Oseen-Frank-type theories of ordered media as the Γ -limit of a non-local mean-field free energy
aa r X i v : . [ m a t h . A P ] M a r Oseen-Frank-type theories of ordered media as the Γ-limit of anon-local mean-field free energy
Jamie M. Taylor ∗ Abstract
In this work we recover the Oseen-Frank theory of nematic liquid crystals as aΓ-limit of a particular mean-field free energy as the sample size becomes infinitelylarge. The Frank constants are not necessarily all equal. Our analysis takes place ina broader framework however, also providing results for more general systems such asbiaxial or polar molecules. We start from a mesoscopic model describing a competitionbetween entropy and a non-local pairwise interactions between molecules. We assumethe interaction potential is separable so that the energy can be reduced to a modelinvolving a macroscopic order parameter. We assume only integrability and decayproperties of the macroscopic interaction, but no regularity assumptions are required.In particular, singular interactions are permitted. The analysis becomes significantlysimpler on a translationally invariant domain, so we first consider periodic domains withincreasing domain of periodicity. Then we tackle the problem on a Lipschitz domain withnon-local boundary conditions by considering an asymptotically equivalent problem onperiodic domains. We conclude by applying the results to a case which reduces to theOseen-Frank model of elasticity, and give expressions for the Frank constants in termsof integrals of the interaction kernel.
Keywords:
Liquid crystals, Mean-field free energy, Oseen-Frank, Gamma convergence
In the mean-field free energy we consider particles inhabiting some configurational space,which for the sake of this work will be taken as a location x ∈ Ω ⊂ R and some orientation-like component m ∈ N , where ( N , µ ) is a finite measure space. For the motivating caseof liquid crystals, particles will be rigid objects with some symmetry, so N will be takentypically as SO(3) for general molecules or S for molecules with axial symmetry. In thecase of a system at constant concentration, we can describe the system by probabilitydistributions on the state space, that is some f : Ω → P ( N ). In this case f describes the ∗ Address for correspondence: Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road,Oxford, OX2 6GG. Email: [email protected] x . We then consider minimisers of the mean-field freeenergy functional [13] Z Ω ×N f ( x, m ) ln f ( x, m ) d ( x, m ) − Z Ω ×N Z Ω ×N K ( x − y, m, m ′ ) f ( x, m ) f ( y, m ′ ) d ( x, m ) d ( y, m ′ ) . (1)The left-hand term represents the entropic part and favours a disordered system. Theright-hand term represents pairwise molecular interactions, with K ( z, m, m ′ ) correspondingto the interaction energy of two molecules oriented as m, m ′ ∈ N respectively with centresof mass spanned by z ∈ R . The pairwise interactions will typically favour an ordered state,and it is the competition between these two terms leads to phase transitions mediated bytemperature and concentration, although for the sake of this work these parameters havebeen absorbed into the interaction K .The Oseen-Frank like elastic theories we will be considering are models to describesimilar systems at a much larger, macroscopic scale [16]. They admit various interpretations,and the following discussion is relevant for the case considered in this work. We define amacroscopic order parameter b as an average of some microscopic quantity. To be morespecific, we assume a finite dimensional vector space V , and a : N → V , and define for all x in our domain b ( x ) = Z N f ( x, m ) a ( m ) dµ ( m ) . (2)Typically b will live in some open subset of V . We assume that the energy of the system atthe macroscopic level can be described as Z Ω W ( ∇ b ( x )) + ψ ( b ( x )) dx, (3)where W is the elastic energy and ψ is the bulk energy. By considering scaling argumentsit can be argued that for a sufficiently large sample, the bulk energy overwhelms the elasticenergy so we consider the constrained case of b in the minimising manifold, M = arg min( ψ ).Thus rather than considering b in V (or some open subset of it), we consider only therestricted class b : Ω → M and minimise a purely elastic energy Z Ω W ( ∇ b ( x )) dx. (4)These are the limiting models that we will be obtaining in this work, although they will beobtained from the non-local free energy (1) rather than the local form (3). The local andnon-local energies (1,3) are not unrelated themselves however, and Taylor approximationarguments can provide a heuristic justification of the local models from the mean-fieldenergy [10]. In the case of uniaxial nematic liquid crystals, M ∼ R P , or in simpler cases M is taken as S . In the case of biaxial nematics, M is taken as SO(3).The exact form of W will vary depending on the system being considered, but theLandau expansion argument can provide a phenomenological basis. The undistorted state2 ∇ b = 0) is typically the ground state and W is assumed to be sufficiently regular, so atleast for configurations near the ground state we can replace W with its second order Taylorapproximation, which is constrained to respect the symmetry of the system. We will showin this work that certain quadratic energies for b : Ω → M can be justified from the mean-field free energy on large domains via Γ-convergence, under assumptions on the interactionkernel K . Explicitly, we consider a rescaled version of the energy in (1) on a domain ǫ Ωand take the asymptotics as ǫ →
0, obtaining the elastic energy for b ∈ W , (Ω , M ) as14 Z Ω L ∇ b ( x ) · ∇ b ( x ) dx (5)for an appropriate tensor L . The analysis will be presented in a very general framework,but we will conclude the paper by considering as a concrete case a set of interaction kernelsthat give rise to the Oseen-Frank elastic energy for uniaxial nematic liquid crystals withtwo distinct elastic constants. For analogous results on the convergence of solutions tominimisers of the local functional (3) in the case of nematic liquid crystals the reader isdirected to [14].In the case where N = S and a ( p ) = p ⊗ p − I , an asymptotic description of minimisersof (1) is given by Liu and Wang [11], providing a first step towards a rigorous link betweenthe mean-field and Oseen-Frank models. Within their work they consider a restricted classof interaction potentials, and show that critical points of (1) converge to weakly harmonicmaps in an appropriate sense. Weakly harmonic maps arise as solutions to the Euler-Lagrange equation for the minimisation of Z Ω |∇ n ( x ) | dx (6)over n ∈ W , (Ω , S ) subject to a Dirichlet boundary condition. Furthermore, they showthat minimisers of (1) converge to minimising harmonic maps, that is minimisers of (6)over its admissible class.Our work extends their ideas into several directions. Firstly, our convergence resultis stated as a Γ-limit of the energy, providing information about the behaviour of thesystem out of equilibrium (see [4] for a thorough reference on techniques of Γ-convergence).Secondly, we permit much more varied configuration spaces M than just S , and moregeneral kernels K . This allows a description of more general molecules such as those withbiaxial or polar character, and furthermore reclaims in a particular case the Oseen-Frankenergy with multiple elastic constants, so that we obtain as our limiting energy14 Z Ω K (div n ) + K | n × curl n | + K ( n · curl n ) dx (7)with K = K = K for n ∈ W , (Ω , S ) with Dirichlet boundary conditions. We notethat the harmonic map problem corresponds to K = K = K , the so-called one-constantapproximation . Finally, within our work we are able to drop almost any regularity assump-tion, with the translational part of a separable interaction potential only required to bemeasurable, non-negative definite and satisfy a certain polynomial decay.3n the mean-field free energy we a priori only have that b ∈ L ∞ (Ω , Q ). This lack ofregularity means that we cannot impose Dirichlet boundary conditions in the sense of tracesas if b were a Sobolev function. To this end, we consider a “thick” boundary conditionspecified on a set of positive volume, so that there is a subdomain Ω ǫ ⊂ Ω, so that b isconstrained to a fixed value on Ω \ Ω ǫ . Under the assumptions we consider (see Section 3.1.1)this gives, for each ǫ >
0, a neighbourhood of ∂ Ω in which b has constrained values. In thelimit as ǫ →
0, we will see this reduce to a standard Dirichlet condition.In our analysis we also consider the effect of electrostatic interactions. This has a moredetailed discussion in Section 3.1.2. In the absence of external charges and at equilibrium,this adds a term to the energy given by − Z Ω A ( b ( x )) ∇ φ ( x ) · ∇ φ ( x ) dx, (8)where A : Q → R × represents the anisotropy of the media, and the electrostatic φ isrequired to satisfy ∇ · ( A ( b ) ∇ φ ) = 0 and a Dirichlet boundary condition. As φ is uniquelydefined by b , the boundary condition and the differential equation, we can view this term asa function of b alone, which is in fact continuous so does not pose difficulties in our analysis. In Section 1.2 we introduce some preliminary definitions and assumptions for the model. Inparticular, by assuming a separability of the interaction kernel and considering a restrictedclass of probability distributions, we start from the mean-field free energy (1) and obtaina simpler free energy in the order parameter b with equivalent minimisers. The orderparameter b is constrained to live in a bounded convex set Q , relating to its definition asan average. The energy for b ∈ L ∞ (Ω , Q ) is given as Z Ω ψ s ( b ( x )) dx − Z Ω Z Ω K ( x − y ) b ( x ) · b ( y ) dx dy. (9)Here ψ s is a macroscopic analogue of entropy, as defined in [18].Next in Section 2 we consider an energy which formally corresponds to the case whereΩ = R , and the system is πǫ -periodic in the three coordinate directions. We denote the3-torus as T = [0 , π ] with opposite faces identified. Since the integrals are unbounded,we consider the energy per periodic cell ǫ T , Z ǫ T ψ s ( b ( x )) dx − Z ǫ T Z R K ( x − y ) b ( x ) · b ( y ) dy dx. (10)While this is a physically unrealistic situation, the mathematical analysis admits a simplertreatment than the case of bounded domains, and will be essential for considering generalbounded domains. Under a rescaling, the energy for ǫ > F ǫ ( b ) = 1 ǫ Z T ψ ( b ( x )) dx + 12 ǫ Z T Z R K (cid:18) x − yǫ (cid:19) · ( b ( x ) − b ( y )) ⊗ dy dx. (11)4ere ψ : Q → R is an appropriate bulk potential. We use the notation here and throughoutthe paper for a symmetric operator A and vector v that A · ( v ) ⊗ = Av · v , denoting theinner product of tensors between A and v ⊗ v = v ⊗ . Up to additive and multiplicativeconstant, the expressions (10) and (11) are equal, however the form in (11) provides a moreconvenient form for our analysis. Loosely speaking it writes the energy as a bulk termweighted by ǫ , and a finite difference quotient that will, in the limit, become a gradient.In Section 2.1 we provide some of the necessary