Report Number 09/10. Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond. Apala Majumdar, Arghir Zarnescu

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1 Report Number 09/0 Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond by Apala Majumdar, Arghir Zarnescu Oxford Centre for Collaborative Applied Mathematics Mathematical Institute 4-9 St Giles Oxford OX LB England

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3 Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond Apala Majumdar and Arghir Zarnescu December 6, 008 Abstract We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in W,, to a global minimizer predicted by the Oseen-Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen-Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau-De Gennes global minimizer. We also study the interplay between biaxiality and uniaxiality in Landau-De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions. Introduction Nematic liquid crystals are an intermediate phase of matter between the commonly observed solid and liquid states of matter []. The constituent nematic molecules translate freely as in a conventional liquid but whilst flowing, tend to align along certain locally preferred directions i.e. exhibit a certain degree of long-range orientational order. Nematic liquid crystals break the rotational symmetry of isotropic liquids; the resulting anisotropic properties make liquid crystals suitable for a wide range of physical applications and the subject of very interesting mathematical modelling [8]. There are three main continuum theories for nematic liquid crystals [8]. The simplest mathematical theory for nematic liquid crystals is the Oseen-Frank theory []. The Oseen-Frank theory is restricted to uniaxial nematic liquid crystal materials liquid crystal materials with a single preferred direction of molecular alignment) with constant degree of orientational order. The state of a uniaxial nematic liquid crystal is described by a unit-vector field, nx) S, which represents the preferred direction of molecular alignment. In the simplest setting, the liquid crystal energy reduces to: F OF [n] = Ω n i,k x)n i,k x) dx, ) the standard Dirichlet energy for vector-valued maps into the unit sphere. The equilibrium configurations the physically observable configurations) correspond to minimizers of the F OF -energy, subject to the imposed boundary conditions. In particular, the minimizers of F OF are examples of S -valued harmonic maps [8, ]. The Oseen-Frank theory has been extensively studied in the literature, see the review [5], and there are rigorous results on the existence, regularity and singularities of Oseen-Frank minimizers. The Oseen-Frank theory is limited in the sense that it can only account for point defects in liquid crystal systems but not the more complicated line and surface defects that are observed experimentally. A second, Mathematical Insitute, University of Oxford, 4-9 St. Giles, OX LB, U.K.

4 more comprehensive theory is the continuum Ericksen theory [8]. The Ericksen theory is also restricted to uniaxial liquid crystal materials but can account for spatially varying orientational order i.e. the state of the liquid crystal is described by a pair, s,n) R S, where s R is a real scalar order parameter that measures the degree of orientational ordering and n represents the direction of preferred molecular alignment. In the simplest setting, the corresponding energy functional is given by F E [s,n] = sx) nx) + k sx) + W 0 s) dx ) Ω where k is a material-dependent elastic constant and W 0 s) is a bulk potential. The Ericksen theory is based on the premise that s vanishes wherever n has a singularity and this theory can account for all physically observable defects. However Ericksen recognizes that his theory is but a simplified description of a possibly more complex situation see [8]): There is the third possibility, that the three eigenvalues of Q are all distinct, giving what are called biaxial nematic configurations. Theories fitting MACMILLAN S [] format permit any of the three types of configurations to occur. Certainly it is not unreasonable to think that flows or other influences could convert a rather stable nematic configuration to one of the biaxial type, etc. I [9] am one of those who have argued that, near isotropic-nematic phase transitions, it should be quite easy to induce such changes. Accounting for such possibilities does add significant complications to the equations and the problems of analyzing them. Experimental information concerning the biaxial configurations is still quite slim and, for me, it is too early to think seriously about them. So, I will develop a theory representing a kind of compromise. The most general continuum theory for nematic liquid crystals is the Landau-De Gennes theory [, 5] which can account for uniaxial and biaxial phases biaxiality implies the existence of more than one preferred direction of molecular alignment). Indeed, this theory was one of the major reasons for awarding P.G. De Gennes a Nobel prize for physics in 99. In the Landau-De Gennes framework, the state of a nematic liquid crystal is modelled by a symmetric, traceless matrix Q M, known as the Q-tensor order parameter. A nematic liquid crystal is said to be a) isotropic when Q = 0, b) uniaxial when the Q-tensor has two equal non-zero eigenvalues; a uniaxial Q-tensor can be written in the special form Q = s n n ) Id ; s R \ {0}, n S ) and c) biaxial when Q has three distinct eigenvalues; a biaxial Q-tensor can always be represented as follows see Proposition ) Q = s n n ) Id + r m m ) Id s,r R; n,m S. 4) The Landau-De Gennes energy functional, F LG [Q], is a nonlinear integral functional of Q and its spatial derivatives. We work with the simplest form of F LG [Q], with Dirichlet boundary conditions, Q b refer to )), on three-dimensional domains Ω R. We take F LG [Q] to be [7] F LG [Q] = Ω L Q x) + f B Qx)) dx 5) where f B Q) is the bulk energy density that accounts for bulk effects, Q = i,j,k= Q ij,kq ij,k is the elastic energy density that penalizes spatial inhomogeneities and L > 0 is a material-dependent elastic constant. We take f B Q) to be a quartic polynomial in the Q-tensor components, since this is the simplest form of f B Q) that allows for multiple local minima and a first-order nematic-isotropic phase transition [, ]. This form of f B Q) has been widely-used in the literature and is defined as follows f B Q) = αt T ) tr Q ) b tr Q ) + c trq ) 4

