆΩ ( ɛ, ɛ and (y, n A GS,. Assume that the trace of (y, y 2 on ˆΩ is in W,2 ( ˆΩ and it is a restriction of a diffeomorphism ϕ of an open set Σ in R 2 containing ˆΩ such that det ϕ > in Σ. Let Ω = ϕ ( ˆΩ. Then a minimizer (ỹ, ñ as in Theorem 3 satisfies: (i (ỹ (ỹ, ỹ 2 has a continuous representative in ˆΩ, ỹ ( ˆΩ Ω and the H measure of Ω \ỹ ( ˆΩ is zero. (ii ỹ is one-to-one almost everywhere in ˆΩ, (ỹ has a representative in W, (Ω, and the usual formula for [(ỹ ] in terms of (ỹ (x, x 2 holds almost everywhere in Ω. (See Corollary 5.4. The proof of Theorem 3 uses Theorem, Lemma 2, and sequential weak lower semi-continuity properties of I in locally bounded and weakly closed subsets of A (see Theorems 4.2 and 4.3. Corollary 4 follows from the fact that ỹ A + 2,2 ( ˆΩ by Theorem 3, and results of Sverak in [8] on regularity and invertibility properties under suitable assumptions on the boundary values for deformations in the space A + p,q(d = {ϕ : D R N : ϕ L p (D, adj ϕ L q (D, det ϕ > a.e. in D} where D is an open set in R N. Remark.. The existence Theorem 3 solves the following model problem (described in [] for the stretching experiments of thin incompressible elastomers in Ω = ˆΩ ( ɛ, ɛ assuming n = (ˆn, with ˆn in S : Assume Ω, Γ, and Γ 2 are in the thin film framework. Let T = T (ɛ be the family of functions (y, n in A GS, (Ω with (y, n of the form n = (n (x, x 2, n 2 (x, x 2, (ˆn, and 7

8 y = (ŷ (x, x 2, ŷ 2 (x, x 2, x 3 γ(x, x 2 (ŷ ( ˆx, ŷ ( ˆx, x 3 γ( ˆx almost everywhere in Ω. Assume (y, n is in T. Minimize the average energy of I in Ω among functions in T A GS, (Ω, i.e. functions in T satisfying y = y on Γ and n = n on Γ 2 (or weak anchoring of n to n in Γ 2. Note that by definition, functions in T A GS, (Ω satisfy γ(x, x 2 = det ŷ(x,x 2 > and ξ(x, x 2 = almost everywhere in ˆΩ, and thus the average energy of I in Ω is given by: (6 Ω I (y, n, y, n = ˆΩ ˆΩ µ 2 ( l ŷ 2 + ɛ2 3 ( det ŷ( ˆx 2 + det ŷ( ˆx 2 r ( ŷ T ˆn 2 ɛ 2 + r 2 3 ( det ŷ( ˆx, ˆn 2 + r 2 ( ŷˆn 2 r r 2 ( ŷˆn, ˆn 2 } + κ ˆn 2. Theorem 3 provides existence of minimizers for this problem in a wide class of boundary conditions determined by (y, n in this set. An example corresponding to the stretching experiments is given by Ω = ( a, a ( b, b ( ɛ, ɛ, n = e 2, Γ = {±a} ( b, b ( ɛ, ɛ, and Γ 2 = Γ or Γ 2 =, with boundary conditions y = y = (cx, x 2, c x 3 and n = n = e 2 on Γ 2 where c > (or weak anchoring of n to n on Γ 2. An explicit formula for the quasi-convexification of the energy in (6 for the case r 2 = ɛ = κ = was calculated by DeSimone and Dolzmann in [2]. (See also []. (They also computed the quasi-convexification of a modified BTW energy with r 2 a small positive number. We remark that if the set T above is further restricted by requiring γ(x, x 2, we obtain from Theorem 3 the existence of minimizers for the two-dimensional coupled BTW/Frank energy, in which the deformations (y, n have the form ((ŷ( ˆx, x 3, (ˆn( ˆx, with (ŷ, ˆn W,2 ( ˆΩ; R 2 S and det ŷ = a.e. in ˆΩ. Nematic elastomers have been extensively studied in several scientific fields. Investigators have obtained minimizers for various models and function spaces. We present here a brief (and non-exhaustive summary of some of the relevant mathematical approaches related to the problem of existence of minimizers corresponding to incompressible elastomers. We have already described fundamental work by Conti, DeSimone and Dolzmann (see [], [], [2], [3], [4]. Recently in [8] Cesani and DeSimone formulated an energy that couples a modification of the BTW energy involving a fourth order strain tensor of elastic moduli and the Landau-de Gennes energy and proved existence in some cases. In [9], they computed the quasiconvex envelope of an energy density for nematic elastomers in the small strain (geometrically linear regime and plane strain conditions. Anderson, Carlson and Fried [2], developed a continuum theory for incompressible nematic elastomers using energy balance arguments. They proposed a two-constant energy density that extends the neo-classical theory and incorporates the classical Oseen-Frank nematic liquid crystal theory. 8