estimates and compactness results forour analysis. We do not need to impose any kind of regularity property on K for ouranalysis, and our assumptions permit singularities, provided they are integrable. The keypoint is to find appropriate smooth, periodic functions ϕ ǫ so that ϕ ǫ → δ in D ′ ( T ), so thatwe can bound Z Ω |∇ ( ϕ ǫ ∗ b )( x ) | dx ≤ C ǫ Z T Z R K (cid:18) x − yǫ (cid:19) · ( b ( x ) − b ( y )) ⊗ dy dx (12)as seen in Proposition 2.3. From this we obtain weak- W , compactness of ϕ ǫ ∗ b ǫ if F ǫ ( b ǫ )is bounded. We then show that ϕ ǫ ∗ b ǫ − b ǫ → L if F ǫ ( b ǫ ) is bounded. Furthermore theweighting on the bulk term implies that if the energy is bounded, then b ǫ → ψ − (0) = M ,which gives us our compactness theorem for the energy: Theorem 1.1 (Compactness) . Let b ǫ ∈ L ∞ ( T , Q ) be such that F ǫ ( b ǫ ) is uniformlybounded. Then there exists some b ∈ W , ( T , M ) and a subsequence ǫ j so that b ǫ j L → b .Once we have our compactness result, the Γ-convergence result is more straightfor-ward. We can re-write the bilinear term in the energy as a difference quotient, so that if D − ǫz b = ǫ | z | ( b ( x ) − b ( x − ǫz )),1 ǫ Z T Z T K (cid:18) x − yǫ (cid:19) · ( b ( x ) − b ( y )) ⊗ dy dx = Z T Z R | z | K ( z ) · ( D − ǫz b ( x )) ⊗ dz dx. (13)Using the well behaved properties of difference quotients on W , ( T , R k ), we show that if b ∈ W , ( T , V ), then the limit of this as ǫ → Z T Z R K ( z ) · ( z · ∇ b ( x )) ⊗ dz dx = Z T L ∇ b ( x ) · ∇ b ( x ) dx (14)for an appropriate tensor L (Proposition 2.9). This fact allows us to use constant recoverysequences to show the limsup inequality for our Γ-limit.For the liminf inequality, the key point is to observe that for g ( z ) = λ min ( K ( z )), thatour method of convergence and energy bounds imply that the map( z, x )
7→ | z | g ( z ) D − ǫ j z b ǫ j ( x ) (15)5onverges weakly in L ( R × T ) to ( z, x )
7→ − g ( z ) z ·∇ b ( x ) as ǫ j →
0. From this point, theliminf inequality becomes a straightforward application of standard lower semicontinuitytheorems and is given in Proposition 2.11. Combining these results gives us our nexttheorem:
Theorem 1.2 (Γ-convergence with periodic domains) . We have that F ǫ Γ → F , where F ( b ) = 14 Z T L ∇ b ( x ) · ∇ b ( x ) dx (16)if b ∈ W , ( T , M ), and + ∞ otherwise. The method of convergence is L -strong conver-gence.In Section 3 we now turn to the case where Ω is a bounded domain in R . As men-tioned previously we take a “thick” boundary condition so that for ǫ > b is only freeto vary on a subdomain Ω ǫ ⊂ Ω, with Ω ǫ sufficiently separated from ∂ Ω. We also requirethat Ω ǫ → Ω in a precise way. The difficulty that would arise on a bounded domain isthat convolutions are generally poorly behaved at the boundary of a domain, even if con-volving against a smooth function. To this end, we extend any admissible b ∈ L ∞ (Ω , Q )to some b ∈ W , ([0 , π ] , R k ), abusing notation to identify b with its extension. Since W , ([0 , π ] , R k ) functions can be identified with functions in W , ( T , R k ), we are nowable to embed our problem into the periodic case, subject to a constraint corresponding tothe boundary condition. The bulk of Section 3.2 is dedicated to showing that, by identifyingeach admissible b with its extension in W , ( T , R k ),1 ǫ Z Ω ψ s ( b ( x )) dx − ǫ Z Ω Z Ω K (cid:18) x − yǫ (cid:19) b ( x ) · b ( y )= 1 ǫ Z Ω ǫ ψ ( b ( x )) − c dx + Z T Z T K ǫ ( x − y )( b ( x ) − b ( y )) · ( b ( x ) − b ( y )) dx dy + R ǫ ( b ) + m ǫ (17)for some constants m ǫ and a function R ǫ ( b ) which tends to zero uniformly in b (Theo-rem 3.9). This means we can instead consider an asymptotically equivalent energy in asimpler, periodic domain. We also include electrostatic interactions which provide a con-tinuous, although non-local, term depending on b . In this case, we have the energy to beminimised G ǫ in b and the electrostatic potential φ , uniquely determined by b and Maxwell’sequations as G ǫ ( b, φ ) = 1 ǫ Z Ω ǫ ψ ( b ( x )) dx + 12 ǫ Z T Z T K ǫ ( x − y ) · ( b ( x ) − b ( y )) ⊗ dx dy − Z Ω A ( b ( x )) ∇ φ ( x ) · ∇ φ ( x ) dx, (18)where we have the constraint on φ that ∇ · ( A ( b ) ∇ φ ) = 0 , (19)6lus Dirichlet boundary conditions. We can write this as G ǫ ( b, φ ) = 1 ǫ Z Ω ǫ ψ ( b ( x )) dx + 12 ǫ Z T Z T K ǫ ( x − y ) · ( b ( x ) − b ( y )) ⊗ dx dy + E ( b ) , (20)where E ( b ) is continuous with respect to L convergence.We then apply our estimates and lower semicontinuity results from the periodic case toobtain the Γ-convergence result for bounded domains, Theorem 1.3.
The functionals G ǫ Γ → G , where G ( b ) = Z T L ∇ b ( x ) · ∇ b ( x ) dx + 12 Z Ω A ( b ) − D ( x ) · D ( x ) dx + E ( b ) (21)if b ∈ W , ( T , R k ) with b ( x ) ∈ M for almost every x ∈ Ω and b = b on T \ Ω, andis + ∞ otherwise. The mode of convergence is L strong convergence. Furthermore let b ǫ L → b , and denote the solutions of the maximisation problem defining E ( b ǫ ) as Φ ǫ . Thendiv( A ( b ǫ )Φ ǫ ) = 0 and Φ ǫ | ∂ Ω = φ . Then Φ ǫ W , → Φ, where div ( A ( b ) ∇ Φ) = 0 and Φ | ∂ Ω = φ .We conclude the paper by considering the case where N = S , V = Sym (3) is thespace of traceless symmetric matrices, and a ( p ) = p ⊗ p − I . In this case the orderparameter is often denoted Q and referred to as the Q-tensor. By enforcing symmetryconstraints on the kernel K , we see a very small class of bilinear forms are permissible,and obtain in these cases the classical Frank elastic constants. In this case we have that M = (cid:8) s ∗ (cid:0) n ⊗ n − I (cid:1) : n ∈ S (cid:9) , and in the case when Q ∈ W , (Ω , M ) can be writtenas Q ( x ) = s ∗ (cid:0) n ( x ) ⊗ n ( x ) − I (cid:1) for n ∈ W , (Ω , S ), the elastic component of the energyreduces to 14 Z Ω K ( ∇ · n ) + K | n · ∇ × n | + K ( n · ∇ × n ) , (22)with K = K = K . That is to say that the so-called one-constant approximation K = K = K does not hold. The exact relations between K i and the kernel K are given inProposition 4.4, with some numerically found values for a case inspired by the Londondispersion forces relation given in Remark 4.5. Assumption 1.4 (Separability of the interaction kernel) . We assume there exists a finitedimensional real Hilbert space V , functions a ∈ L ∞ ( N , V ) and K : R → B ( V, V ) so thatthe interaction kernel K can be separated as K ( x − y, m, m ′ ) = K ( x − y ) a ( m ) · a ( m ′ ) . (23)Without loss of generality we will often assume that V = R k for some k .7his is a simplifying assumption and will allow us to reduce our problem to that of afinite dimensional order parameter. Throughout the work we will need bounds on termsinvolving K , so we define g ( z ) = λ min ( K ( z )) . (24) Assumption 1.5.
We assume the following technical assumptions on a, K, g .1. g is non-negative everywhere and bounded away from zero on some open set.2. g ∈ L ( R ) and has finite second moment, so R R g ( z ) | z | dz < + ∞ .3. There exists a constant M > λ max ( K ) ≤ M g .4. K is a measurable, even function, so K ( z ) = K ( − z ) for all z ∈ R .5. a satisfies the property that for all ( c, ξ ) ∈ R × V \ { (0 , } , µ ( { m ∈ N : c + ξ · a ( m ) =0 } ) = 0.Assumptions 1,2 and 3 will be required for the various coercivity and compactnessestimates on the energy. In particular, 3 means that K can be estimated above and belowby the same scalar function.Assumption 4 with the estimates gives that K ∈ L ( R , B ( V, V )), and the symmetryassumption is required for several proofs. In the case of uniaxial nematics with head-to-tailsymmetry, the evenness of K corresponds to mirror symmetry of interactionsFinally 5, known as the pseudo-Haar condition, allows a more elegant analysis of therelationship between the microscopic and macroscopic problem. It is essentially a strongform of linear independence. In the common cases where N is a connected analytic manifoldand each component of a , denoted a i , is an analytic function, it is equivalent to the set offunctions { , a , ..., a k } being linearly independent [18, Proposition 3]. We then define thefollowing objects Definition 1.6.
Let
Q ⊂ V be the set of admissible moments defined by Q = (cid:26)Z N a ( m ) f ( m ) dm : f ∈ P ( N ) (cid:27) . (25)For b ∈ Q , let U ( b ) = (cid:8) f ∈ P ( N ) : R N a ( m ) f ( m ) dm = b (cid:9) . Then we define the singularpotential as ψ s ( b ) = min f ∈ U ( b ) Z N f ( m ) ln f ( m ) dm. (26)We recall from [18] the following results on Q , ψ s , which strongly require the pseudo-Haar condition. Proposition 1.7. Q is an open, bounded, non-empty convex set.2. ψ s : Q → R is strictly convex, C ∞ , and satisfies lim b → ∂ Q ψ s ( b ) = + ∞ .8. The minimisation problem defining ψ s admits a unique solution denoted ρ b ∈ P ( N ),given by ρ b ( m ) = 1 Z exp(Λ b · a ( m )) (27)for some C ∞ bijection Λ b : Q → V .4. For every Lipschitz function F : Q → R , the function b ψ s ( b ) + F ( b ) admits aminimum.In light of the blow up of ψ s at the boundary of Q , with abuse of notation we identifyit with its extension to Q given by ψ s ( b ) = + ∞ for b ∈ ∂ Q .Formally, given f ∈ L (Ω × N , [0 , + ∞ ]), satisfying R N f ( x, m ) dm = 1 and R N f ( x, m ) dm = b ( x ) for x ∈ Ω, we have that Z Ω ×N f ( x, m ) ln f ( x, m ) d ( x, m ) − Z Ω ×N Z Ω ×N K ( x − y ) a ( m ) · a ( m ′ ) f ( x, m ) f ( y, m ′ ) d ( x, m ) d ( y, m ′ )= Z Ω ×N f ( x, m ) ln f ( x, m ) d ( x, m ) − Z Ω Z Ω K ( x − y ) b ( x ) · b ( y ) dx dy ≥ Z Ω ψ s ( b ( x )) dx − Z Ω Z Ω K ( x − y ) b ( x ) · b ( y ) dx dy, (28)with equality if and only if f ( x, m ) = ρ b ( x ) ( m ) for almost every x, m . This allows theproblem to be reduced entirely to one phrased in L ∞ (Ω , Q ). The exact argument requirestechnical care, since for an L (Ω × N ) function it is not immediately clear how to interpret R N f ( x, m ) a ( m ) dm or R N f ( x, m ) dm . In [11] this was rigorously interpreted via duality inthe case when N = S and a ( p ) = p ⊗ p − I , so that R Ω ×N (cid:0) f ( x, m ) a ( m ) − b ( x ) (cid:1) ϕ ( x ) dx = 0for all ϕ ∈ D (Ω). Their argument extends in a straightforward manner to the case we areconsidering, so the details will be omitted here and the energy Z Ω ψ s ( b ( x )) dx − Z Ω Z Ω K ( x − y ) b ( x ) · b ( y ) dx dy (29)will be taken as a black box. In fact the precise form of ψ s becomes irrelevant to much ofthe analysis in this work, with only a lower bound and lower semicontinuity on Q beingimportant.We use T to denote the 3-torus, often identified with [0 , π ] . We note however that W , ([0 , π ] , R ) = W , ( T , R ), although L p ([0 , π ] , R ) = L p ( T , R ). Again with abuse ofnotation, we identify functions in L ( R ) which are 2 π -periodic in the coordinate directionswith L ( T ). For h ∈ L ( T ), the L norm is only defined as the integral over a singledomain of periodicity, i.e. || h || L ( T ) = || h || = Z T | h ( x ) | dx. (30)We now define notation for simplifying integrals.9 efinition 1.8. Let h ∈ L ( R ). For ǫ >
0, define h ǫ ( z ) = 1 ǫ X k ∈ Z h (cid:18) z + 2 πkǫ (cid:19) . (31)The following results are readily verified, with only sketch proofs provided. Proposition 1.9.