5 where α, b, c R are material-dependent positive constants, T is the absolute temperature and T is a characteristic liquid crystal temperature. We work in the low-temperature regime T < T for which αt T ) < 0. Keeping this in mind, we recast the bulk energy density as follows: f B Q) = a tr Q ) b tr Q ) + c 4 trq ), 6) where a,b,c R + are material-dependent and temperature-dependent positive constants. The equilibrium configurations the physically observable configurations) then correspond to minimizers of F LG [Q], subject to the imposed boundary conditions. In the first part of the paper, we study the the limit of vanishing elastic constant L 0 for global minimizers, Q L), of F LG [Q]. This study is in the spirit of the asymptotics for minimizers of Ginzburg- Landau functionals for superconductors []. The limit L 0 is a physically relevant limit since the elastic constant L is typically very small, of the order of 0 Joule/metre. [7]. We define a limiting harmonic map Q 0) as follows Q 0) = s + n 0) n 0) Id ) where s + is defined in 0), n 0) is a minimizer of the Oseen-Frank energy, F OF [n] in ), subject to the fixed boundary condition n = n b C Ω, S ) and Q b and n b are related as in ). Our main results are: There exists a sequence of global minimizers { Q } L k) such that Q L k) L k 0 + Q 0) strongly in the Sobolev space W, Ω, R 9 ). The sequence { Q L k) } as above converges uniformly to Q 0) as L k 0, in the interior of Ω, away from the possible) singularities of Q 0). The bulk energy density, f B Q L k ) ), converges uniformly to its minimum value away from the possible) singularities of Q 0) ; the uniform convergence of the bulk energy density holds in the interior and up to the boundary, away from the possible) singularities of Q 0). These results show that the predictions of the Oseen-Frank theory described by the limiting map Q 0) ) and the Landau-De Gennes theory agree away from the singularities of Q 0). The global minimizers, Q L), are real analytic see Proposition ) and have no singularities as such. However, one of the most intriguing features of nematic liquid crystals are the optical defects that appear in the Schlieren textures []. From a physical point of view, these defects are regions of rapid changes in the configurational properties of a nematic liquid crystal []. We conjecture that certain types of optical defects in Q L k) for small L k ), when they exist, may be localized near the analytic singularities of the limiting map Q 0), since Q L k) can have strong variations only near the singularities of Q 0) more precisely, the gradient, Q L k), cannot be bounded independently of L k on any set containing a singularity of Q 0) ). There is existing literature on the location of singularities in harmonic maps [] and this may allow one to predict the location of optical) defects in a global Landau-De Gennes minimizer. Our convergence results analyze the limit of vanishing elastic constant L 0. Physical situations are modelled by small but non-zero values of the elastic constant L. Thus our convergence results show that for L sufficiently small, the limiting harmonic map Q 0) provides but a rough description of Q L) i.e. Q L) can be thought of as having a leading uniaxial part plus a small biaxial perturbation, away from the singularities of Q 0). This small biaxial perturbation is of order O L) where L << see Section 5 for details). However, numerical simulations show that biaxiality may become prominent in the vicinity of defects [, 8]. In the second part of our paper, we study biaxiality and their role in global minimizers Q L), noting that biaxiality if it exists) is one of the main differences between Q L) and the limiting approximation Q 0). More precisely, in Propositions and, we obtain estimates for the size of the regions where Q L k) can deviate significantly from Q 0) and on the size of admissible strongly biaxial regions in Q L), in terms of the biaxiality parameter β defined in )) and the material-dependent constants. While Proposition may be relevant to the properties of Q L) near the singular set of Q 0), Proposition is relevant to the equilibrium properties away from the singular set of Q 0).

6 Using a simple nearest-neighbour projection argument see Corollary ), we show that the leading eigendirection, corresponding to the leading uniaxial part see Section 5 for definitions) is smooth on any compact set K not containing any singularity of Q 0). Further, in Proposition 5, we also show that Q L) is either a) uniaxial everywhere except for possibly a set of measure zero where Q can be isotropic) or b) Q L) is biaxial everywhere and can be uniaxial or isotropic only on sets of measure zero. It is known that as long as the number of distinct eigenvalues does not change, the eigenvectors of Q L) enjoy the same degree of regularity as Q L) itself [6]. In Corollary, we show that the eigenvectors are necessarily smooth everywhere except for possibly a zero-measure set where the number of distinct eigenvalues changes and therefore, if the eigenvectors of Q L) suffer any discontinuities, these discontinuities must be localized on the uniaxial-biaxial, uniaxial-isotropic or biaxial-isotropic interfaces. This result may be relevant to the interpretation of optical data from experiments and we hope to explore this connection in future work. Finally, we note that the Landau-De Gennes theory for uniaxial liquid crystal materials has strong analogies with the D version of the Ginzburg-Landau theory for superconductors []. The Ginzburg-Landau energy functional for a three-dimensional vector field, u : Ω R, is typically of the form F GL [u] = Ω u + u ) dx 7) 4ɛ where ɛ > 0 is a very small parameter. The functional F GL has been rigorously studied in the limit ɛ 0 which is analogous to the limit L 0 in our problem. The new mathematical complexities in the Landau-De Gennes theory for nematic liquid crystals come from the high dimensionality of the target space and also from the possibility of biaxiality in global energy minimizers. Future challenges include a better understanding of the qualitative properties of global minimizers for small but non-vanishing values of L, a better description of Q L) near the singularities of the limiting harmonic map Q 0), the regularity of the eigenvectors and eigenvalues, along with a deeper understanding of the appearance and role of biaxiality in global minimizers. The paper is organized as follows - in Section, we introduce the conventions and notations that are used in the rest of the paper. In Section, we state two representation formulae for Q-tensors that are useful for subsequent computations in later sections. In Section 4, we study the properties of global energy minimizers in the limit L 0 and prove the convergence results. In Section 5, we discuss the consequences of our convergence results and their relevance to the bulk energy density, the biaxiality parameter, the eigenvalues and the eigenvectors of a global Landau-De Gennes minimizer. In Section 6, we derive estimates for the bulk energy density, obtain bounds for the size of admissible strongly biaxial regions and discuss the interplay between biaxiality and uniaxiality in a global energy minimizer. Preliminaries We take our domain, Ω R, to be bounded and simply-connected with smooth boundary, Ω. Let S 0 M denote the space of Q-tensors, i.e. S 0 def = { Q M ; Q ij = Q ji, Q ii = 0 } where we have used the Einstein summation convention; the Einstein convention will be assumed in the rest of the paper. The corresponding matrix norm is defined to be Q def = trq = Q ij Q ij. As stated in the introduction, we take the bulk energy density term to be f B Q) = a tr Q ) b tr Q ) + c 4 trq ) ) where a,b,c R are material-dependent and temperature-dependent positive constants. One can readily verify that f B Q) is bounded from below see Proposition 8, []), and we define a non-negative bulk energy 4