9 Recently in [7], Calderer and Luo proved existence of minimizers of the two-dimensional coupled BTW-Frank energy with r 2 = for boundary value problems in a domain D R 2, with y in W,2 (D; R 2 and n in W,2 (D; S for incompressible nematic elastomers. They also studied a related linearized system with Lagrange multipliers and computed numerical simulations of rotations of the director field (semisoft elasticity for these problems. Our Theorem 3 extends their existence result in two dimensional problems to the case r 2 > and to generalized incompressible shear deformations in three dimensional problems. 2. PRELIMINARY CALCULATIONS In this section, we derive the simplified expression of the trace formula (3 stated in Section.2 and use it to rewrite the coupled BTW/Frank energy. This expression will be used in our analysis to obtain convexity properties of the energy. Recall that by (, L = L(n is a symmetric positive definite matrix with eigenvalues l and, for n in S 2. It is easily checked that the eigenspace corresponding to l is spanned by n, and the eigenspace corresponding to is the two-dimensional subspace of vectors in R 3 that are orthogonal to n. Similarly, L = L(n has these same properties with l, l and n replacing l, and n. By definition of L, setting N = n n we have Tr ( L ( y T L ( y = Tr ( l {I + (r N } ( y T L ( y = l Tr ( ( y T L ( y + (l l Tr ( N ( y T L ( y. Recalling that for a real matrix A we have A 2 = Tr (AA T, by (2 we have Tr ( ( y T L ( y = Tr ( ( ( y T L 2 L 2 ( y = Tr ( ( y T (L ( 2 T L 2 ( y = L 2 2 ( y. Since n = implies that (N 2 = N, and since Tr (AB = Tr (BA for all matrices A, B M k, we can also write Tr ( N ( y T L ( y = Tr ( N ( y T L ( yn ([ ] T [ ] = Tr L 2 ( yn L 2 ( yn = 2 2 ( yn and hence L Tr ( L ( y T L ( y = l L 2 2 ( y + (l l L 2 ( yn 2. Thus, the trace formula (3 can be written as (7 F BWT (n, y = µ { l L /2 ( y 2 + (l 2 l L 2 } 2 ( yn By formula (2 for L(n and L(n /2 (which uses the fact that n = n =, and since 9

10 AN 2 = An 2 and N AN 2 = n, An 2 when N = n n, N = n n, n = n = and A M 3, we have L 2 2 ( y = = L 2 ( yn 2 = ( ] r [ y 2 + ( y T N 2 y 2 + ( y T n 2 ( yn 2 + ( l ( l, ( yn, n 2. Inserting this in (7, we obtain (8 F BWT (n, y = µ 2 = { l ( y 2 + l ( y T n 2 l + (l l ( yn 2 + (l l l ( l { ( y 2 ( y T n 2 + r µ 2 (r ( ( yn, n 2 l [ ( ( yn 2 ]} ( yn, n 2. r For ease of notation, define (9 r := r >, r 2 := r = l l l. From the calculations (7-(9 (with y replaced by A and the definition of r and r, we have: Lemma 2.. For L = L(n and L = L(n with n S 2, A M 3, and n as described above, the trace formula (3 satisfies ( F BWT (n, A := µ 2 l { L(n /2 A 2 + r2 L(n 2 AN 2 }, ( l { A [ 2 r A T n 2 + r 2 An 2 r An, n 2]}. = µ 2 Note that since < r < and r 2, the first four terms inside the brackets above have coefficients that alternate in sign between positive and negative when r 2 >. Adding the Frank energy density, the coupled energy density is given by:

11 ( W (y, n, y, n F BTW (n, y + κ n 2 = µ { L(n 2 l /2 ( y 2 + r2 L(n 2 ( y(n n 2} + κ n 2 ( l = µ 2 { y 2 r ( y T n 2 + +r 2 ( yn 2 r r 2 ( yn, n 2} + κ n 2 for all (y, n in A = {(y, n W,2 (Ω; R 3 W,2 (Ω; R 3 : n = a.e. in Ω}. 3. AN EXTENSION OF THE COUPLED ENERGY DENSITY In this section we prove that the trace formula F BWT defined on S 2 M 3 has a continuous extension F e defined on R 3 M 3 such that F e (m, A is convex and coercive with respect to A in M 3 for each fixed m R 3. Our definition of F e is constructive, since formula ( is not coercive in A for every n in R 3 \S 2. Our approach is to define a function T : R 3 M 3 such that T(m = L(m /2 when m =, and show that the function F e (m, A obtained by replacing L(n /2 by T(m in the trace formula ( is convex and coercive in A for each m R 3. It follows that the coupled energy density W (y, n, y, n : R 3 S 2 M 3 M 3 R for I defined in ( has a continuous extension, W e (y, m, y, m : R 3 R 3 M 3 M 3 R such that W e (y, m, A, B is convex and coercive in (A, B for each fixed (y, m R 3 R 3. We will use this result in Section 4 to establish weak lower-semicontinuity and existence of minimizers of the extended energy, I e in appropriate admissible subsets of A, and related properties for I that will be used in Section 5 to prove our existence theorem 3 for I. > is fixed, let us define additional param- Recalling that the anisotropy parameter r l eters r 3, r 4 and r 5 by (2 r 3 := r, Note that r 4 := + r 3, and 2 r 5 := r 4. r 3 (3 < r 3 <, < r 3 < r 4 <, r 5 > and Define the positive function σ : R 3 R by (4 σ(m := m 2 + r 5 r 4 ( 2 r 4 m 2, m 2 + r 5 r 4 + r 5 = r 3.