Let h ∈ L ( R ). Then1. If h ≥ || h ǫ || L ( T ) = || h || L ( R ) .2. Even if h is negative somewhere, h ∈ L ( T ).3. If u ∈ L ∞ ( T ), then 1 ǫ Z R h (cid:16) xǫ (cid:17) u ( x ) dx = Z T h ǫ ( x ) u ( x ) dx. (32) Proof.
To prove result 1, first we must verify h ǫ is 2 π -periodic in the coordinate directions,but this follows simply by translating k in the definition of h ǫ . To show h ǫ is integrable if h ≥
0, we see that Z T | h ǫ ( x ) | dx = 1 ǫ Z T X k ∈ Z h (cid:18) x + 2 πkǫ (cid:19) dx = 1 ǫ Z S k ∈ Z (2 πk + T ) h (cid:16) xǫ (cid:17) dx = 1 ǫ Z R h (cid:16) xǫ (cid:17) dx = Z R h ( x ) dx = || h || . (33)To see result 2, we note that | h ǫ | < | h | ǫ , and the result follows by result 1. Result 3 followsby the same argument described in lines 2-4 of (33) applied to R T h ǫ ( x ) u ( x ) dx .We also recall the definition of Γ-convergence [4]. Definition 1.10.
Let X be a metric space and let F ǫ , F : X → [ −∞ , ∞ ]. We say that F ǫ Γ → F , with respect to the topology on X , if the following hold.1. (liminf inequality) For all x ǫ → x , lim inf ǫ → F ǫ ( x ǫ ) ≥ F ( x ).2. (limsup inequality) For every x ∈ X , there exists a sequence x ǫ → x so thatlim ǫ → F ǫ ( x ǫ ) = F ( x ). 10 Periodic domains
First, rather than considering domains ǫ Ω, we take Ω = R and assume that b is πǫ -periodicin the three coordinate directions. This is problematic in that the integrals will be infiniteunless all energy vanishes, so instead we consider what is formally the energy per periodiccell. That is, Z ǫ [0 , π ] ψ s (˜ b (˜ x )) d ˜ x − Z ǫ [0 , π ] Z R K ( x − y ) ˜ b (˜ x ) · ˜ b (˜ y ) d ˜ y d ˜ x. (34)While such an arrangement may seem physically unrealistic, it will provide the mathematicalframework to understand the more physically reasonable case to be addressed in Section 3.We first perform a change of variables, x = ǫ ˜ x ∈ [0 , π ] , y = ǫ ˜ y ∈ [0 , π ] , b ( x ) = ˜ b ( ǫx ) so b ∈ L ∞ ( T , Q ). Then the energy becomes1 ǫ Z T ψ s ( b ( x )) dx − ǫ Z T Z R K (cid:18) x − yǫ (cid:19) b ( x ) · b ( y ) dx dy. (35)As in Definition 1.8 denoting K ǫ ( z ) = ǫ P k ∈ Z K (cid:0) z +2 πkǫ (cid:1) , we re-write the integral as1 ǫ Z T ψ s ( b ( x )) dx − ǫ Z T Z T K ǫ ( x − y ) b ( x ) · b ( y ) dy dx. (36)Now we note that for vectors u, v and a symmetric linear operator A , 2 Au · v = Au · u + Av · v − A ( u − v ) · ( u − v ). Then we re-write the energy as1 ǫ Z T ψ s ( b ( x )) dx − ǫ Z T Z R K ǫ ( x − y ) b ( x ) · b ( x ) + K ǫ ( x − y ) b ( y ) · b ( y ) dx dy + 14 ǫ Z T Z T K ǫ ( x − y ) · ( b ( x ) − b ( y )) ⊗ dx dy = 1 ǫ Z T ψ s ( b ( x )) dx − ǫ Z T (cid:18)Z T K ǫ ( x − y ) dy (cid:19) b ( x ) · b ( x ) dx + 14 ǫ Z T Z T K ǫ ( x − y ) ( b ( x ) − b ( y )) ⊗ dx dy. (37)Now we note that R T K ǫ ( x − y ) dy = R R K ( z ) dz = K , a tensor independent of x . Thus ifwe define c = min b ∈Q ψ ( b ) − K b · b , and ψ ( b ) = ψ s ( b ) − Kb · b − c , then the energy, rescaledby an additive constant and dividing through by ǫ , is F ǫ ( b ) = 1 ǫ Z T ψ ( b ( x )) dx + 14 ǫ Z T Z T K ǫ ( x − y ) ( b ( x ) − b ( y )) ⊗ dx dy. (38)Furthermore min b ∈Q ψ ( b ) = 0. It is this functional F ǫ as written that we turn our attention to.11 .1 Estimates and compactness We first provide estimates that will give us our required compactness results. The generalidea is to show that if b ∈ L ∞ ( T , Q ), then for an appropriate periodic mollifier ϕ ǫ , the W , norm of ϕ ǫ ∗ b can be estimated by an appropriate constant times F ǫ ( b ). Furthermore,we will be able to estimate || ϕ ǫ ∗ b − b || = O ( ǫ F ǫ ( b )). This will give that if F ǫ ( b ǫ )is bounded, then ϕ ǫ ∗ b ǫ is admits a W , converging subsequence, and the sequence isasymptotically equivalent in L to b ǫ , providing our compactness theorem. First we provideseveral preliminary estimates. Lemma 2.1.
Let h ∈ L ( T ) be even and non-negative, and b ∈ L ∞ ( T , R k ). Then Z T Z T ( h ∗ h )( x − y ) | b ( x ) − b ( y ) | dx dy ≤ || h || Z T Z T h ( x − y ) | b ( x ) − b ( y ) | dx dy. (39) Proof.
First, we note that given any vector space, the triangle inequality gives us | u − v | ≤| u − w | + | w − v | . Squaring both sides gives | u − v | ≤ | u − w | + | w − v | + 2 | u − w || v − w | ,and applying Young’s inequality we have | u − v | ≤ | u − w | + 2 | w − v | . Thus Z T Z T ( h ∗ h )( x − y ) | b ( x ) − b ( y ) | dx dy = Z T Z T Z T h ( x − y − z ) h ( z ) | b ( x ) − b ( y ) | dz dx dy ≤ Z T Z T Z T h ( x − y − z ) h ( z ) (cid:0) | b ( x ) − b ( x − z ) | + | b ( x − z ) − b ( y ) | (cid:1) dz dx dy =2 || h || Z T Z T h ( z ) | b ( x ) − b ( x − z ) | dz dx + 2 Z T Z T Z T h ( ξ − y ) h ( z ) | b ( ξ ) − b ( y ) | dz dξ dy =2 || h || Z T Z T h ( x − z ) | b ( x ) − b ( x − z ) | dz dx + 2 || h || Z T Z T h ( ξ − y ) | b ( ξ ) − b ( y ) | dy dξ =4 || h || Z T Z T h ( x − y ) | b ( x ) − b ( y ) | dx dy. (40)Note a change of variables, replacing x by taking ξ = x − z was used.In order provide estimates of ϕ ǫ ∗ b , the mollifier need be somehow comparable to theconvolution kernel K . The exact relation, and existence of such a mollifier is given in thenext result. Lemma 2.2.
Let g ∈ L ( T ) be even, non-negative and non-zero. Then there exists someeven periodic mollifier ϕ ǫ ∈ C ∞ ( T ) and c > ϕ ǫ ≤ c g ǫ and |∇ ϕ ǫ | ≤ cǫ g ǫ .12 roof. Let ˜ ϕ ∈ C ∞ ( R ) be non-negative and satisfy ˜ ϕ ≤ g and |∇ ˜ ϕ | ≤ g . This can bedone since we assume that there exists an open set on which g is strictly bounded awayfrom zero. Define ϕ ( x ) = ˜ ϕ ( x )+ ˜ ϕ ( − x )2 || ˜ ϕ || . Then a straightforward algebraic exercise gives that ϕ ǫ = P k ∈ Z ǫ ϕ (cid:0) x +2 πkǫ (cid:1) satisfies the required properties. Proposition 2.3.
Let ϕ ǫ be as in Lemma 2.2. Then for all b ∈ L ∞ ( T , Q ), Z T |∇ ( ϕ ǫ ∗ b ) | ≤ || g || c ǫ (cid:18)Z T Z T K ǫ ( x − y ) · ( b ( x ) − b ( y )) ⊗ dy dx (cid:19) . (41) Proof.
This is predominantly an algebraic exercise. First note that since ϕ ǫ is even, itsderivative is odd. Define the inner-product convolution ˆ ∗ for u, v : T → R n , u ˆ ∗ v : T → R by u ˆ ∗ v ( x ) = Z T u ( x − y ) · v ( y ) dy. (42)Then we proceed as Z T |∇ ( ϕ ǫ ∗ b ǫ ) | = Z T Z T Z T (cid:18) ∇ ϕ ǫ ( x − y ) · ∇ ϕ ǫ ( x − z ) (cid:19)(cid:18) b ǫ ( y ) · b ǫ ( z ) (cid:19) dy dz dx = Z T Z T b ( y ) · b ( z ) (cid:18)Z T −∇ ϕ ǫ ( y − x ) · ∇ ϕ ǫ ( x − z ) dx (cid:19) dy dx = Z T Z T − b ( z ) · b ( y ) ( ∇ ϕ ǫ ˆ ∗∇ ϕ ǫ ) ( y − z ) dy dz = 12 Z T Z T (cid:0) | b ( z ) − b ( y ) | − | b ( x ) | − | b ( y ) | (cid:1) ( ∇ ϕ ǫ ˆ ∗∇ ϕ ǫ ) ( y − z ) ≤ Z T Z T | b ( z ) − b ( y ) | |∇ ϕ ǫ | ∗ |∇ ϕ ǫ | ( y − z ) dy dx − Z T | b ( z ) | dz Z T ( ∇ ϕ ǫ ˆ ∗∇ ϕ ǫ ) ( y ) dy ≤ c ǫ Z T Z T | b ( z ) − b ( y ) | ( g ǫ ∗ g ǫ )( y − z ) dy dz ≤ || g || c ǫ Z T Z T | b ( z ) − b ( y ) | g ǫ ( y − z ) dy dz. (43)Note that we used R T ( ∇ ϕ ǫ ˆ ∗∇ ϕ ǫ ) ( y ) dy = R T ∇ ϕ ǫ ( x ) dx · R T ∇ ϕ ǫ ( y ) dy = 0, since ϕ ǫ isperiodic. In this case, Z T |∇ ( ϕ ǫ ∗ b ǫ ) | ≤ || g || c ǫ Z T Z T | b ǫ ( z ) − b ǫ ( y ) | g ǫ ( y − z ) dy dx ≤ || g || c ǫ (cid:18)Z T Z T K ǫ ( x − y ) · ( b ǫ ( x ) − b ǫ ( y )) ⊗ dy dx (cid:19) . (44)13 orollary 2.4. Let b ǫ ∈ L ∞ ( T , R k ) for ǫ >
0. Assume that Z T Z T K ǫ ( x − y ) · ( b ǫ ( x ) − b ǫ ( y )) ⊗ dy dx (45)is uniformly bounded. Then up to a subsequence, b ǫ j → b in L with b ∈ W , ( T , R k ). Proof.