7 density, f B Q), that differs from f B Q) by an additive constant as follows: f B Q) = f B Q) min Q S 0 f B Q). 8) It is clear that f B Q) 0 for all Q S 0 and the set of minimizers of f B Q) coincides with the set of minimizers for f B Q). In Proposition 8, we show that the function f B Q) attains its minimum on the set of uniaxial Q-tensors with constant order parameter s + as shown below f B Q) = 0 Q Q min where Q min = {Q S 0,Q = s + n n ) } Id,n S 9) with s + = b + b 4 + 4a c 4c. 0) We work with Dirichlet boundary conditions, referred to as strong anchoring in the liquid crystal literature []. The boundary condition Q b Q min is smooth and is given by Q b = s + n b n b ) Id, n b C Ω; S ). ) We define our admissible space to be A Q = { Q W, Ω;S 0 ) ;Q = Q b on Ω, with Q b as in ) }, ) where W, Ω;S 0 ) is the Sobolev space of square-integrable Q-tensors with square-integrable first derivatives [9]. The corresponding W, -norm is given by Q W, Ω) = Ω Q + Q dx ) /. In addition to the W, -norm, we also use the L -norm in this paper, defined to be Q L Ω) = ess sup x Ω Qx). We study global minimizers of a modified Landau-De Gennes energy functional, F LG [Q], in the admissible space A Q. The functional FLG [Q] differs from F LG [Q] in 5) by an additive constant and is defined to be L F LG [Q] = Q ij,kx)q ij,k x) + f B Qx)) dx. ) Ω For a fixed L > 0, let Q L) denote a global minimizer of F LG [Q] in the admissible class, A Q. The existence of Q L) is immediate from the direct methods in the calculus of variations [9]. The bulk energy density, f B Q), is bounded from below, the energy density is convex in Q and therefore, F LG [Q] is weakly sequentially lower semi-continuous. Moreover, it is clear that F LG [Q] and F LG [Q] have the same set of global minimizers for a fixed set of material-dependent and temperature-dependent constants { a,b,c,l }. The global minimizer Q L) is a weak solution of the corresponding Euler-Lagrange equations [] L Q ij = a Q ij b Q ik Q kj δ ) ij trq ) + c Q ij trq ) i,j =,,. 4) where the term b δ ij trq ) is a Lagrange multiplier that enforces the tracelessness constraint. It follows from standard arguments in elliptic regularity that Q L) is actually a classical solution of 4) and Q L ) is smooth and real analytic see also Section 6.). Finally, we introduce a limiting uniaxial harmonic map Q 0) : Ω Q min ; Q 0) is defined to be a global minimizer not necessarily unique) of F LG [Q] in the restricted class, A Q {Q : Ω S 0, Qx) Q min a.e. x Ω}. Then Q 0) is necessarily of the form Q 0) = s + n 0) n 0) ) Id, 5) 5

8 where n 0) is a global minimizer of F OF [n] see [], [4]), n 0 x) dx = min n A n Ω Ω nx) dx 6) in the admissible class A n = { n W, Ω; S ) ; n = n b on Ω } and n b and Q b are related as in ). This limiting harmonic map Q 0) is therefore obtained from an energy minimizer, n 0, not necessarily unique) within the Oseen-Frank theory for uniaxial nematic liquid crystals with constant order parameter for more results about the relation between n 0) and Q 0) see []). It follows from standard results in harmonic maps [] that Q 0) has at most a finite number of isolated point singularities points where n 0) has singularities). In the following sections we will elaborate on the relation between Q L) and Q 0). Representation formulae for Q-tensors We have: Proposition A matrix Q S 0 can be represented in the form with n and m unit-length eigenvectors of Q, n m = 0 and Q = sn n Id) + rm m Id) 7) 0 r s or s r 0 8) The scalar order parameters r and s are piecewise linear combinations of the eigenvalues of Q. Proof. From the spectral decomposition theorem we have Q = λ n n + λ n n + λ n n 9) where λ,λ,λ are eigenvalues of Q and n,n,n are the corresponding unit eigenvectors, pairwise perpendicular. We have I = i= n i n i and the tracelessness condition implies that λ + λ + λ = 0. Thus Q = λ n n + λ n n λ + λ )I n n n n ) We consider six regions R + i,i =,...,6 in the λ,λ ) - plane which cover exactly half of the whole plane. This corresponds to the representation 7) with 0 r s. The other half of the plane is covered by the regions R i,i =,...,6, which are obtained by reflecting R+ i through the origin 0,0)) and the regions R i correspond to the representation 7),with r,s 0. We let R + = {λ,λ ) R, λ λ,λ 0}. In this case r def = λ + λ and s def = λ + λ with n def = n,m def = n. One can directly verify that for r,s thus defined, we have r = λ + λ s = λ + λ. Interchanging λ with λ in the definition of r and s and m with n, we obtain the region R + = {λ,λ );λ λ /;λ 0}. Let R + = {λ, λ ) R,λ 0,λ λ }. Taking r def = λ λ,s def = λ λ, n def = n, m def = n, one can check that r = λ λ s = λ λ. 6