12 and define the function T : R 3 M 3 by (5 T(m := ] r 4 [I m m. l m 2 + r 5 By (2 and (3, for a unit-length vector, definition (5 coincides with the second term in (2, i.e. for every m R 3 with m = we have T(m = L(m /2. Using these definitions, we have: Lemma 3.. Define the extension of the trace formula for m R 3 and A M 3, by (6 F e (m, A := µ 2 l { T(mA 2 + r 2 T(mA(n n 2}. Then for all (m, A R 3 M 3, we have (7 and F e (m, A = µ 2 ( l { A 2 σ(m A T m 2 + r 2 An 2 r 2 σ(m A T m, n 2 }, (8 F e (m, A = F BWT (m, A when m =. In particular, if we define the extended coupled energy density W e (u, C for u = (y, m R 3 R 3 and C = (A, B M 3 M 3 by (9 W e (u, C = W e (y, m, A, B = F e (m, A + κ B 2 then W e (y, m, A, B = µ l { A 2 σ(m A T m 2 + r 2 An 2 r 2 σ(m A T 2 } m, n 2 (2 +κ B 2, and W e (y, m, A, B = W (y, m, A, B for all m R 3 such that m =. Proof. Let (m.a R 3 M 3. Using (5 and setting τ := τ(m = T(mA 2 = A τ[m m]a 2 [ ] 2 = ai j τ[(m ma] i j = i, j 2 (a i j 2 2τa i j m i m k a k j + τ 2 m i m k a k j i, j = A 2 2τ a i j m i a k j m k + τ2 i j = A 2 2τ A T m 2 + τ 2 m 2 A T m 2 = A 2 τ A T m 2 (2 τ m 2 i 2 k k k r 4 m 2 + r 5 we have that m 2 i (A T m 2 j j

13 and therefore (2 A 2 = T(mA 2 + We have shown, using (4, that ( r 4 A T m 2 r 4 2 m 2. m 2 + r 5 m 2 + r 5 T(mA 2 = A 2 σ(m A T m 2. The above calculation with A replaced by AN yields T(mAN 2 = AN 2 σ(m (AN T m 2 The sum of these equations and (6 proves (7. = An 2 σ(m N A T m 2 = An 2 σ(m n, A T m 2. Note that by definitions (9 and (2, for every vector m R 3 such that m =, we have σ(m r, and therefore, using (, we obtain (8. The rest of the lemma follows from this. The following theorem describes convexity and coercivity properties satisfied by the extended trace formula F e : Theorem 3.2. The extended trace formula F e (m, A given by equation (6 satisfies, for m R 3 and A M 3, the following properties: (i F e (m, A is a continuous function of (m, A and F e (m, A. (ii F e (m, A is strictly convex with respect ( to A, for every fixed m R 3. (iii There exists a positive constant c = c r, r, µ 2 ( l l independent of m and A, such that F e (m, A c A 2 for every m R 3. (iv F e (m, A is uniformly coercive( with respect to A for each fixed m R 3, i.e., there exists a positive constant c = c r, µ 2 ( l l independent of m and A such that c A 2 F e (m, A for every m R 3. (v F e (m, A is quadratic with respect to A, i.e. for each i, j, α, β in {, 2, 3} there exist real functions M i, j α,β : R3 R such that F e (m, A = 3 i, j,α,β= M i, j α,β (ma αia β j. Moreover, F e (m, A satisfies the Legendre-Hadamard condition, i.e., there exists a positive constant ν > depending only on r and µ 2 ( l l such that for every ξ, η R 3 we have 3 (M i, j α,β (mξ iξ j η α η β ν ξ 2 η 2. i, j,α,β= 3

14 Proof. We will prove each statement: (i F e is clear from its expression in (6. The continuity follows from (6 and definition (5. (ii Let m R 3 and A, A 2 M 3. Set N = n n. For λ [, ] we have, using that the function X KXJ 2 is strictly convex for any fixed matrices K, J M 3 : F e (m, λa + ( λa 2 = µ { 2 l T(m(λA + ( λa r 2 T(m(λA + ( λa 2 N 2} µ { ( 2 l λ T(mA 2 + ( λ T(mA r 2 λ T(mA N 2 + ( λ T(mA 2 N 2} = λf e (m, A + ( λf e (m, A 2, with equality holding if and only A = A 2. (iii Using the fact that for any vector v R 3, v v = v 2, we have by the definition of T(m in (5 that if m R 3, r 4 T(m = I m m l m 2 + r 5 ( I + r 4 m m l := c. ( 3 + r4 l m 2 + r 5 Therefore for every (m, A R 3 M 3, F e (m, A µ [ 2 l c 2 A 2 + r 2 c 2 A 2] := c 2 A 2 and we have the desired result with (iv Let m, A R 3 M 3. Since we have by (2 that which by (2 yields (22 A 2 T(mA 2 + c := c 2 = µ 2 l c 2 ( + r 2 = µ ( l ( 3 + r 4 2 r >. 2 m 2 m 2 + r 5 ( r 4 2 m 2 2 r m 2 4 >, + r 5 Consider now the real valued function Since < r 4 <, if z [, ], we have ( r 4 A 2 m 2 r 4 2 m 2. m 2 + r 5 m 2 + r 5 f (z = r 4 z(2 r 4 z = (r 4 z 2. f (z (r 4 2 := c 3 <, 4