We can take a subsequence so that b ǫ j ∗ ⇀ b ∈ L ∞ ( T , R k ). Let ϕ ǫ be as in Lemma 2.2.Then by the previous result we have that ϕ ǫ j ∗ b ǫ j is bounded in W , , and thereforeadmits a weakly converging subsequence (not relabelled). Furthermore, for any h ∈ L , h ϕ ǫ j ∗ b ǫ j − b ǫ j , h i = h b ǫ j , ϕ ǫ j ∗ h − h i , which tends to zero since ϕ ǫ j ∗ h → h in L . Thereforethe weak-* limit of b ǫ j and the weak limit of ϕ ǫ j ∗ b ǫ j coincide, with the former known tobe in W , . Now since ϕ ǫ ∗ b ǫ j ⇀ b in W , , we can take a subsequence (not relabelled) sothat ϕ ǫ ∗ b ǫ j → b in L . Then finally we see that Z T | ϕ ǫ j ∗ b ǫ j ( x ) − b ǫ j ( x ) | dx = Z T (cid:12)(cid:12)(cid:12)(cid:12)Z T ϕ ǫ j ( x − y )( b ǫ j ( y ) − b ǫ j ( x )) dy (cid:12)(cid:12)(cid:12)(cid:12) dx = Z T (cid:12)(cid:12)(cid:12)(cid:12)Z T ϕ ǫ j ( x − y ) (cid:16) ϕ ǫ j ( x − y ) ( b ǫ j ( y ) − b ǫ j ( x )) (cid:17) dy (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ Z T Z T ϕ ǫ j ( x − y ) | b ǫ j ( y ) − b ǫ j ( x ) | dx dy ≤ c Z T Z T g ǫ j ( x − y ) | b ǫ j ( x ) − b ǫ j ( y ) | dx dy ≤ ǫ j c F ǫ j ( b ǫ j ) (46)which, since F ǫ j ( b ǫ j ) is uniformly bounded, must then be of order ǫ j . Therefore b ǫ j admitsthe same L limit as ϕ ǫ j ∗ b ǫ j , which is known to be b ∈ W , ( T , R k ). Proposition 2.5. If F ǫ j ( b ǫ j ) is uniformly bounded, then any cluster point b as given inCorollary 2.4 must satisfy b ( x ) ∈ M = ψ − (0) pointwise almost everywhere. Proof.
Since b ǫ j → b in L , we must have thatlim inf j →∞ Z T ψ ( b ǫ j ( x )) dx ≥ Z T ψ ( b ( x )) dx. (47)The left-hand side is of order ǫ j , so therefore we have that R T ψ ( b ( x )) dx = 0, so that ψ ( b ( x )) = 0 almost everywhere.Combining the previous results is our main compactness theorem. Theorem 2.6 (Compactness) . Let b ǫ ∈ L ∞ ( T , Q ) be such that F ǫ ( b ǫ ) is uniformlybounded. Then there exists some b ∈ W , ( T , M ) and a subsequence ǫ j so that b ǫ j L → b .14 .2 The Γ -limit Now that we have our compactness result, we turn to evaluating the Γ-limit itself. The keyidea is to rewrite the bilinear form as a bilinear form involving finite difference quotients,which will in the limit become gradients. To this end, we must first recall some propertiesof translation and finite difference operators on function spaces.
Definition 2.7.
Let z ∈ R , and b ∈ L ∞ ( T , Q ). Then define the translation and finitedifference operators, T z : L ∞ ( T , Q ) → L ∞ ( T , Q ) and D z : L ∞ ( T , Q ) → L ∞ ( T , R k )respectively by T z b ( x ) = b ( x + z ) D z b ( x ) = 1 | z | ( b ( x + z ) − b ) = 1 | z | ( T z − I ) b ( x ) . (48)We now recall some elementary results of these operators, which in various forms canbe found in standard references (e.g. [9]), although proofs are included for completeness. Proposition 2.8.
Fix u ∈ L ( T , R ). Then1. T z u → u in D ′ ( T ).2. The map from T to L ( T , R ) given by z → T z u is continuous.3. If u ∈ W , ( T , R ), then D tz u → b z · ∇ u in D ′ ( T ) as t → + .4. If u ∈ W , ( T , R ), then || D tz u || ≤ || b z · ∇ u || and || D z u − b z · ∇ u || ≤ sup | ξ |≤| z | || ( T ξ − I ) ∇ u || , which tends to zero as | z | → Proof.
1. Let φ ∈ D ( T ). Then |h T z u, φ i − h u, φ i| = (cid:12)(cid:12)(cid:12)(cid:12)Z T u ( x + z ) φ ( x ) − u ( x ) φ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z T u ( x ) (cid:0) φ ( x − z ) − φ ( x ) (cid:1) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z T | u ( x ) | | φ ( x − z ) − φ ( x ) | dx ≤|| u || (cid:18)Z T Lip ( φ ) | z | dx (cid:19) = O ( | z | ) . (49)2. It is immediate that || T z u || = || u || by translational invariance on T . Then since T z u → u in D ′ ( T ) and || T z u || is bounded, this implies T z u L ⇀ u in L . But also,lim z → || T z u || = lim z → || u || = || u || , so that T z u L ⇀ u and || T z u || → || u || , so T z u L → u .15. Again let φ ∈ D ( T ). We first note that D ty φ ( x ) → b y ∇ · φ ( x ) uniformly in x as t → + . h D z u, φ i = 1 | z | Z T ( u ( x + z ) − u ( x )) φ ( x ) dx = 1 | z | Z T u ( x ) ( φ ( x − z ) − φ ( x )) dx → Z T u ( x ) (cid:0) − b z · ∇ φ ( x ) (cid:1) dx = Z T b z · ∇ u ( x ) φ ( x ) dx. (50)4. To show that || D z u || ≤ || b z · ∇ u || , we see it suffices to only consider when b ∈ C ( T )and then the result follows by density. In this case we see that | D z u ( x ) | = 1 | z | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z | z | ∇ u ( x + t b z ) · b z dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | z | Z | z | dt ! Z | z | |∇ u ( x + t b z ) · b z | dt ! ⇒ Z T | D z u ( x ) | dx ≤ | z | Z T Z | z | |∇ u ( x + t b z ) · b z | dt dx = ||∇ u ( x ) · b z || . (51)To show the inequality, we again consider only u ∈ C ( T ) and the result follows bydensity. Z T | D z u ( x ) − b z · ∇ u ( x ) | = Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | z | Z | z | b z · ∇ u ( x + t b z ) − b z · ∇ u ( x ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ≤ | z | Z T Z | z | |∇ u ( x + t b z ) − ∇ u ( x ) | dt dx = Z | z | || ( T t b z − I ) ∇ u || ≤ sup | ξ |≤| z | || ( T ξ − I ) ∇ u || . (52)That this tends to zero as | z | → ξ → T ξ ∇ u is continuouson a compact set and T = I . Proposition 2.9.
Let b ǫ ∈ W , ( T , R k ) with b j W , → b . Thenlim ǫ → Z T Z T ǫ K ǫ ( x − y ) · ( b ǫ ( x ) − b ǫ ( y )) ⊗ dy dx = Z T L ∇ b ( x ) · ∇ b ( x ) dx, (53)16here L ∇ b ( x ) · ∇ b ( x ) = X i,j =1 (cid:18)Z R K ( z ) z i z j dz (cid:19) ∂b∂x i ( x ) · ∂b∂x j ( x ) . (54) Proof.
We fully write out L , and recall the definition of K ǫ as an integral over R , and thisgives (cid:12)(cid:12)(cid:12)(cid:12)Z T Z T ǫ K ǫ ( x − y ) · ( b ǫ ( x ) − b ǫ ( y )) ⊗ dy − L ∇ b ( x ) · ∇ b ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z R ǫ K (cid:18) x − yǫ (cid:19) · ( b ǫ ( x ) − b ǫ ( y )) ⊗ dy − Z R K ( z )( z · ∇ b ( x )) · ( z · ∇ b ( x )) dz dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z T Z R ǫ K ( z ) · ( b ǫ ( x ) − b ǫ ( x − ǫz )) ⊗ − K ( z )( z · ∇ b ( x )) ⊗ dz dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z T Z R | K ( z ) || z | (cid:12)(cid:12)(cid:12) ( D − ǫz b ǫ ( x )) ⊗ − ( b z · ∇ b ( x )) ⊗ (cid:12)(cid:12)(cid:12) dz dx = Z R | K ( z ) || z | Z T (cid:12)(cid:12)(cid:12) ( D − ǫz b ǫ ( x )) ⊗ − ( b z · ∇ b ( x )) ⊗ (cid:12)(cid:12)(cid:12) dx dz ≤ Z R | K ( z ) || z | (cid:18)Z T | D − ǫz b ǫ ( x ) − b z · ∇ b ( x ) | dx (cid:19) (cid:18)Z T | D − ǫz b ǫ ( x ) + b z · ∇ b ( x ) | dx (cid:19) dz = Z R | K ( z ) || z | || D − ǫz b ǫ − b z · ∇ b || || D − ǫz b ǫ + b z · ∇ b || dz ≤ Z R | K ( z ) || z | (cid:18) || D − ǫz b ǫ || + ||∇ b || (cid:19)(cid:18) || D − ǫz ( b ǫ − b ) || + || D − ǫz b − b z · ∇ b || (cid:19) dz ≤ ( ||∇ b ǫ || + ||∇ b || ) Z R | K ( z ) || z | ||∇ b ǫ − ∇ b || + sup t ≤ ǫ | z | || ( T − t b z − I ) ∇ b || ! dz. (55)Now we note that since ∇ b ǫ L → ∇ b and || ( T − t b z − I ) ∇ b || is bounded uniformly in t, b z , theintegrand is bounded by a constant times | K ( z ) || z | which is integrable, so we may applydominated convergence. Then using that sup t ≤ ǫ | z | || ( T − t b z − I ) ∇ b || → ǫ → b ǫ , the result holds.In order to show the liminf inequality, a key step is that, as a function on R × T ,( z, x ) g ( z ) | z | D ǫz Q ǫ ( x ) has the limit g z · ∇ Q ( x ) in the sense of distributions. Theresult is far more trivial if we integrate this against test functions which are separable as φ ( z, x ) = φ ( x ) φ ( z ). By means of the next lemma, we will show that the span of suchtest functions are dense in D ( R × T ) with uniform convergence, allowing us to test onlyagainst such separable functions. Lemma 2.10.
Let U denote the subalgebra of D ( R × T ) given by the span of separable17unctions, so that if φ ∈ U , φ ( x, z ) = m P i =1 φ i ( x ) φ i ( z ) for some φ i ∈ D ( T ) = C ∞ ( T ) and φ i ∈ D ( R ). Then U is dense with respect to uniform convergence in D ( R × T ). Proof.
The result is essentially Stone-Weierstrass. Let φ ∈ D ( R × T ). Without loss ofgenerality, assume supp( φ ) ⊂ B × T . Let U ′ denote the algebra given by the span ofall smooth functions on B × T which are separable, but not necessarily with compactsupport. We note that this set separates points. This can be seen by taking classical bumpfunctions with φ ( x ) = 1 if x = x and 0 ≤ φ ( x ) < φ ( z ) = 1if z = z and φ ( z ) ∈ [0 ,
1) otherwise, then their product separates ( x , z ) and every otherpoint. Furthermore this is clearly an algebra, and contains a non-zero constant function.Therefore it is dense in C ( B × T ) by Stone-Weierstrass. Now let φ ∈ C ∞ ( R ) be a cutofffunction, so that φ ( z ) = 1 if z ∈ B , φ ( z ) ∈ (0 ,
1) for 1 < | z | < φ ( z ) if | z | > φ j ∈ U ′ and φ j → φ in L ∞ , then ˜ φ j ( x, z ) = φ ( x ) φ j ( x, z ) must also satisfy ˜ φ j → φ in L ∞ , but also ˜ φ j ∈ U . Proposition 2.11. If L ∞ ( T , R k ) ∋ b ǫ j L → b ∈ W , ( R , R k ), thenlim inf j →∞ Z T Z T ǫ K ǫ ( x − y ) · ( b ǫ j ( x ) − b ǫ j ( y )) ⊗ dy dx ≥ Z T L ∇ b ( x ) · ∇ b ( x ) dx. (56) Proof.