9 λ λ = λ λ = λ R + 5 R + R 4 λ = λ R 6 R R λ R + R + R + 6 R + 4 R 5 R λ = λ Figure : The λ,λ plane and its regions The region R + 4 is obtained from interchanging λ and λ. We have R 5 + = {λ,λ ) R,λ 0, λ λ λ } with r def = λ λ, s def = λ λ, n = n and m def = n. Again, it is straightforward to check that Interchanging λ with λ, we obtain the region R + 6. r = λ λ λ λ. Finally the remaining half of the λ,λ )-plane is covered by the regions R i obtained from R + i by changing the signs of the inequalities and keeping the definitions of r and s unchanged). For example, R is defined to be R = { λ, λ ) R } ;λ 0, λ λ with r = λ + λ and s = λ + λ. One can then directly check that s r 0. The remaining five regions R i for i =...6 can be defined analogously. Remark The representation formula 7) is known in the literature [5]. In Proposition, we state that it suffices to consider the two cases given by 8); we have not found references for this fact. In Proposition, we state a second representation formula for admissible Q S 0 and its relation to the representation formula 7). The representation formula 0) is known in the literature [] and will be used in Section 5. For reader s convenience we provide a quick proof. 7

10 Proposition A second representation formula) A matrix Q S 0 can be represented as: Q = S n n ) Id + R m m p p) 0) The vectors n, m and p are unit-length and pairwise perpendicular eigenvectors of Q with corresponding eigenvalues λ,λ, λ. The scalar order parameters S and R are given by S = λ R = λ + λ ). ) Proof. We have the spectral decomposition of Q, namely Q = λ n n + λ m m + λ p p with n,m, p pairwise perpendicular unit-length eigenvectors of Q and Id = n n + m m + p p. Combining the last two relations and taking S = λ,r = λ + λ ) we obtain the claim. 4 The limiting harmonic map 4. The uniform convergence in the interior Firstly, we recall that for a Q S 0 the biaxiality parameter βq) see for instance []) is defined to be ) trq βq) = 6 trq ) ) The significance of βq) as a measure of biaxiality is due to the following Lemma i) The biaxiality parameter βq) [0, ] and βq) = 0 if and only if Q is purely uniaxial i.e. if Q is of the form, Q = s n n Id) for some s R,n S. ii) The biaxiality parameter, βq), can be bounded in terms of the ratio r s, where s, r) are the scalar order parameters in Proposition. These bounds are given by ) β r s + ) β. ) Equivalently, β + β R S + β β where S, R) are the order parameters in Proposition. Further βq) = if and only if r = s or if and only if R S =. iii)for an arbitrary Q S 0, we have that 4) Q β ) trq Q β ). 5) 6 6 Proof: The proof of Lemma is deferred to the Appendix. The next proposition gives us apriori L bounds, independent of L. 8

11 Proposition Let Ω R be a bounded and simply-connected open set with smooth boundary. Let Q L) be a global minimizer of the Landau-De Gennes energy functional ), in the space ). Then where s + is defined in 0). Q L) L Ω) s + 6) Proof. Proposition has been proven in []; we reproduce the proof here for completeness. The proof proceeds by contradiction. In the following we drop the superscript L for convenience. We assume that there exists a point x Ω where Q attains its maximum and Qx ) > s +. On Ω, s +). If Q is a Q = s + by our choice of the boundary condition Q b note that if Q Q min then Q = global minimizer of F[Q] then Q is a classical solution see Section 6. for regularity) of the Euler-Lagrange equations L Q ij = a Q ij b Q ip Q pj ) trq δ ij + c trq ) Q ij. 7) Since the function Q : Ω R must attain its maximum at x Ω, we necessarily have that ) Q x ) 0 8) We multiply both sides of 7) by Q ij and obtain ) L Q = a trq b trq + c trq ) + L Q. 9) We note that where a trq ) b trq ) + c trq ) ) f Q ) 0) f Q ) = a Q b 6 Q + c Q 4, ) since trq ) Q 6 from 5). One can readily verify that f Q ) > 0 for Q > s + ) which together with 9) and 0) imply that ) Q x) > 0 ) for all interior points x Ω, where Qx) > s +. This contradicts 8) and thus gives the conclusion. In what follows, let e L Qx)) denote the energy density e L Qx)) def = Q + f B Qx)) L. We consider the normalized energy on balls Bx,r) Ω = {y Ω; x y r} We have: FQ,x, r) def = e L Qx))dx = r Bx,r) r Bx,r) f B Q) L + Q dx. 4) 9

12 Lemma Monotonicity lemma) Let Q L) be a global minimizer of F LG [Q] in ). Then FQ L), x,r) FQ L),x,R), x Ω,r R, so that Bx, R) Ω 5) Proof. The proof follows a standard pattern see for instance [0]) and is a consequence of the Pohozaev identity. We assume, without loss of generality, that x = 0 and 0 < R < d0, Ω), where d denotes the Euclidean distance. Since Q L) is a global energy minimizer, it is a classical solution see Section 6. for regularity) of the system 4): [ Q ij = f ] B Q) + b δ ij L Q ij trq ) 6) In 6) and in what follows, we drop the superscript L for convenience. We multiply 6) by x k k Q ij, sum over repeated indices and integrate over B0,R) to obtain the following 0 = Q ij,ll x) x k k Q ij x) B0,R) L = Q ij,ll x) x k k Q ij x)dx B0,R) f B Qx)) x k k Q ij x) δ Q ij L b ij } {{ } I trq x)) x k k Q ij x)dx f B Qx)) x k k Q ij dx B0,R) L Q ij }{{} II 7) where we have used the tracelessness condition Q ii = 0. Integrating by parts, we have that: I = = Q ij,l δ lk Q ij,k x) + x k Q ij,kl x))dx + B0,R) B0,R) = Q ij,l x)q ij,l x)dx + B0,R) B0,R) Q ij,l x)q ij,l x) x k x k R dx + Q ij,ll x)x k k Q ij x)dx B0,R) B0,R) B0,R) Q ij,l x k Q ij,k x l R dx Q ij,lx)q ij,l x)dx Q ij,k x) x k ) dx 8) R II = Hence 7) becomes: B0,R) f B Qx)) x k k Q ij x)dx = L Q ij L = f B Qx)) dx + L L B0,R) B0,R) B0,R) k fb Qx)) x k dx f B Qx)) xk x k R dx 9) Q ij,l x)q ij,l x) + f B Qx)) B0,R) L = R Q ij,k x)q ij,k x) dx + R B0,R) B0,R) Q ij,k x) x k ) dx + B0,R) + f B Qx)) L dx f B Qx)) dx 40) L 0