15 m 2 and thus we conclude, taking z := [,, that for every m R 3 m 2 + r 5 r 4 m 2 (2 r 4 m 2 c m 2 + r 5 m 2 3 <. + r 5 In particular, by definition (4 of the function σ we have that for every m R 3 (23 m 2 σ(m c 3 and from (22 this also yields A 2 T(mA 2 + c 3 A 2. Since c 3 <, we can define c 4 := c 3 > to conclude from the last inequality that c 4 A 2 T(mA 2. From this and definition (6 ( µ l c 4 A 2 µ 2 2 l T(mA 2 F e (m, A, ( l ( µ and we obtain the desired conclusion defining c := c 4 2 = µ 2 ( r 3 l 2 2 l >. (v Using formula (7 we can define for m R 3 and i, j, α, β 3 the functions M i, j α,β (m = µ l { } δi j δ αβ. σ(mδ i j m α m β + r 2 δ i2 δ j2 δ αβ r 2 σ(mδ i2 δ j2 m α m β. 2 Now using also (23 we can calculate for every ξ, η R 3 : 3 ( F e (m, A = M i, j α,β (mξ iξ j η α η β = µ l ( ξ 2 + r 2 ξ 2 2 ( η 2 σ(m η m 2 2 i, j,α,β= l µ ( ξ 2 + r 2 ξ 2 2 η 2 ( σ(m m 2 2 l µ ( ξ 2 + r 2 ξ 2 2 η 2 ( c 3 2 µ l ξ 2 η 2 ( c 3. 2 The Legendre-Hadamard condition follows taking ν := µ 2 l ( c 3 >. The following is a direct consequence of Theorem 3.2 and the definition of the extended coupled energy density W: Corollary 3.3. The extended coupled energy density W e (u, C = W e (y, m, A, B given by equation (9 satisfies, for general u = (y, m R 3 R 3 and C = (A, B M 3 M 3, the following properties: (i W e (u, C is a continuous function of (u, C and W e (u, C. (ii W e (u, C is strictly convex with respect to C, for every fixed u R 3 R 3. 5

16 ( (iii For the constant c = c r, r, µ 2 ( l l > independent of u and C defined in Theorem 3.2, we have W e (u, C C 2 max{c, κ} for every (u, C R 3 R 3 M 3 M 3. (iv W e (u, C is uniformly ( coercive with respect to C for each fixed u R 3 R 3, i.e. for the constant c = c r, µ 2 ( l l > independent of u and C defined in Theorem 3.2, we have C 2 min{ c, κ} W e (u, C for every (u, C R 3 R 3 M 3 M 3. (v W e (u, C is quadratic in C for every u R 3 R 3, and for N i, j α,β = κδ i jδ αβ and M i, j α,β (m as in Theorem 3.2, we have W e (u, C = W e (y, m, A, B = 3 i, j,α,β= ( M i, j α,β (ma αia β j + N i, j α,β b αib β j. Moreover, W e satisfies the Legendre-Hadamard condition, i.e., for the positive constant ν > defined in Theorem 3.2, we have 3 ( M i, j α,β (uξ iξ j η α η β + N i, j α,β (uξ iξ j θ α θ β ξ 2 ( η 2 + θ 2 min{ν, κ} i, j,α,β= for every ξ, η, θ R WEAK LOWER-SEMICONTINUITY OF THE EXTENDED ENERGY FUNCTIONAL IN R 3 In this section, we prove weak lower semicontinuity for the extended coupled energy, (24 I e (u = I e (y, m = W e (u, udx = [F e (m, y + κ m 2 ] dx Ω defined for all u = (y, m W,2 (Ω; R 3 R 3, and related results for the coupled energy density for elastomers, (25 I (u = W (u, udx = [F BTW (n, y + κ n 2 ] dx defined for all u = (y, n in Ω (26 A = {(y, n W,2 (Ω; R 3 R 3 : n = a.e. in Ω}, where Ω is a simply connected bounded Lipschitz domain in R 3. Note that by Lemma 3., if u = (y, m is in A, then W e (u, u = W (u, u almost everywhere in Ω and thus I e (y, m = I (y, m. From the calculus of variations and the properties of W e proved in Corollary 3.3, we have the following result: Theorem 4.. (Sequential Weak Lower-Semicontinuity of I e The extended coupled energy functional I e is sequentially weakly lower semicontinuous in W,2 (Ω; R 3 R 3 ; that is, if {u ν (y ν, n ν } W,2 (Ω; R 3 R 3 and converges weakly in W,2 (Ω; R 3 R 3 to some ũ = (ỹ, ñ W,2 (Ω; R 3 R 3, then I e (ỹ, ñ lim inf ν I e(y ν, n ν. 6 Ω Ω