Assume the liminf is finite, else there is nothing to prove, and take a subsequenceconverging to liminf without relabelling. Since the liminf is finite, we can take a constant
C > C ≥ Z T Z T K ǫ j ( x − y ) (cid:0) b ǫ j ( x ) − b ǫ j ( y ) (cid:1) ⊗ dx dy = Z R Z T K ( z ) (cid:0) b ǫ j ( x ) − b ǫ j ( x − ǫ j z ) (cid:1) ⊗ dx dz = Z R Z T | z | K ( z ) · (cid:0) D − ǫ j z b ǫ j ( x ) (cid:1) ⊗ dx dz ≥ Z R Z T | z | g ( z ) | D − ǫ j z b ǫ j ( x ) | dx dz. (57)This implies that ( z, x )
7→ | z | g ( z ) D − ǫ j z b ǫ j ( x ) is bounded in L ( R × T ). Thus we take aweakly converging subsequence (not relabelled). We now seek to find the weak limit. Wedo this by finding the limit in the sense of distributions. In light of the previous lemma, itsuffices to test against separable functions. 18et φ ∈ D ( R × T ), with φ ( x, z ) = φ ( x ) φ ( z ). Then (cid:12)(cid:12)(cid:12)(cid:12)Z R × T | z | g ( z ) D − ǫ j z b ǫ j ( x ) φ ( x, z ) − g ( z ) z · ∇ x φ ( x, z ) b ( x ) d ( x, z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z R × T | z | g ( z ) (cid:18) b ǫ j ( x ) D ǫ j z φ ( x, z ) − b ( x ) b z · ∇ x φ ( x, z ) (cid:19) d ( x, z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R × T | z | g ( z ) (cid:18) | b ǫ j ( x ) | | φ ( z ) | | D ǫ j z φ ( x ) − b z · ∇ φ ( x ) | + |∇ x φ ( x, z ) | | b ǫ j ( x ) − b ( x ) | (cid:19) d ( x, z ) ≤ (cid:18)Z R | z | g ( z ) dz Z T | b ǫ j ( x ) | dx (cid:19) (cid:18)Z R | φ ( z ) | || D ǫ j z φ − b z · ∇ φ || dz (cid:19) + ||∇ x φ || (cid:18)Z R | z | g ( z ) dz (cid:19) || b ǫ j − b || ≤ C Z R | φ ( z ) | sup | ξ |≤ ǫ j z || ( T ξ − I ) ∇ φ || dz ! + C || b ǫ j − b || . (58)To deal with the term in the integral, we recall that sup | ξ | <ǫ j z || ( T ξ − I ) ∇ φ || is bounded andconverges to zero pointwise as ǫ j →
0, so we can apply dominated convergence to givethat the integral tends to zero in the limit. The second term tends to zero trivially, thus | z | g ( z ) D − ǫ j z b ǫ j ( x ) L ⇀ − g ( z ) b z · ∇ b ( x ). We take the convention that when K ( z ) = 0, g K ( z ) = 0 so that the proceeding equalities hold. Now we can apply standard lowersemicontinuity results for convex functionals under weak convergence (see e.g. [6]), to givethat lim inf j →∞ ǫ j Z T Z T K ǫ ( x − y ) · ( b ǫ j ( x ) − b ǫ j ( y )) ⊗ dy dx = lim inf j →∞ Z R × T | z | K ( z ) · ( D − ǫ j z b ǫ j ( x )) ⊗ d ( z, x )= lim inf j →∞ Z R × T (cid:18) g ( z ) K ( z ) (cid:19) · (cid:16) | z | g ( z ) D − ǫ j z b ǫ j ( x ) (cid:17) ⊗ d ( z, x ) ≥ Z R × T g ( z ) K ( z ) · (cid:16) | z | g ( z ) b z · ∇ b ( x ) (cid:17) ⊗ d ( z, x )= Z R × T K ( z ) · ( z · ∇ b ( x )) ⊗ d ( z, x )= Z T L ∇ b ( x ) · ∇ b ( x ) dx. (59)19 heorem 2.12 (Γ-convergence with periodic domains) . We have the Γ-convergence F ǫ Γ →F , where F ( b ) = 14 Z T L ∇ b ( x ) · ∇ b ( x ) dx (60)if b ∈ W , ( T , M ), and + ∞ otherwise. The method of convergence is L -strong conver-gence. Proof.
For the liminf inequality, since ψ is non-negative, using Proposition 2.11,lim inf j →∞ F ǫ j ( b ǫ j ) ≥ lim inf ǫ → ǫ Z T Z T K ǫ ( x − y ) · ( b ǫ j ( x ) − b ǫ j ( y )) ⊗ dx dy ≥ Z T L ∇ b ( x ) · ∇ b ( x ) dx. (61)For the limsup inequality, we are fortunate in that we can take constant recovery sequencesfrom Proposition 2.9, so that if b ∈ W , ( T , M ), then ψ ( b ) = 0 almost everywhere, and F ǫ j ( b ) = 14 ǫ Z T Z T K ǫ ( x − y ) · ( b ( x ) − b ( y )) ⊗ dx dy → Z T L ∇ b ( x ) · ∇ b ( x ) dx. (62)As per the compactness result, if b W , ( T , M ), then F ǫ j ( b ǫ j ) must blow up if b ǫ j → b . Remark 2.13.
Our systems will more often than not have to respect the principle offrame indifference. In order to interpret frame indifference we must be able to act rotations R ∈ SO(3) on the order parameter space V . We denote this group action as ( R, b ) [ R ] b .For example if V = R , then [ R ] b = Rb , if V = Sym (3) as in the Q-tensor theory, then[ R ] b = RbR T . The kernel will need to satisfy frame indifference also, so that [ R ] K ( z )[ R ] T = K ( Rz ), giving K ( Rz )[ R ] b · [ R ] b = K ( z ) b · b . Given b : Ω → Q , the act of rotating the systemby R ∈ SO(3) implies that ∇ b is replaced by [ R ] ∇ bR T . Therefore frame indifference requiresthat L (cid:0) [ R ] AR T (cid:1) · (cid:0) [ R ] AR T (cid:1) = LA · A for all A in the tangent space of M . Should we haveframe indifference of K this holds however, as seen in the next proposition. Proposition 2.14.
Assume SO(3) is defines group action on V by R [ R ] ∈ B ( V, V ),and K satisfies K ( Rz ) = [ R ] K ( z )[ R ] T . Then the quadratic form A LA · A for A in thetangent space of M is frame invariant, so that L ([ R ] AR T ) · ([ R ] AR T ) = LA · A .20 roof. L [ R ] AR T · [ R ] AR T = Z R K ( z ) (cid:0) [ R ] AR T z (cid:1) · (cid:0) [ R ] AR T z (cid:1) dz = Z R (cid:0) [ R ] T K ( z )[ R ] (cid:1) A ( R T z ) · A ( R T z ) dz = Z R K ( R T z ) A ( R T z ) · A ( R T z ) dz = Z R K ( y ) Ay · Ay dy = LA · A, (63)by performing a change of variables y = R T z . Since we consider periodic domains and limiting energies purely quadratic in the gradient,we have that global minimisers are in all cases trivial, i.e. any constant map. However,particularly due to the role of the topology of the ground state manifold M , we expect local minimisers to provide interesting behaviour, in particular since the energy is typically notconvex. We must however do some work to say anything about local minimisers. Standardresults of Γ-convergence tell us that isolated local minimisers of F can be approximated bylocal minimisers of F ǫ [3, Theorem 4.1.1], but the translational symmetry means that theonly isolated local minimiser is the undistorted state. We now aim to reproduce the resultfor our case with continuous symmetry. Heuristically, we are applying the same proof asthe given reference to the equivalence classes by identifying translations of functions. Definition 2.15.
Let F : L ∞ ( T , Q ) → R ∪ { + ∞} . We say that b ∈ L ∞ ( T , Q ) is anisolated L -local minimiser of F , modulo translations, if there exists δ > b ∈ L ∞ ( T , Q ), if || b − b || < δ and F ( b ) ≤ F ( b ), then b = T z b for some z ∈ R . Remark 2.16.
Note that if b is an isolated L -local minimiser modulo translations, or a(potentially non-strict) L -local minimiser, then so is T z b for all z , which follows immediatelyfrom the invariance of the energy and that T z is an automorphism of L ( T , Q ). Lemma 2.17.
For ǫ >
0, and any closed ball ¯ B ⊂ L ( T , Q ) we have that min b ∈ ¯ B F ǫ ( b ) exists. Proof.
We proceed with a standard direct method argument. We first recall that F ǫ can bewritten as 1 ǫ Z T ψ s ( b ( x )) dx − ǫ Z T Z T K ǫ ( x − y ) b ( x ) · b ( y ) dy dx. (64) F ǫ trivially admits a lower bound on ¯ B , and ¯ B is weakly compact in L , as a convex,strongly closed set bounded in L -norm. Thus it only remains to show lower semicontinuity.The convexity of ψ s gives lower semicontinuity of the entropic term. The bilinear form21emains to be considered. If ( x, y ) K ǫ ( x − y ) is in L ( T × T ), then the convolution b R T K ǫ ( · − y ) b ( y ) dy defines a compact linear operator from L to L , which wouldprove the result. However our assumptions only guarantee that ( x, y ) K ǫ ( x − y ) is an L function. We use instead that if b ǫ j L ⇀ b , then b ǫ j L ⇀ b also. In light of [7, Corollary 4.1], wesee that b K ǫ ∗ b is compact from L to L ∞ if and only if { T − y K ǫ : y ∈ T } is relativelycompact in L . This however holds, since T is compact, and y T − y K ǫ is continuous in L ( T ). Proposition 2.18.
Let b be an isolated L -local minimiser of F modulo translations.Then there exists some sequence ǫ j → L -local minimisers of F ǫ j converging to b in L . Proof.