13 We have R R B0,R) Q ij,l x)q ij,l x) + f B Qx)) L dx ) = R + R B0,R) B0,R) Q ij,l x) Q ij,l x) Q ij,l x) Q ij,l x) The right-hand side of 4) is positive from 40) and hence the conclusion. + f B Qx)) L + f B Qx)) L dx dx. 4) Lemma W, convergence to harmonic maps) Let Ω R be a simply-connected bounded open set with smooth boundary. Let Q L) be a global minimizer of F LG [Q] in the admissible class A Q defined in ). Then there exists a sequence L k 0 so that Q L k) Q 0) strongly in W, Ω; S 0 ), where Q 0) is the limiting harmonic map defined in 5). Proof. Our proof follows closely, up to a point, the ideas of Proposition in []. Firstly, we note that the limiting harmonic map Q 0) belongs to our admissible space A Q and since Q 0) x) Q min, a.e. x Ω see Section ) we have that f B Q 0) x) ) = 0 a.e. x Ω. Therefore Ω QL) ij,k x)ql) ij,k x) dx Ω QL) ij,k x)ql) ij,k x) + L f B Q L) x)) dx Ω Q0) ij,k x)q0) ij,k x)dx 4) The Q L) s are subject to the same boundary condition, Q b, for all L. Therefore 4) shows that the W, -norms of the Q L) s are bounded uniformly in L. Hence there exists a weakly-convergent subsequence Q Lk) such that Q Lk) Q ) in W,, for some Q ) A Q as L k 0. Using the lower semicontinuity of the W, norm with respect to the weak convergence, we have that Q ) x) dx Q 0) x) dx 4) Ω Relation 4) shows that f Ω B Q Lk) x))dx L k Ω Q0) ij,k x)q0) ij,k x)dx and hence fb Q Lk) x)) dx 0 as L k 0. Taking into account that f B Q) 0, Q S 0 we have that, on a subsequence L kj, f B Q L k ) j x)) 0 for almost all x Ω. From Proposition 8, we know that f B Q) = 0 if and only if Q Q min i.e. if Q = s + n n Id) for n S. On the other hand, the sequence Q Lk) converges weakly in W, and, on a subsequence, strongly in L to Q ). Therefore, the weak limit Q ) is of the form Q ) x) = s + n ) x) n ) x) ) Id, n ) x) S, a.e. x Ω 44) Ω It was proved in [] see also [4]) that if Q ) W, and the domain Ω is simply-connected, we can assume, without loss of generality, that n ) W, Ω, S ) and its trace is n b. Then 44) implies Q ) x) = s + n ) x) for a.e. x Ω. Also, recalling the definition of Q 0) from Section we have Q 0) x) = s + n 0) x) for a.e. x Ω. Combining 4) with 6) and the observations in the previous paragraph, we obtain Ω n) x) dx = Ω n0) x) dx and Ω Q) x) dx = Ω Q0) x) dx. Then: Ω Q 0) x) dx liminf L kj 0 Ω Q L k ) j x) dx lim sup Q L k ) j x) dx Q 0) x) dx, L kj 0 Ω Ω which demonstrates that lim Lkj 0 Q L k j ) L = Q 0) L. This together with the weak convergence Q L k j ) Q 0) suffices to show the strong convergence Q L k j ) Q 0) in W,. The following has an elementary proof, that will be omitted:

14 Lemma 4 The function f B : S 0 R + is locally Lipschitz. We can now prove the uniform convergence of the bulk energy density in the interior, away from the singularities of the limiting harmonic map Q 0). Proposition 4 Let Ω R be a simply-connected bounded open set with smooth boundary. Let Q L) W, Ω,S 0 ) denote a global minimizer of F LG [Q] in the admissible class A Q. Assume that we have a sequence {Q L k) } k N so that Q L k) Q 0) in W, Ω,S 0 ) as L k 0. For any compact K Ω such that Q 0) has no singularity in K we have and the limit is uniform on K. lim f B Q Lk) x)) = 0 x K 45) L k 0 Proof. Lemma shows that the strong limit Q 0) is a limiting harmonic map, as defined in Section, Q 0) = s + n 0) x) n 0) x) Id) where n0) x) W, Ω, S ) a global energy minimizer of the harmonic map problem, subject to the boundary condition n = n b on Ω. Let α L k = f B Q L k) x 0 )), for x 0 K an arbitrary point. Proposition and Lemma 4 imply that there exists a constant β independent of x 0 ) so that for any x,y Ω,L > 0. We then have f B Q L) x)) f B Q L) y)) β Q L) x) Q L) y) 46) α L k f B Q L k) x)) + β Q L k) x) Q L k) x 0 ) f B Q L k) x)) + β Q L k) L K ) x x 0 f B Q L k) x)) + Cβ Lk x x 0, x K 47) where K Ω is a compact neighborhood of K to be precisely defined later. In the last relation above we use Lemma A. from [] and the apriori bound given by Proposition. For reader s convenience we recall that Lemma A. in [] states that if u is a scalar-valued ) function such that u = f on Ω R n then ux) C f L Ω) u L Ω) + dist x, Ω) u L Ω) where C is a constant that depends on n only. In our case the constant C depends on the dimension, n =, on a,b, c and on the distance sup y K dy, Ω) only. From 47) we have that α L k Cβρ k Lk f B Q L k) x)), x K, x x 0 < ρ k 48) We argue similarly as in [] and divide by L k and integrate over B ρk x 0 ) to obtain: ρ k α Lk Cβρ k ) L k Lk B ρk x 0 ) f B Q L k x)) L k dx 49) Take an arbitrary ε > 0. Recall that K is a compact set that does not contain singularities of Q 0). Then there exists a larger compact set K, so that K K, that does not contain singularities either, and a constant C K such that Q 0) x) < C K, x K. For R 0 small enough, with R 0 < distk, Ω) and such that Bx 0,R 0 ) K, x 0 K we have Q 0) x) dx 4π 6 C K R 0 ε, x 0 K 50) R 0 B R0 x 0)