17 Proof. A classical result by Acerbi and Fusco (see [], Theorem II.4 states that if Ω is a bounded open set in R N and ϕ(x, u, C : R N R m M m N R is measurable in x, continuous in (u, C, quasiconvex in C for any x R N and u R m, and satisfies for some M >, then the functional u ϕ(x, u, C M( + C 2, Ω ϕ(x, u(x, u(x dx is sequentially weakly lower-semicontinuous in W,2 (Ω; R m. By (4, (2, and Corollary 3.3 the sufficient properties described above hold for the energy density ϕ(x, u, C = W e (u, C with m = 6, and N = 3, and therefore we conclude that the energy functional I e is weakly lower semicontinuous in W,2 (Ω; R 3 R 3. In order to consider boundary value problems for I of interest for applications, we assume for the remainder of this section that Υ and Υ 2 are fixed open subsets of Ω such that Υ, and each of Υ and Υ 2 is a finite union of open connected subsets of Ω whose closures are disjoint. Let u = (y, n A and define (27 A = A (y, n := {u = (y, n A: y = y in Υ, n = n in Υ 2 } The boundary conditions in (27 should be interpreted in the weak sense, meaning that y y W,2 (Ω\Υ ; R 3 and if Υ 2, n n W,2 (Ω\Υ 2; R 3, where for a closed subset T Ω, W,2 (Ω\T denotes the set of functions in W,2 (Ω that are a limit of a sequence of functions belonging to C (Ω\T. (If Υ 2 =, there are no boundary conditions on n. Given any R >, let B R = {(y, m W,2 (Ω; R 3 R 3 : (y, m W,2 (Ω;R 3 R 3 R}. Note that since W,2 (Ω\T is a closed convex subset of W,2 (Ω for T = Υ and for T = Υ 2, a bounded sequence in W,2 (Ω\T has a subsequence that converges weakly in W,2 (Ω to (Ω\T. Also, by the Sobolev Imbedding Theorem for bounded Lipschitz domains in R 3, a bounded sequence in W,2 (Ω has a subsequence that converges weakly in W,2 (Ω, strongly in L 4 (Ω, and after passing to another subsequence, pointwise almost everywhere in Ω. It follows that A B R and A B R are sequentially weakly compact in W,2 (Ω; R 3 R 3. a function in W,2 Define (28 A = {(y, n A : det y = a.e. in Ω}. For fixed sets Υ and Υ 2 Ω as described above and u = (y, n A, define A = A A. For the problem of minimizing I in appropriate subsets of A, we shall need: 7

18 Theorem 4.2. Fix Υ and Υ 2 Ω as described above. Let S be a nonempty subset of W,2 (Ω; R 3 R 3 such that for each R >, S B R is sequentially weakly closed in W,2 (Ω; R 3 R 3. (For example, S = A is such a set. Assume (y, n S A, and {(y ν, n ν } is a minimizing sequence for I in S A (or any sequence in S A such that {I (y ν, n ν } is bounded above. Then there exists a function (ỹ, ñ S A and a subsequence {(y νk, n νk } such that {(y νk, n νk } is bounded in W,2 (Ω; R 3 R 3, (y νk, n νk (ỹ, ñ weakly in W,2 (Ω; R 3 R 3, (y νk, n νk (ỹ, ñ in L 2 (Ω and pointwise a.e. in Ω, and I (ỹ, ñ lim inf k I (y νk, n νk. In particular, I achieves a minimum in S A. Proof. Let {u ν = (y ν, n ν } be a sequence in S A such that {I (u ν } is bounded above. Using Corollary 3.3, we have u ν L 2 ( c I e (u ν = c I (u ν K < where c and K are independent of ν. Since n ν = a.e. in Ω, we have (29 n ν 2 W,2 (Ω Ω + n ν 2 L 2 (Ω Ω + u ν 2 L 2 (Ω. Also, since Υ is a nonempty relatively open set in Ω and the trace of y ν y is zero in Υ, the Sobolev-Poincare inequality for bounded Lipschitz domains implies that y ν y L 2 (Ω C( (y ν y L 2 (Ω where C > depends only on Υ and Ω. Thus y ν W,2 (Ω c( + y ν L 2 (Ω where c > depends only on Ω, Υ, and y, and we have y ν W,2 (Ω c ( (3 + y ν L 2 (Ω c ( + u ν L 2 (Ω for a constant c > as above. By (29 and (3 we conclude that u ν W,2 (Ω c 2 ( + uν L 2 (Ω c2 ( + K < for all ν, where c 2 and K are independent of ν. Thus {u ν } S A B R for some R >. Hence there exists a function ũ = (ỹ, ñ W,2 (Ω; R 3 R 3 and a subsequence {u νk } {u ν } such that u νk ũ in W,2 (Ω; R 3 R 3, u νk ũ in L 2 (Ω, and u νk ũ pointwise a.e. in Ω. 8