The main ingredients of the proof are that constant recovery sequences can be used,and that T z is an automorphism for all z ∈ R .Assume that δ > || b − b || < δ and F ( b ) ≥ F ( b ) implies b = T z b for some z . Let ¯ B be the closed ball of radius δ about b . Now let b ǫ ∈ arg min ¯ B F ǫ foreach ǫ >
0, which must be a non-empty set. It must hold that F ǫ ( b ǫ ) ≤ F ǫ ( b ), and since F ǫ ( b ) → F ( b ) this implies F ǫ ( b ǫ ) is bounded. We thus take a subsequence b ǫ j so that b ǫ j L → b ∗ ∈ ¯ B . Then we have F ( b ∗ ) ≤ lim inf j →∞ F ǫ j ( b ǫ j ) ≤ lim inf j →∞ F ǫ j ( b )= lim ǫ → F ǫ ( b )= F ( b ) . (65)Thus b ∗ is so that || b ∗ − b || ≤ δ < δ and F ( b ∗ ) ≤ F ( b ), so we must have that thereexists some z with T z b = b ∗ . Since T z is continuous, this means that T − z b ǫ j → T − z b ∗ = T − z T z b = b . By invariance of the energy, F ǫ j ( T − z b ǫ j ) = F ǫ j ( b ǫ j ) = min ¯ B F ǫ j . Furthermore,for j sufficiently large, since T − z b ǫ j → b , this means T − z b ǫ j ∈ int ( B ). This means that T − z b ǫ j is a minimiser of F ǫ j on an open set in L , i.e. it is an L -local minimiser. Therefore T − z b ǫ j is a sequence of L -local minimisers of F ǫ j converging to b in L . We now consider the free energy on a bounded domain, with electrostatic interactions. LetΩ ⊂ [0 , π ] be a Lipschitz domain, so that there exists δ > d (Ω , ∂ [0 , π ]) = δ .If Ω [0 , π ] but is still bounded, then we simply rescale and translate our domain. We22onsider Lipschitz subdomains Ω ǫ ⊂ Ω, so that there exists δ ǫ with d (Ω ǫ , ∂ Ω) > δ ǫ , andassume c ǫ α > δ ǫ > c ǫ α for some 1 > α >
0. We have a boundary data, b ∈ W , (Ω , M ),and work in the admissible set A ǫ = (cid:8) b ∈ L ∞ (Ω , Q ) : ( b − b ) | Ω \ Ω ǫ = 0 (cid:9) . (66)Since Ω is a Lipschitz domain, we can extend b to ˜ b ∈ W , ([0 , π ] , R k ). Furthermore,since Ω and ∂ [0 , π ] are separated, by multiplying by a smooth function which is zero R \ [0 , π ] and unity on Ω, we can assume that the extension is in W , ([0 , π ] , R k ).Even more so, if P Q is the projection from R k to Q , then P Q ˜ b ∈ W , ([0 , π ] , Q ), and( P Q ˜ b − b ) | Ω = 0 Therefore we identify ˜ b with a periodic extension, P Q ˜ b ∈ W , ( T , Q ).With abuse of notation, we identify all functions b ∈ A ǫ with their extension ˜ b ∈ W , ( T , R k ) with ˜ b ∈ T \ Ω = P Q ˜ b . Importantly, the values outside of Ω ǫ are fixed.We also impose a decay assumption on g . We assume there exists C > p > − α )( p − > | z | is sufficiently large, then g ( z ) ≤ C | z | − p . We assume that the electrical anisotropy of the media is described by a Lipschitz, tensor-valued function function A : Q → R × , so that the following holds: • A ( b ) = A ( b ) T for all b ∈ Q . • We have that min b ∈Q λ min ( A ( b )) >
0. Since A is continuous and Q compact,max b ∈Q λ max ( A ( b )) < + ∞ necessarily.Then the electrostatic contribution to the energy on a domain ǫ Ω is given by − Z ǫ Ω D ( x ) · E ( x ) dx, (67)where D is the displacement field, assumed to depend on the electric field as D ( x ) = A (˜ b ( x )) E ( x ) for the order parameter field ˜ b ∈ L ∞ (cid:0) ǫ Ω , R k (cid:1) . Maxwell’s equations at equi-librium and in the absence of external charges gives that ∇× E = 0, ∇· D = 0. Therefore wecan write this in terms of an electrostatic potential ˜ φ ∈ W , (cid:0) ǫ Ω , R (cid:1) subject to a Dirichletcondition ˜ φ | ǫ ∂ Ω = ˜ φ , as max ˜ φ ∈ ˜ φ + W , ( ǫ Ω , R ) − Z ǫ Ω A (˜ b ( x )) ∇ ˜ φ ( x ) · ∇ ˜ φ ( x ) dx. (68)In this case we see that ∇· D = 0 is simply the Euler-Lagrange equation for this maximisationproblem, which is satisfied by only the unique maximiser. By performing a change ofvariables x ǫy , ˜ b ( x ) = b ( ǫx ), ˜ φ ( x ) = φ ( ǫx ), electrostatic term becomes1 ǫ E ( b ) = max φ ∈ φ + W , (Ω , R ) − ǫ Z Ω A ( b ( y )) ∇ φ ( y ) · ∇ φ ( y ) dx. (69)23his brings in a technical point in that our problem to be solved will no longer by aminimisation problem, but rather a saddle-point problem in b and φ . However, by viewing E as just a function of b , and hiding the dependence on φ , we can overcome these issues.We have the following results that simplify the analysis Proposition 3.1.
Let E ( b ) be as given in (69). Then the following hold:1. E ( b ) is uniformly bounded from below.2. If b k L → b , then E ( b k ) → E ( b ). That is, E is continuous with respect to strong L convergence.3. If Φ b denotes the unique solution of the maximisation problem in (69) for a given b ,then if b k L → b , Φ b k W , → Φ b . Proof.
These results follow from [5, Appendix A], since A is Lipschitz, implying that if b k L → b , then A ( b k ) L → A ( b ).Thus the saddle-point energy, with x dependence suppressed for brevity when unam-biguous, is Z ǫ Ω ψ s (˜ b ) dx − Z ǫ Ω Z ǫ Ω K ( x − y )˜ b ( x ) · ˜ b ( y ) dy dx − Z ǫ Ω A (˜ b ) ∇ ˜ φ · ∇ ˜ φ dx. (70)Performing a change of variables as we did in (35), this means we considermin b max φ ǫ Z Ω ψ s ( b ) dx − ǫ Z Ω Z Ω K (cid:18) x − yǫ (cid:19) b ( x ) · b ( y ) dy dx − ǫ Z Ω A ( b ) ∇ φ · ∇ φ dx = min b ǫ Z Ω ψ s ( b ) dx − ǫ Z Ω Z Ω K (cid:18) x − yǫ (cid:19) b ( x ) · b ( y ) dy dx + 1 ǫ E ( b ) (71)We thus multiply through by ǫ as before and consider min b ∈A ǫ G ′ ǫ ( b ), with G ′ ǫ ( b ) = 1 ǫ Z Ω ψ ( b ( x )) dx + 12 ǫ Z Ω Z Ω K (cid:18) x − yǫ (cid:19) b ( y ) dy dx + E ( b ) . (72) Γ -convergence With the model set up, we now turn to investigating the Γ-limit. The key idea is that given b ∈ A ǫ , we can find an appropriate extension ˜ b ∈ L ∞ ( T , R k ), so that G ′ ǫ ( b ) is asymptoticallyequivalent to F ǫ (˜ b ) with a constraint. At this point, the result follows essentially from theresults for periodic domains. The bulk of this subsection is devoted to showing that the24error-term” is asymptotically small so that such an embedding into the periodic domainis successful. Heuristically, the error term is directly related to the non-locality of thebilinear form. By partitioning the periodic domain into T \ Ω and Ω, the bilinear formobtains a cross-term involving an integral over T \ Ω × Ω, which would not exist for a localenergy functional. In order to show this cross-term does not effect the minimisation in anasymptotic sense, we exploit the fact that K decays faster than d ( T \ Ω , Ω ǫ ) approacheszero. We now proceed to find the appropriate estimates. Proposition 3.2.
There exists ˜
C > ǫ > x ∈ Ω ǫ , ǫ R R \ Ω 1 ǫ K (cid:0) x − yǫ (cid:1) dx ≤ ˜ Cǫ (1 − α )( p − I . Proof.
First note that if x ∈ Ω ǫ , d ( x, R \ Ω) > δ ǫ ≥ c ǫ α . Therefore Z R \ Ω ǫ K (cid:18) x − yǫ (cid:19) dx = Z ǫ ( x − R \ Ω) K ( z ) dz ≤ M Z R \ B δǫǫ g ( z ) I dz ≤ πM C − p (cid:2) r − p (cid:3) ∞ δǫǫ I = 4 πM C − p (cid:18) ǫδ ǫ (cid:19) p − I ≤ ˜ Cǫ (1 − α )( p − I. (73)The estimate on g holds for δ ǫ ǫ ≥ c ǫ − α sufficiently large, i.e. ǫ sufficiently small. Corollary 3.3.
There exists some remainder term R ǫ : A ǫ → R which tends to zerouniformly in its argument so that12 ǫ Z Ω ǫ (cid:18)Z Ω K (cid:18) x − yǫ (cid:19) dy (cid:19) b ( x ) · b ( x ) dx = 12 ǫ Z Ω ǫ K b ( x ) · b ( x ) dx + R ǫ ( b ) , (74)with K = R R K ( z ) dz . Proof.
This follows since (cid:12)(cid:12)(cid:12)(cid:12) ǫ Z Ω ǫ (cid:18)Z Ω K (cid:18) x − yǫ (cid:19) dy (cid:19) b ( x ) · b ( x ) dx − Z Ω ǫ K b ( x ) · b ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ǫ Z Ω ǫ Z R \ Ω K (cid:18) x − yǫ (cid:19) dy ! b ( x ) · b ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤| Ω ǫ | || b || ∞ ǫ sup x ∈ Ω ǫ Z R \ Ω ǫ (cid:12)(cid:12)(cid:12)(cid:12) K (cid:18) x − yǫ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dx dy ≤ ˜ C | Ω ǫ | || b || ∞ ǫ (1 − α )( p − − . (75)25ince Q is bounded and by assumption (1 − α )( p − > L (Ω ǫ ) ≤ L (Ω), the resultfollows. Proposition 3.4.
There exists a positive constant ˜
C >
0, so that for ǫ > b ∈ A ǫ , Z Ω Z Ω (cid:18) K ǫ ( x − y ) − ǫ K (cid:18) x − yǫ (cid:19)(cid:19) ( b ( x ) − b ( y )) · ( b ( x ) − b ( y )) dx dy ≤ ˜ Cǫ p − . (76) Proof.
Since b is bounded it suffices to provide an estimate the term involving K . In thiscase, Z Ω Z Ω (cid:18) K ǫ ( x − y ) − ǫ K (cid:18) x − yǫ (cid:19)(cid:19) ( b ( x ) − b ( y )) · ( b ( x ) − b ( y )) dx dy ≤ || b || ∞ Z Ω Z T (cid:12)(cid:12)(cid:12)(cid:12) K ǫ ( x − y ) − ǫ K (cid:18) x − yǫ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dx dy =2 || b || ∞ Z Ω Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X k ∈ Z \{ } ǫ K (cid:18) x − y + 2 πkǫ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx dy ≤ || b || ∞ Z Ω Z T X k ∈ Z \{ } ǫ (cid:12)(cid:12)(cid:12)(cid:12) K (cid:18) x − y + 2 πkǫ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dx dy =2 || b || ∞ Z Ω Z R \ T ǫ (cid:12)(cid:12)(cid:12)(cid:12) K (cid:18) x − yǫ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dx dy ≤ M || b || ∞ Z Ω Z R \ T ǫ g (cid:18) x − yǫ (cid:19) dx dy ≤ M || b || ∞ Z Ω Z ǫ ( x − R \ T ) g ( z ) dx dy (77)Now we note that for all x ∈ Ω, R \ B δ ǫ ⊂ ǫ ( x − R \ T ). Then the remaining integral canbe estimated as in the previous result, although it is somewhat simpler since Ω is uniformlybounded away from R \ [0 , π ] . This gives that for small ǫ > C | Ω | ǫ p − . (78) Definition 3.5.
We define the second remainder R ǫ : A ǫ → R by R ǫ ( b ) = 1 ǫ Z Ω Z Ω (cid:18) K ǫ ( x − y ) − ǫ K (cid:18) x − yǫ (cid:19)(cid:19) ( b ( x ) − b ( y )) · ( b ( x ) − b ( y )) dx dy. (79)Note that R ǫ → roposition 3.6. There is a constant ˜
C > ǫ > b ∈ A ǫ ,1 ǫ Z Ω ǫ Z T \ Ω K ǫ ( x − y ) · ( b ( x ) − b ( y )) · ( b ( x ) − b ( y )) dx dy ≤ ˜ Cǫ (1 − α )( p − − . (80) Proof.