15 We fix an R 0 as before. As Q L k) Q 0) in W,, we have that there exists an L 0 > 0 so that: R 0 B R0 x 0 ) Q Lk) x) dx < R 0 B R0 x 0 ) Q 0) x) dx + ε, for L k < L 0, x 0 K 5) The arguments in [] fail to work in our case as we have a three dimensional domain, unlike in the quoted paper, where the domain is two dimensional. In our case, using the monotonicity formula from Lemma and taking ρ k < R 0 we obtain: B ρk x 0 ) f B Q L k) x)) L k dx ρ k Q Lk) x) + f B Q Lk) x)) ε dx ρ k R 0 B R0 x 0 ) L k + ε ) 5) for L k < L with L small enough so that f B Q R 0 B L k ) x)) R0 x 0 ) L k dx < ε note that there exists such an L as the proof of Lemma shows that f B Q L k ) x)) Ω L k dx = o) as L k 0). We take ρ k = α L k Lk Cβ. Then, from 49) and 5) we obtain α L k < 8 Cβ) ε for L k < min{ L 0, L }. As ε > 0 is arbitrary and the estimate on α Lk = f B Q L k) x 0 )), x 0 K is obtained in a manner independent of x 0, we have the claimed result. We also need the following Lemma 5 There exists ε 0 > 0 so that: f B Q) C i,j= f ) B Q) + b δ ij Q ij trq ) C f B Q) Q S 0 such that Q s + n n Id) ε 0, for some n S 5) where s + = b + b 4 +4a c 4c and the constant C is independent of Q, but depends on a,b,c. Proof. Recall from Proposition 8, [] that f B Q) 0 and f B Q) = 0 Q = s + n n Id) with s + = b + b 4 +4a c 4c and n S. Let the eigenvalues of Q be x,y, x y. We define Fx,y) def = a x + y + xy) + b xyx + y) + c x + y + xy) and D def def = min x,y) R Fx,y). Then Fx,y) = Fx,y) D = f B Q). Then F = 0 only at three pairs x,y) namely s +, s + ), s +, s + ) and s +, s + ). On the other hand we have f ) B + b δ ij Q ij trq ) = a 4 trq ) + b4 6 a c )trq )) i,j= where we used the identity trq 4 ) = trq) ) +c 4 trq )) + a b trq ) b c trq )trq ) 54), valid for a traceless symmetric matrix) by If we denote hq) = i,j= fb Q) Q ij + b δ ij trq )) we have hq) = Hx,y) where H : R R is given

16 Hx,y) def = a 4 x + y + xy) + 4 b4 6 a c )x + y + xy) + 8c 4 x + y + xy) We claim that there exist ε,ε,ε > 0 so that +b c xyx + y)x + y + xy) 6a b xyx + y) Fx,y) Hx,y) C C Fx,y), x, y) B ε s +, s + ),B ε s +, s + ),B ε s +, s + ) 55) which gives the conclusion. We prove the inequality 55) only for x,y) B ε s+, s+ ); the other two cases can be dealt with similarly. Careful computations show: H s +, s + ) = H x s +, s + ) = H y s +, s + ) = 0 H y s +, s + ) = H x s +, s + ) = 4b4 + 6a c ) b4 + a c + b b4 + a c 4c 4 H x y s +, s + ) = b4 a c ) b4 + a c + b b4 + 4a c 4c 4 F s +, s + ) = F x s +, s + ) = F y s +, s + ) = 0 F y s +, s + ) = F x s +, s + ) = 4c b4 + a c + b b 4 + 4a c ) Let x 0,y 0 ) = s+, s+ ). We have F x y s +, s + ) = a Hx, y) Fx,y) = H x,y) + R H x, y) F x,y) + R Fx,y) where H x,y) = x x 0 ) H x x 0, y 0 ) + x x 0 )y y 0 ) H x y x 0,y 0 ) + y y 0 ) H y x 0,y 0 ) and F x,y) = x x 0 ) F x x 0, y 0 ) + x x 0 )y y 0 ) F x y x 0,y 0 ) + y y 0 ) F y x 0,y 0 ) with R H, R F the remainders in the Taylor expansions around x 0,y 0 ). From the definition of Taylor expansions, we have that there exists ε 0 > 0 so that on B ε x 0,y 0 ) we have R H x, y) H x, y) and R Fx,y) F x,y), x,y) B ε x 0,y 0 ) 56) On the other hand we have F x,y) 8b 4 + 6a c ) H x,y) F x,y)8b 4 + 6a c ) x, y) R 57) hence, combining 56) and 57), we get: 4