19 Since A B R is sequentially weakly closed in W,2 (Ω; R 3 R 3, we have ũ A B R. By Theorem 4. and since I = I e in A, I [ũ] lim inf k I [u νk ]. Finally, since S B R is sequentially weakly closed, ũ S A. The existence of minimizers for I in S A follows from this. The above result can be modified to include weak anchoring of n to n on Υ 2 rather than strong anchoring as above, which is of interest in many applications. This is formulated by adding a weak anchoring term known as the Rapini-Papoular surface energy. Given Υ and Υ 2 as above, assume ω >. Let (y, n A. Define and A w = A w(ω = {(y, n A : y = y on Υ } I w (y, n = I (y, n ω n, n ds. Υ 2 A slight change in the proof of Theorem 4.3 using the compactness of the trace map from W,2 (Ω; R 3 into L 2 ( Ω; R 3 gives: Theorem 4.3. Fix Υ, Υ 2, and S as in Theorem 4.2. Assume ω > and (y, n S A. Assume {(y ν, n ν } is a minimizing sequence for I w in S A w (or any sequence in S A such that {I w (y ν, n ν } is bounded above. Then there exists a function (ỹ, ñ S A and a subsequence {(y νk, n νk } such that {(y νk, n νk } is bounded in W,2 (Ω; R 3 R 3, (y νk, n νk (ỹ, ñ weakly in W,2 (Ω; R 3 R 3, (y νk, n νk (ỹ, ñ in L 2 (Ω and pointwise a.e. in Ω, and I (ỹ, ñ lim inf k I (y νk, n νk. In particular, I ω achieves a minimum in S A. Remark 4.. The proof of Theorem 4.2 can be generalized to show that if (y, m W,2 (Ω; R 3 R 3 and η > is fixed, the relaxed energy Ĩe defined by Ĩ e (y, m = [F e (m, y, m + η ( 2 m 2 2 ] achieves a minimum in Ω {(y, m W,2 (Ω; R 3 R 3 : y = y in Υ and m = m in Υ 2.} The energy Ĩe is sequentially weakly lower-semicontinuous in W,2 (Ω; R 3 R 3 by Theorem 4. and the fact that the imbedding from W,2 (Ω; R 3 into L 4 (Ω; R 3 is compact. 9

20 Note that Theorem 4.2 is not sufficient to conclude the existence, given (y, n in A, of a minimizer of I in A, that is, a function (ỹ, ñ A satisfying I (ỹ, ñ = inf I (y, n, (y,n A since we have not shown that A BR is a sequentially weakly closed set in W,2 (Ω; R 3 R 3. In fact, counterexamples (see [3] show that in a bounded Lipschitz domain Ω in R N with N 3, the mapping: y det y is not sequentially weakly continuous from W,p (Ω; R N to L (Ω with p N, that is, y ν y in W,p (Ω; R N does not imply that det y ν det y in L (Ω. On the other hand, in Section 5 we will use Theorems 4.2 and 4.3 to prove existence of minimizers for I among states (y, n in A in which the deformations y are incompressible generalized shear deformations (denoted by A subject to our boundary conditions, GS, with an analogous result for I w in the case of weak anchoring on n. 5. ENERGY-MINIMIZING GENERALIZED SHEAR DEFORMATIONS FOR INCOMPRESSIBLE ELASTOMERS Throughout this section, we assume that Ω ˆΩ ( ɛ, ɛ, Γ ˆΓ ( ɛ, ɛ, and Γ 2 ˆΓ 2 ( ɛ, ɛ are in the reference film framework as defined by conditions H and H2 in Section.2. Recall that the set of generalized shear deformations in Ω, denoted by GS = GS(Ω, is the set of all y in W,2 (Ω; R 3 such that (3 y = (ŷ (x, x 2, ŷ 2 (x, x 2, ξ(x, x 2 + x 3 γ(x, x 2 (ŷ(x, x 2, ξ(x, x 2 + x 3 γ(x, x 2 for some ŷ, ŷ 2, ξ, γ W,2 ( ˆΩ with with γ(x, x 2 > almost everywhere in ˆΩ. Depending on the boundary conditions, this allows the deformation gradient y to correspond to simple or anti plane shears, as well as invertible diffeomorphisms in the x, x 2 plane composed with nonhomogeneous translations and dilations in the x 3 -coordinate direction. For y GS the deformation gradient y is ŷ,x ŷ,x2 (32 y = ŷ 2,x ŷ 2,x2 ξ x + x 3 γ x ξ x2 + x 3 γ x2 γ and adj y = [ γŷ 2,x2 γŷ,x2 γŷ 2,x ] [ γŷ,x ] [ ŷ,x ξ x2 ŷ,x2 ξ x +x 3 (ŷ,x γ x2 ŷ,x2 γ x ŷ 2,x ξ x2 ŷ 2,x2 ξ x +x 3 (ŷ 2,x γ x2 ŷ 2,x2 γ x ] ŷ,x ŷ 2,x2 ŷ,x2 ŷ 2,x where ŷ i,x j denotes ŷ i x j. 2