The heuristics follow almost identically to the previous arguments, so for brevityonly a sketch is provided for the estimate.1 ǫ Z Ω ǫ Z T \ Ω K ǫ ( x − y )( b ( x ) − b ( y )) · ( b ( x ) − b ( y )) dx dy ≤ || b || ∞ Mǫ Z Ω ǫ Z R \ Ω g ( x − y ) dx dy = O ( ǫ (1 − α )( p − − ) (81)by the same argument as before, and tends to zero uniformly in b . Definition 3.7.
Define the third remainder, R ǫ : A ǫ → R by R ǫ ( b ) = 1 ǫ Z Ω ǫ Z T \ Ω K ǫ ( x − y )( b ( x ) − b ( y )) · ( b ( x ) − b ( y )) dx dy. (82)Note that under our assumptions, R ǫ → ǫ → Definition 3.8.
Let U , U ⊂ R be measurable, and b ∈ A ǫ . Define the bilinear form B ǫ as B ǫ ( b, U , U ) = Z U Z U K ǫ ( x − y )( b ( x ) − b ( y )) · ( b ( x ) − b ( y )) dx dy. (83) Theorem 3.9.
There exists a function R ǫ : A ǫ → R which tends to zero uniformly as ǫ →
0, and constants m ǫ so that if c = inf b ∈Q ψ s ( b ) − K b · b , then G ′ ǫ ( b ) = 1 ǫ Z Ω ǫ ψ s ( b ( x )) − K b ( x ) · b ( x ) − c dx + 14 ǫ Z T Z T K ǫ ( x − y )( b ( x ) − b ( y )) · ( b ( x ) − b ( y )) dx dy + E ( b ) + R ǫ ( b ) + m ǫ . (84) Proof.
For the bulk term, we we immediately have Z Ω ψ s ( b ( x )) − (cid:18) ǫ Z Ω K (cid:18) x − yǫ (cid:19) dy (cid:19) b ( x ) · b ( x ) dx = Z Ω ǫ ψ s ( b ( x )) − K b ( x ) · b ( x ) dx + m ǫ + R ǫ ( b ) . (85)The constant m ǫ = R Ω \ Ω ǫ K b ( x ) · b ( x ) dx depends only on the boundary data b | Ω \ Ω ǫ .27o deal with the non-local term, first estimate this as1 ǫ Z Ω Z Ω ǫ K (cid:18) x − yǫ (cid:19) ( b ( x ) − b ( y )) · ( b ( x ) − b ( y )) dx dy = 1 ǫ Z Ω Z Ω K ǫ ( x − y )( b ( x ) − b ( y )) · ( b ( x ) − b ( y )) dx dy + R ǫ ( b )= 1 ǫ B ǫ ( b, Ω , Ω) + R ǫ ( b ) (86)Then we decompose B ǫ ( b, Ω , Ω) as B ǫ ( b, Ω , Ω) = B ǫ ( b, T , T ) − B ǫ ( b, T \ Ω , Ω) − B ǫ ( b, T \ Ω , T \ Ω)= B ǫ ( b, T , T ) − B ǫ ( b, T \ Ω , Ω ǫ ) − B ǫ ( T \ Ω , Ω \ Ω ǫ ) − B ǫ ( b, T \ Ω , T \ Ω)= (cid:20) B ǫ ( b, T , T ) − B ǫ ( b, T \ Ω , Ω ǫ ) (cid:21) − (cid:20) B ǫ ( T \ Ω , Ω \ Ω ǫ ) + B ǫ ( b, T \ Ω , T \ Ω) (cid:21) = B ǫ ( b, T , T ) − R ǫ ( b ) − m ǫ , (87)with m ǫ = 2 B ǫ ( T \ Ω , Ω \ Ω ǫ ) + B ǫ ( b, T \ Ω , T \ Ω) only dependent on the boundary values.We now combine these three results, to give that1 ǫ Z Ω ψ s ( b ( x )) dx − ǫ Z Ω Z Ω K (cid:18) x − yǫ (cid:19) b ( x ) · b ( y ) dx dy = 1 ǫ Z Ω ǫ ψ s ( b ( x )) − K b ( x ) · b ( x ) − c dx − c | Ω ǫ | + m ǫ R ǫ ( b )+ 1 ǫ B ǫ ( b, Ω , Ω) + R ǫ ( b )= 1 ǫ Z Ω ǫ ψ s ( b ( x )) − K b ( x ) · b ( x ) − c dx − c | Ω ǫ | + m ǫ + R ǫ ( b )+ 1 ǫ B ǫ ( b, T , T ) − ǫ R ǫ ( b ) − m ǫ ǫ + R ǫ ( b ) . (88)Therefore by defining m ǫ = m ǫ − c | Ω ǫ | − m ǫ ǫ ,R ǫ = R ǫ − R ǫ + R ǫ , (89)the result holds.In light of this result, it suffices to consider the Γ-limit of the asymptotically equivalentenergy, G ǫ ( b ) = 1 ǫ Z Ω ǫ ψ ( b ( x )) dx + 12 ǫ Z T Z T K ǫ ( x − y ) · ( b ( x ) − b ( y )) ⊗ dx dy + E ( b ) (90)28 roposition 3.10. Assume that b ǫ ∈ A ǫ with G ǫ ( b ǫ ) uniformly bounded. Then there is asubsequence b ǫ j so that b ǫ j L → b ∈ W , ( T , R k ), with b ( x ) ∈ M for almost every x ∈ Ω. Inparticular, b ǫ j L → b ∈ W , (Ω , M ) with b | ∂ Ω = b | ∂ Ω . Proof.
Since the electrostatic and bulk terms admit lower bounds, this implies that1 ǫ Z T Z T K ǫ ( x − y ) · ( b ǫ ( x ) − b ǫ ( y )) ⊗ dx dy (91)is also uniformly bounded. Therefore by Corollary 2.4, we can extract a subsequence b ǫ j L → b ∈ W , ( T , R k ). Furthermore, for x ∈ Ω \ Ω ǫ , b ( x ) ∈ M almost everywhere. Since M = ψ − (0), this gives 1 ǫ j Z Ω ǫj ψ ( b ǫ j ( x )) dx = 1 ǫ j Z Ω ψ ( b ǫ j ( x )) dx (92)which is bounded. Thus since the energy is bounded, this implies that ψ ( b ǫ j ( x )) → b ( x ) ∈ M almosteverywhere in Ω. In the sense of traces, b | ∂ Ω = b | ∂ Ω , since b − b is a W , functionsupported on the closure of Ω.Before turning to the Γ-convergence result, we establish a lemma required for our con-struction of recovery sequences. Lemma 3.11.
Let U ⊂ R n be a bounded Lipschitz domain. Then for every ǫ > A ⊂ Ω so that L n ( U \ A ) < ǫ and U \ A is Lipschitz. Proof.
The result is essentially Vitali’s covering theorem. We denote x + B r the closed ballof radius r > x ∈ R . Since U is open, we can define a function R : U → (0 , ∞ )so by R ( x ) = sup { r > x + B r ⊂ Ω } . Then if r ≤ R ( x ), x + B r is compactly supportedin U . Furthermore { x + B r : r < R ( x ) , x ∈ U } is a Vitali covering of U . Therefore we cantake a countable pairwise disjoint subcollection ( x i + B r i ) i ∈ N , whose union covers U up to aset of measure zero. Even more so if A N = N S i =1 ( x i + B r i ), then A N is a finite union of disjointclosed balls, compactly supported in U , with lim N →∞ L n ( U \ A N ) →
0. Thus by taking A = A N for sufficiently large N , we have that L n ( U \ A ) < ǫ . Furthermore, since A N is a union ofdisjoint balls compactly supported in U , we have that ∂ ( U \ A ) = ∂U ∪ (cid:18) N S i =1 x i + ∂B r i (cid:19) ,which is a disjoint union of Lipschitz surfaces. Therefore U \ A is a Lipschitz domain. Theorem 3.12.
The functionals G ǫ Γ → G , where G ( b ) = Z T L ∇ b ( x ) · ∇ b ( x ) dx + 12 Z Ω A ( b ) − D ( x ) · D ( x ) dx + E ( b ) (93)29f b ∈ W , ( T , R k ) with b ( x ) ∈ M for almost every x ∈ Ω and b = b on T \ Ω, andis + ∞ otherwise. The mode of convergence is L strong convergence. Furthermore let b ǫ L → b , and denote the solutions of the maximisation problem defining E ( b ǫ ) as Φ ǫ . Thendiv( A ( b ǫ )Φ ǫ ) = 0 and Φ ǫ | ∂ Ω = φ . Then Φ ǫ W , → Φ, where div( A ( b ) ∇ Φ) = 0 and Φ | ∂ Ω = φ . Proof.
The electrostatic term E ( b ) is continuous with L convergence, so does not effectthe result. The liminf inequality follows directly by the same argument as Theorem 2.12.For the limsup inequality, we can no longer use constant recovery sequences, since for b ∈ W , ( T , R k ) with b − b = 0 on T \ Ω, it is not in general true that b | Ω \ Ω ǫ = b . Weovercome this with an interpolation by harmonic functions.Let b ∈ W , ( T , R k ) with b (Ω) ⊂ M and b | T \ Ω = b . Let E ǫ ⊂ Ω ǫ be a compactset, so that U ǫ = Ω ǫ \ E ǫ satisfies L ( U ǫ ) ≤ c ǫ with U ǫ \ E ǫ Lipschitz, which exists byLemma 3.11. Let b ǫ ( x ) = b ( x ) for x ∈ T \ U ǫ , and let b ǫ satisfy R U ǫ ∇ b ǫ ( x ) · ∇ u ( x ) dx = 0 ( ∀ u ∈ W , ( U ǫ )) ,b ǫ ( x ) = b ( x ) ( x ∈ ∂U ǫ ) . (94)Since U ǫ is Lipschitz and b − b ǫ ∈ W , ( U ǫ , R k ), the extension by zero is in W , ( T ),therefore b ∈ W , ( T , R k ). First we show b ǫ W , → b . The maximum principle gives that b ǫ admits a uniform bound, so b ǫ and b are both bounded and only differ on U ǫ . This set haswhich has vanishing measure as ǫ →
0, so it holds that b ǫ L → b . To show that the gradientsconverge, we use that b ǫ is harmonic and b − b ǫ can be used to test the weak form of thePDE in (94). Suppressing the x dependence for readability, ||∇ b ǫ − ∇ b || = Z T |∇ b ǫ − ∇ b | dx = Z U ǫ ( ∇ b ǫ − ∇ b ) · ( ∇ ( b ǫ − b )) dx = Z U ǫ ∇ b · ( ∇ b ǫ − ∇ b ) dx ≤ (cid:18)Z U ǫ |∇ b | dx (cid:19) ||∇ b ǫ − ∇ b || ⇒ ||∇ b ǫ − ∇ b || ≤ (cid:18)Z U ǫ |∇ b | dx (cid:19) . (95)Since the measure of U ǫ tends to zero and b ∈ W , , the integral on the right hand sidetends to zero as ǫ →
0, so that ∇ b ǫ L → ∇ b and b ǫ W , → b . Therefore by Proposition 2.9, wehave thatlim ǫ → ǫ Z T Z T K ǫ ( x − y ) · ( b ǫ ( x ) − b ǫ ( y )) ⊗ dx dy = Z T L ∇ b ( x ) · ∇ b ( x ) dx. (96)30t remains to show that the bulk energy of b ǫ tends to zero. Again since b ǫ is harmonic wehave a maximum principle, and since the boundary data is in M , we have b ǫ ( x ) ∈ Conv( M )almost everywhere. We can now obtain an estimate on ψ b ( b ( x )). First, we note that since ψ b blows up uniformly at ∂ Q , we must have that M is compactly supported in Q . Thengiven b ǫ ( x ) ∈ Conv( M ) ⊂ Q , using the convexity of ψ s and non-negativity of K we have ψ ( b ǫ ( x )) = ψ s ( b ( x )) − K b ( x ) · b ( x ) ≤ max M ψ s = c , (97)which is a uniform bound on ψ b ( b ǫ ). Therefore1 ǫ Z Ω ψ b ( b ǫ ) dx = Z U ǫ ψ s ( b ǫ ) dx ≤ c ǫ Z U ǫ dx ≤ c c ǫ ǫ = O ( ǫ ) . (98)Therefore lim ǫ → ǫ R Ω ψ b ( b ǫ ) dx →
0. Combining these, we have b ǫ → b , and G ǫ ( b ǫ ) → G ( b ),giving the limsup inequality. The results for the convergence of the electrostatic potentialare in Proposition 3.1. Remark 3.13.