17 Fx,y) 4b 4 + 6a c ) Hx, y) Fx,y)4b 4 + 6a c ), x,y) B ε s +, s + ) 58) which yields claim 55) for x,y) B ε s +, s + ). The other two cases can be analyzed analogously. We continue by proving a Bochner-type inequality that is crucial for the derivation of uniform in L) Lipschitz bounds, away from the singularities of the limiting harmonic map. This type of inequalities were first used to the best of our knowledge) in the context of harmonic maps see [9] and the references there) and later adapted to other, more complicated contexts see for instance [6]). The main difficulty in the proof of Proposition 5 to follow) is the derivation of the next lemma. Lemma 6 There exists ε 0 > 0 and a constant C > 0, independent of L, so that for Q L) a global minimizer of F LG [Q] in the admissible space A Q, we have e L Q L) )x) Ce LQ L) x)) 59) provided there exists a ball B ρx) x) for some ρx) > 0 such that Q L) y) s + my) my) Id) < ε 0 with my) S, for all y B ρx) x). Proof. In the following we drop the superscript L for convenience. We have: ) Qij,k Q ij,k = Q ij,k )Q ij,k Q ij,kl Q ij,kl [ f ] [ B k Qx)) + b δ ij L Q ij L trq ) Q ij,k = f ] B k Qx)) Q ij,k 60) L Q ij On the other hand: [ ] L f B Qx) = f ) [[ B k Qx)) k Q ij = f ] ] B k Qx)) + b δ ij L Q ij L Q ij L trq ) k Q ij = f ) B Qx)) + b δ ij L Q ij L trq ) Q ij f ) B k Qx)) Q ij,k }{{} L Q ij }{{} =Z def = Z f ) B k Qx)) Q ij,k 6) L Q ij We take ε > 0 a small number, to be made precise later. For any such ε we can pick ε 0 > 0 small enough so that if the eigenvalues of Qx) are λ,µ, λ µ) then one of the three numbers λ + s+ ) + µ + s+ ) + λ + µ + s+ ), λ + s+ ) + µ s+ ) + λ + µ s+ ), λ s+ ) + µ + s+ ) + λ + µ s+ ) is less than or equal to ε this can be done because the eigenvalues are continuous functions of matrices, [7], and the matrix s + n n Id) has eigenvalues s +, s + and s + ). Note moreover that we need to choose ε 0 to be smaller than the choice of ε 0 ) in Lemma 5 as we will need to use that lemma in the remainder of this proof. For the matrix Qx), let us denote its eigenvectors by n x),n x),n x) and let λ x),λ x), λ x) = λ x) λ x) denote the corresponding eigenvalues. From the preceeding discussion, we can, without loss of generality, assume that λ + s + ) + λ + s + ) + λ + λ + s + ) < ε 6) 5

18 We define the matrix Q x def = s + n x) n x) s + n x) n x) + s + n x) n x) Note that there exists a mx) S so that Q x = s + mx) mx) Id)). Taking into account 6) and the fact that Qx) and Q x have the same eigenvectors, we have : trqx) Q x ) = λ + s + ) + λ + s + ) + λ + λ + s + ) < ε 6) fb Using the of Taylor expansion of Q ij Q mn Qx)) around Q x we obtain: f B Qx)) = f B Q x ) + f B Q x )Q pq x) Q x Q ij Q mn Q ij Q mn Q ij Q mn Q pq) + R ijmn Q x,qx)) 64) pq where R ijmn Q x,qx)) is the remainder. From 64) we have: f ) B k Qx) Q ij,k = f B Q mn,k Q ij,k = L Q ij L Q ij Q mn = f B Q x )Q mn,k Q ij,k f B Q x )Q pq x) Q x L Q ij Q mn L Q ij Q mn Q pq)q ij,k Q mn,k pq }{{} 0 δ L i,j,m,n= C 0δ L Σ i,j,m,n= L Rijmn Qx),Q x )Q ij,k Q mn,k ) Q pqx) Q x f B Q ij Q mn Q pq Q x ) pq) + C 0δ R ijmn ) Qx),Q x L ) + δ Q 4 i,j,m,n= [ C ) f 0 Q x ) + ] Q pq x) Q x Q ij Q mn Q pq) pq δ Q 4 C δ L trqx) Qx ) + δ Q 4 65) where 0 < δ < and C 0, C 0,C are independent of L and x. For the first term in the second line above we use the fact that the Hessian matrix of a function f B Q) is non-negative definite at a global minimum which holds true in our case as well, as one can easily check, even though we have f B Q) restricted to the linear space S 0 ). Let us recall from the proof of the previous lemma) the definitions of F and F. Then, for a matrix Q S 0 with eigenvalues λ,λ, λ λ ) we have f B Q) = Fλ,λ ) 66) We claim that for ε > 0 small enough there exists C independent of L,λ, λ so that C λ + s + ) + λ + s + ) + λ + λ + s + )) Fλ, λ ) for all λ,λ ) so thatλ + s + ) + λ + s + ) + λ + λ + s + ) < ε. 67) 6