21 The set of incompressible generalized shear deformations is given by (33 GS = GS (Ω: = {y GS(Ω: det y = a.e. in Ω}. By (32, if y GS(Ω then Therefore by (3 if y GS (Ω, we have (34 γ(x, x 2 = where det y = γ(x, x 2 det ŷ a.e. in Ω. ŷ = det ŷ(x > a.e. in ˆΩ, ( ŷ,x ŷ,x2. ŷ 2,x ŷ 2,x2 The incompressible generalized shear admissible set is defined by and for (y, n fixed in A GS,, A GS, = A GS, (Ω := {(y, n A : y GS }, A GS, = A GS, (Ω := {(y, n A : y GS }. Note that A GS, since (y, n belongs to this set by assumption. Remark 5.. Deformations of incompressible thin nematic elastomers in a subclass of A GS, (Ω were considered by Bladon, Warner and Terentjev in [5] and [6] assuming that ˆΩ is a two-dimensional rectangle, n = e 2 and n(x R 2 {} with ξ(x, x 2. Also in [], Conti, DeSimone, and Dolzmann computed an effective energy in terms of F = ŷ(x, x 2 for a reduced model motivated by the problem of minimizing I in a subclass, ξ, of deformations in A GS, (Ω in order to model the stretching experiments of a thin nematic elastomer. In Theorem 5.3 of this section, we prove existence of minimizers of I in A (or in the GS, intersection of A GS, with any nonempty set S in W,2 (Ω; R 3 R 3 such that S B R is sequentially weakly closed for each R >, assuming that (y, n is in A GS, (or S A GS, (Ω. A consequence is the existence of minimimizers for the two-dimensional reduced energy (6, as well as the more general formulation in which ξ(x, x 2 is not assumed to be zero, subject to boundary conditions on the sides of Ω which has been used to model the stretching of very thin incompressible elastomers. We first prove that the intersection of a closed ball in W,2 (Ω; R 3 and the set of incompressible generalized shear deformations GS (Ω is sequentially weakly compact. (See Theorem 5.2. Our proof uses the following proposition which is a special case of a result due to Müller, Qi and Yan on the weak convergence of determinants of a sequence of deformation gradients in the sense of distributions. (See Lemma 4. in [7] with N = p = q = N N = 2. Proposition 5.. (Müller, Qi and Yan Assume {w ν } is a sequence in W,2 ( ˆΩ; R 2 satisfying w ν w in W,2 ( ˆΩ; R 2 and det w ν a.e. in ˆΩ. Then det w ν det w in L loc ( ˆΩ. Lemma 5.2. Assume Ω satisfies H. Given R >, let B = B(R = {y W,2 (Ω; R 3 : y W,2 (Ω R}. Then B GS (Ω is sequentially weakly compact in W,2 (Ω; R 3. 2

22 Proof. Let {y ν } be a sequence in B GS (Ω. Then y ν (x, x 2, x 3 = (ŷ ν (x, x 2, ξ ν (x, x 2 + x 3 γ ν (x, x 2 with (y ν, (y ν 2, ξ ν, γ ν in W,2 ( ˆΩ. The set of all functions of this form in W,2 (Ω; R 3 (with no sign condition on γ is convex and closed in W,2 (Ω; R 3. Since {y ν } B(R, there exists a constant C = C(R independent of ν such that = Ω { ŷ ν 2 + (y ν ŷ ν 2 + (y ν 3 2 }dx {2ɛ ( ŷ ν 2 + ξ ν ˆΩ 3 ɛ3 γ ν 2 + 2ɛ ( ŷ ν 2 + ξ ν ɛ3 γ ν 2 }d ˆx C. It follows that a subsequence {y νk } converges weakly in W,2 (Ω; R 3 to a function (35 y (x, x 2, x 3 = ((y (x, x 2, (y 2 (x, x 2, ξ (x, x 2 + x 3 γ (x, x 2 (ŷ (x, x 2, (y 3 (x, x 2, x 3 such that y B R, (ŷ, ξ, γ is the weak limit in W,2 ( ˆΩ; R 4 of (ŷ νk, ξ νk, γ νk, and γ. By passing to a subsequence (which we do not relabel, we also have that {(ŷ νk, ξ νk, γ νk } converges strongly in L 2 ( ˆΩ and pointwise almost everywhere in ˆΩ to (ŷ, ξ, γ. By (35, det y (x, x 2, x 3 = γ (x, x 2 det ŷ (x, x 2 a.e. in Ω. Thus the theorem follows if we prove that γ (x, x 2 > and γ (x, x 2 det ŷ (x, x 2 = almost everywhere in ˆΩ, since then we can conclude that y GS (Ω. Since {y ν } GS (Ω: (36 < γ νk (x, x 2 = and thus γ (x, x 2 = lim k γ νk (x, x 2 det ŷ νk (x, x 2 < a.e. in ˆΩ. = lim k det ŷ νk (x, x 2 < a.e. in ˆΩ. The above limits hold pointwise almost everywhere and in L 2 ( ˆΩ. In particular, since γ L 2 ( ˆΩ and hence γ < almost everywhere in ˆΩ, and similarly for each k, < det ŷ νk L ( ˆΩ and hence < det ŷ νk < almost everywhere in ˆΩ, we conclude from the above that (37 < lim k det ŷ νk (x, x 2 = lim k γ νk (x, x 2 = γ (x, x 2 a.e. in ˆΩ, 22