Whilst the energy contains an elastic term given by the integral over T \ Ω,this is only dependent on the prescribed boundary conditions extension and thus gives aconstant which is irrelevant to the minimisation process. In fact we could subtract thisconstant from G ǫ ( b ) to remove this. The condition that b | T \ Ω = b then reduces to aDirichlet condition on the boundary ∂ Ω as L (Ω \ Ω ǫ ) →
0. This gives that the minimisationproblem for the Γ-limit is equivalent to minimising14 Z Ω L ∇ b ( x ) · ∇ b ( x ) dx (99)over b ∈ W , (Ω , M ) with b | ∂ Ω = b | ∂ Ω . For definiteness, we now consider a case which will reduce to the Oseen-Frank elastic model.In this case, we take N = S and a ( p ) = p ⊗ p − ∈ Sym (3). We require our interactionkernel to be frame indifferent, so that K ( Rz ) a ( Rp ) · a ( Rq ) = K ( z ) a ( p ) · a ( q ) for all R ∈ SO(3). We can see that this highly constrains which bilinear forms are available [17], weonly have K ( z ) a ( p ) · a ( q ) = g ( z ) a ( p ) · a ( q ) + g ( z )( a ( p ) b z ) · ( a ( q ) b z ) + g ( z )( b z · a ( p ) b z )( b z · a ( q ) b z ) , (100)31here g i are frame indifferent functions on R . We assume that g i have sufficient integra-bility and decay for our results to hold. It is not necessary that g , g are non-negative,however g will have to be able to compensate. We write this in indices as K ( z ) i i ,j j = g ( z ) δ i ,j δ i ,j + g ( z ) δ i ,j b z i b z j + g ( z ) b z i b z i b z j b z j , (101)with K ( z ) Q · Q = X i ,i ,j ,j =1 K ( z ) i i ,j j Q i i Q j j . (102)We note that K admits a symmetry, so that if σ is a permutation of { , , } , then K ( z ) ij,kl = K ( z ) σ i σ j ,σ k σ l . Furthermore K i i ,j j = K j j ,i i .For the sake of simplifying later integrals involving moments of isotropic functions, weinclude the following lemma. Lemma 4.1.
Let g ∈ L ( R ) be isotropic and have finite fourth moment. Then Z R (cid:0) z − z z (cid:1) g ( z ) , dz = 0 . (103) Proof.
In order to see this, we note that due to the symmetry of indices, we can write Z R (cid:0) z − z z (cid:1) g ( z ) dz = 12 Z R (cid:0) z − z z + z (cid:1) g ( z ) dz. (104)Next, we see that if P ( z ) = z − z z + z , and R is the rotation about e of π , then P ( Rz ) = − P ( z ) for all z ∈ R . This is just an algebraic computation, P ( Rz ) = √ z − √ z ! − √ z − √ z ! √ z + √ z ! + √ z + √ z ! = (cid:18) z + 12 z − z z (cid:19) − (cid:18) z + 12 z + z z (cid:19) (cid:18) z + 12 z − z z (cid:19) + (cid:18) z + 12 z + z z (cid:19) = 14 z − z z + 32 z z − z z + 14 z − z + 3 z z − z + 14 z + z z + 32 z z + z z + 14 z = − z + 6 z z − z = − P ( z ) . (105)32hus by applying the change of variables z Rz into the integral, Z R (cid:0) z − z z (cid:1) g ( z ) dz = 12 Z R P ( z ) g ( z ) dz = 12 Z R P ( Rz ) g ( Rz ) dz = − Z R P ( z ) g ( z ) dz, (106)so the integral is its own negative and hence zero. Proposition 4.2.
Let g ∈ L ( R ) be isotropic and Q ∈ Sym (3). Then Z R g ( z ) b z i b z j dz = δ ij Z R g ( z ) b z dz, Z R g ( z )( b z · Q b z ) dz = 23 Z R g ( z ) b z dz | Q | . (107) Proof.
The first equality is more straightforward. If i = j , then the integral vanishes sinceit is odd in one of the components. If i = j , then the integrand is dependent only on i sinceapplying a rotation that swaps between labelling bases leaves the integral unchanged.For the second inequality, we have that, as a map from Sym (3) to R , R R g ( z )( z · Qz ) dz is immediately frame indifferent and quadratic in Q . Therefore it must be of the form c Tr( Q ) + c Tr( Q ). Since Q is trace free however, it suffices to evaluate c . In this case,we test Q = e ⊗ e − e ⊗ e . Then Q b z · b z = b z − b z and | Q | = 2. Thus2 c = Z R ( b z − b z ) g ( z ) dz = Z R (cid:0)b z + b z − b z b z (cid:1) dz =2 Z R ( b z − b z z ) g ( z ) dz = 43 Z R b z g ( z ) dz (108)using the previous lemma. Corollary 4.3.
For K as given in (100), and K = R R K ( z ) dz , then K Q · Q = k | Q | ,where k = Z R g ( z ) dz + Z R g ( z ) b z dz + 23 Z R g ( z ) b z dz (109) Proof.
This follows immediately from the previous proposition.33n this case the bulk energy is given by ψ s ( Q ) − k | Q | − c . Depending on the valueof k , this is minimised either at Q = 0, or Q = s ∗ (cid:0) n ⊗ n − I (cid:1) for any n ∈ S and s ∗ > k [8]. We consider only the latter case. In this case with a nematic groundstate, the minimising manifold M is readily identified with projective space. By identifying M with R P , if a lifting for Q ∈ W , (Ω , M ) → W , (Ω , S ) exists, we say Q is orientable and obtain the Oseen-Frank energy for a director field n ∈ W , (Ω , S ). This is alwayspossible, for example, when the boundary data Q is orientable and Ω is simply connected[2]. Proposition 4.4.
Let Q = s ∗ (cid:0) n ⊗ n − I (cid:1) for some n ∈ W , (Ω , S ). For K as given in(100), the elastic component of the energy can be written L ∇ Q · ∇ Q = K (div n ) + K | n × curl n | + K ( n · curl n ) (110)with 1( s ∗ ) K =2 G , , + G , , + G , , + G , , − G , , , s ∗ ) K =2 (cid:0) G , , + G , , + G , , − G , , (cid:1) ,K = K . (111)The constants G nijk are defined by G nijk = Z R | z | n − g I ( z ) z i z j z k dz. (112) Proof.
Denote the components of ∇ Q as Q αβ,γ . Then the bilinear form is L ∇ Q · ∇ Q = Z R K ( z ) i i ,j j z i z j Q i i ,i Q j j ,j dz. (113)It is immediate to see that this is a frame indifferent function of ∇ Q , following from theisotropy of K . In fact this holds for generally any Q ∈ W , (Ω , Sym (3)), not just thosewith values in M which are orientable. In this case, by [15] we know that the bilinear formcan be written as L Q αβ,γ Q αβ,γ + L Q αβ,β Q αγ,γ + L Q αβ,γ Q αγ,β . (114)In the case when Q gives an orientable line field, so Q = s ∗ (cid:0) n ⊗ n − I (cid:1) for n ∈ W , (Ω , S ),we have that L ∇ Q · ∇ Q = K (div n ) + K | n × curl n | + K ( n · curl n ) , (115)where the Frank constants K i are related to L i as in [1] by1( s ∗ ) K =2 L + L + L , s ∗ ) K =2 L ,K = K . (116)34t thus suffices to test values of ∇ Q to obtain relations between the tensor L and constants L , L , L , which then give the Frank constants. First we try Q ( x ) = ( e ⊗ e − e ⊗ e ) x ,so ∇ Q = ( e ⊗ e − e ⊗ e ) ⊗ e . Then Q αβ,γ Q αγ,β = Q αβ,β Q αγ,γ = 0 and Q αβ,γ Q αβ,γ = 2.Then 1( s ∗ ) K =2 L = L ∇ Q · ∇ Q = Z R K ( z ) z − K ( z ) z + K ( z ) z dz =2 Z R K z − K ( z ) z dz =2 Z R g ( z ) z + g ( z ) | z | z z + g ( z ) | z | z z − g ( z ) | z | z z z dz =2 (cid:0) G , , + G , , + G , , − G , , (cid:1) (117)Next we trial ∇ Q ( x ) = ( e ⊗ e − e ⊗ e ) ⊗ e . In this case Q αβ,γ Q αγ,β = Q αβ,β Q αγ,γ = 1, Q αβ,γ Q αβ,γ = 2. Therefore1( s ∗ ) K =2 L + L + L = L ∇ Q · ∇ Q = Z R K ( z ) z − K ( z ) z + K z dz = Z R g ( z ) z + g ( z ) | z | z z + g ( z ) | z | z z − g ( z ) | z | z z + g ( z ) z + g ( z ) | z | z + g ( z ) | z | z dz. =2 G , , + G , , + G , , + G , , − G , , . (118) Remark 4.5.
For the sake of illustration, we now consider the simple case where g i ( z ) = c i g ( z ), and g ( z ) = (cid:26) / | z | | z | > .
10 else (119)for constants c i to illustrate. This is consistent with the quadratic part of the London dis-persion forces formula for uniaxial molecules [12], with a cutoff region to avoid singularitiesas employed in the derivation of Maier-Saupe. It should be noted however that while theLondon dispersion forces can be expressed in the form of (100) with g i = c i g as here, theexact constants c i unfortunately give rise to a bilinear form lacking the necessary coercivity35or our analysis (explicitly, c = C , c = − C , c = C for a material constant C > k ≈ c + 1353 c + 811 c . (120)Then by Proposition 4.4 we find that G constants first, which evaluate as G , , ≈ . c , G , , ≈ . c , G , , ≈ . c G , , ≈ . c , G , , ≈ . c , G , , ≈ . c (121)This gives our Frank constants as1( s ∗ ) K ≈ c + 33 c + 14 c , s ∗ ) K ≈ c + 17 c + 4 . c ,K = K . (122)In particular, if c i are all roughly equal, so c i ≈ c for i = 2 ,
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2. Alternatively if the isotropic constant c dominates, so c ≥ c i ≥ i = 2 ,
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The author would like to thank John M. Ball, Epifanio G. Virga, Claudio Zannoni and Gia-como Canevari for insightful discussions that have benefited this work. The research leadingto these results has received funding from the European Research Council under the Euro-pean Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n ◦ References [1]
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