23 where the use of limit in (37 refers to a limit valued in the extended positive real numbers in (, ]. By (36 and the fact that ŷ νk converges weakly in W,2 ( ˆΩ; R 2 to ŷ, we can apply Proposition 5. to conclude that det ŷ νk (x, x 2 converges weakly in L loc ( ˆΩ to det ŷ (x, x 2. From this and Fatou s lemma (using (36 and (37, we have that for any ϕ in C ( ˆΩ (or any nonnegative ϕ L ( ˆΩ with compact support in ˆΩ: ϕ d ˆx = lim ˆΩ γ k ϕ = lim ˆΩ γ k ϕ det ŷ νk d ˆx νk ˆΩ lim k ϕ det ŷ νk d ˆx ˆΩ = ϕ det ŷ d ˆx <. ˆΩ Thus γ Lloc ( ˆΩ. By (37 and the above inequality, we have < It follows that < γ (x, x 2 < and γ (x, x 2 det ŷ (x, x 2 < a.e. in ˆΩ. γ (x, x 2 det ŷ (x, x 2 a.e. in ˆΩ. To prove that equality holds above, for each δ in (, let ˆΩ δ = { ˆx ˆΩ: δ γ ( ˆx δ and lim k γ ν k ( ˆx = γ ( ˆx}. Since γ L 2 ( ˆΩ and we have < γ < a.e. in ˆΩ, it follows that χ ˆΩ δ χ ˆΩ as δ a.e. in ˆΩ. By Egorov s Theorem and the definition of ˆΩ δ, for each δ >, there exists a measurable set E δ ˆΩ δ such that γ νk γ uniformly on E δ and ˆΩ δ \E δ δ. Since γ νk = det ŷ νk a.e. in ˆΩ and γ γ νk uniformly on E δ, using the weak L loc ( ˆΩ convergence of det ŷ νk to det ŷ, we have for any ϕ in C ( ˆΩ χ Eδ ϕ(γ det ŷ d ˆx ˆΩ = lim χ Eδ ϕ(γ det ˆ y νk d ˆx k ˆΩ = lim χ Eδ ϕ( γ d ˆx k ˆΩ γ νk =. Since ϕ(γ det ŷ a.e. in ˆΩ, we conclude from the above equation that ϕ(γ det ŷ = a.e. in E δ for each δ >. Also since Ω\E δ as δ, we have ϕ(γ det ŷ = a.e. in ˆΩ. Recalling that ϕ in C ( ˆΩ was arbitrary, we obtain γ det ŷ = a.e. in ˆΩ. We can now prove the existence result stated as Theorem 3 in the introduction: Theorem 5.3. Assume Ω, Γ and Γ 2 are in the reference film framework. Let S be a nonempty subset of W,2 (Ω; R 3 R 3 such that for each R >, S B R is sequentially 23

24 weakly closed in W,2 (Ω; R 3 R 3. If (y, n S A GS,, there exists a function (ỹ, ñ in S A GS, (Ω such that I (ỹ, ñ = inf (y,n S A GS, I (y, n. In particular, if (y, n A GS, (Ω, there exists (ỹ, ñ in A (Ω such that GS, I (ỹ, ñ = inf I (ỹ, ñ. (y,n A GS, Moreover, an analogous result holds for weak anchoring. More precisely, for Γ, Γ 2, S and (y, n as above, the statement of Theorem 5.3 holds with I replaced by I w, and A GS, (Ω replaced by A w,gs, (Ω A GS, A w. Proof. Let {(y ν, n ν } be a minimizing sequence for I in S A. By Theorem 4.2 with GS, Υ = Γ and Υ 2 = Γ 2, there exists (ỹ, ñ in S A, a constant R >, and a subsequence {(y νk, n νk } B R that converges to (ỹ, ñ weakly in W,2 (Ω; R 3 R 3, strongly in L 2 (Ω; R 3 R 3 and pointwise almost everywhere in Ω, such that I (ỹ, ñ inf I (y, n. (y,n S A GS, Since {(y νk, n νk } is contained in S B R A which is sequentially weakly closed in W,2 (Ω; R 3 R 3 by our assumption on S, we have (ỹ, ñ S B R A. By Lemma 5.2, since {y νk } B R GS (Ω and converges weakly to ỹ in W,2 (Ω; R 3 R 3, we have ỹ GS (Ω. Thus (ỹ, ñ S A (Ω and GS, I (ỹ, ñ = inf I (y, n. (y,n S A GS, The existence of minimizers in A GS, follows by applying the above result with S = A GS,. The result in the case of weak anchoring is proved similarly, using Theorem 4.3 instead of Theorem 4.2. A consequence of this result is: Corollary 5.4. Assume ˆΩ is a bounded simply connected C 3 domain, Ω, Γ ˆΩ ( ɛ, ɛ and Γ 2 are in the reference film framework, and S is a nonempty set as in Theorem 5.3. Assume (y, n S A GS, (Ω, and the trace of (y (y, y 2 on ˆΩ is in W,2 ( ˆΩ; R 2 and it is a restriction on ˆΩ of a homeomorphism ϕ of an open set Σ in R 2 containing ˆΩ onto an open set ϕ (Σ in R 2 which satisfies ϕ A + 2,2 (Σ and ϕ ˆΩ A 2,2( ˆΩ. (For example, ϕ as described in Corollary 4 of the introduction. Let Ω = ϕ ( ˆΩ. Then the minimizer (ỹ, ñ of I (or I w in Theorem 5.3 satisfies: (i (ỹ (ỹ, ỹ 2 has a continuous representative in ˆΩ, ỹ ( ˆΩ Ω and the H measure of Ω \ỹ ( ˆΩ is zero. (ii ỹ is one-to-one almost everywhere in ˆΩ, (ỹ has a representative in W, (Ω, and the usual formula for [(ỹ ] in terms of (ỹ ( ˆx holds almost everywhere in Ω. Proof. The continuity of ỹ ( ˆx in ˆΩ follows from a result of Vodopyanov and Goldstein which holds under our assumptions on the trace (y of ỹ on ˆΩ together with the fact that by Theorem 5.3, ỹ is in W,2 ( ˆΩ; R 2 and satisfies det ỹ > a.e. in ˆΩ. (See [2] or Theorem 5 of [8]. 24

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