19 Careful computations show: F s +, s + ) = F λ s +, s + ) = F λ s +, s + ) = 0 F λ s +, s + ) = F λ s +, s + ) = 4c b4 + a c + b b 4 + 4a c ) F λ λ s +, s + ) = a Using a Taylor expansion around λ,λ ) = s +, s + ) we have Fλ,λ ) = 8c b 4 + a c + b b 4 + 4a c ) [λ + s + ) + λ + s + ) ] + +a λ + s + )λ + s + ) + Rλ, λ ) { 8c b 4 + a c + b ) b 4 + 4a c [λ + s + ) + λ + s + ) ] + a λ + s + )λ + s } + ) where Rλ,λ ) is the remainder in the Taylor expansion, and the inequality holds provided that the remainder R is small enough. We choose ε > 0 to be small enough so that if λ + s+ ) +λ + s+ ) +λ +λ + s+ ) < ε then R is small enough and the inequality above holds. As the quadratic form 6c b 4 + a c + b b 4 + 4a c ) [λ + s+ ) +λ + s+ ) ]+ a λ + s+ )λ + s+ ) is positive definite, there exists a C > 0, depending only on a,b and c such that { 8c b 4 + a c + b ) b 4 + 4a c [λ + s + ) + λ + s + ) ] + a λ + s + C λ + s + ) + λ + s + ) + λ + λ + s ) + ) Combining this last inequality with 68) we obtain the claim 67). )λ + s + } ) λ,λ ) R The relation 67) together with 66) and 6) show that trqx) Q x ) C fb Qx)) for some C independent of L and x, which combined with 65) shows f ) B k Qx) Q ij,k δc 4 L Q ij L f B Qx)) + δ Qx) 4 with C 4 a constant independent of L and x and any δ > 0. This last inequality together with 60) and 6) show: e L + L f ) B + b δ ij Q ij trq ) δc 4 L f B Q) + δ Q 4 i,j= Taking into account Lemma 5 and choosing δ small enough depending only on C 4 and the constant C from Lemma 5) we can absorb the term δc4 L f B Q) on the right hand side into the left hand side and obtain 68) giving the desired conclusion. e L δ Q 4, 7

20 Lemma 7 Let Ω R be a simply-connected bounded open set with smooth boundary. Let Q L k) W, Ω,S 0 ) be a sequence of global minimizers for the energy F LG [Q] in the admissible space A Q. Assume that as L k 0 we have Q L k) Q 0) in W, Ω,S 0 ). Let K Ω be a compact set which contains no singularity of Q 0). There exists C > 0,C > 0, L 0 > 0 all constants independent of L k ) so that if for a K, 0 < r < da, K) we have e Lk Q L k )x)) dx C r B r a) then r sup e Lk Q Lk) ) C. B r a) for all L k < L 0. Proof. Taking into account our assumptions on the sequence Q ) L k), Proposition 4 shows that for k N any given ε 0 smaller than ε 0 in Lemma 5 and also smaller than the ε 0 in Lemma 6, we have that there exists a L 0 so that for L k < L 0 we have Q L k) x) s + nx) nx) Id ) ε 0, x K, for some nx) S 69) We continue reasoning similarly as in [9]. We fix an arbitrary L k < L 0 and an a Ω and take a r > 0 so that 0 < r < min{da, Ω),da,K)}. We let r > 0 and x B r a) be such that max 0 s r r s) max B sa) e L k Q L k) ) = r r ) max B r a) e L k Q L k) ) = r r ) e Lk Q L k) x )) Define e L k) def = max Br a) e Lk Q L k )). Then: max e L k Q Lk) ) max e L k Q Lk) ) x ) a) B / r r B / r+r / r r ) max Br a) e Lk Q L k) ) / r / r + r )/) = 4 max B r a) e L k Q L k) ) = 4e L k) 70) where for the first inequality we use the fact that B /r r ) x ) B / r+r ) a) and for the second inequality, we use the definition of r. ) q e L k ) Let r = / r r ) and define R Lk) x) = Q L k) x + q x e L k ). We let L k = e L k) L k and then e Lk R Lk) ) = RL k) + f B R Lk) ) = Q Lk) L k e L k) Equation 70) then implies max x B r 0) e L k R L k) ) = max x B /r r ) x ) + f B Q L k) ) e L k) e L k) = L k e Lk Q Lk) x)) 4 e L k e Lk Q L k) ) where the equality above follows from the definition of r and R L k) and the inequality above follows from equation 70). Thus, we have 8

21 where R L k) satisfies the following system of elliptic PDEs max e Lk R Lk) ) 4, e Lk R Lk) )0) = 7) B r 0) L k R L k) ij,kk = a R L k) ij b R L k) ik R L k) kj δ ) ij trrl k) ) ) + c R L k) ij trr Lk) ) ) 7) We now claim that r 7) It is clear that r implies the conclusion. Let us assume for contradiction that r >. Then we claim that there exists a constant C > 0, independent of L k, so that C e Lk R Lk) )x) dx B 74) The matrix R L k) satisfies the system 7) which is the rescaled version of 4) ); using relation 69) and the definition of R L k) as well as the fact that r >, we can apply Lemma 6 to e Lk R L k) ) and obtain e Lk R L k) x)) Ce Lk R L k) x)) 7) 4Ce Lk R L k) x)), x B 0) Combining 7) and the Harnack inequality see for instance [], Ch.4, Thm. 9.) along with the above relation we obtain 74). We have e Lk R Lk) x)) dx R Lk) x) + f B R Lk) x)) dx = B r B r 0) L k e L k) = e Lk Q Lk) x))dx e Lk Q Lk) x)) dx / r r B / r r )/x ) r B r/ x ) e Lk Q Lk) x))dx C 75) r B r a) where for the first inequality we use the monotonicity inequality Lemma ) and the assumption that r note that the equation satisfied by R Lk), equation 7) is the same as the equation satisfied by Q Lk), up to a different elastic constant, hence the use of Lemma here is justified). For the equality in relation 75) we use the change of variables y = x + x q e L k and use the relation: e Lk R L k) ) = e L k ) e Lk Q L k) ). For the second inequality in 75) we use the monotonicity inequality and the fact that /r r r. For the third inequality in 75) we use the fact that B r/ x ) B r a) since x a < r < r. The last step in 75) follows from the hypothesis of the Lemma. Choosing C small enough we reach a contradiction with 74) which in turn implies that r and hence the conclusion. We can now prove the uniform convergence of Q L k) away from singularities of the limiting harmonic map Q 0) : Proposition 5 Let Ω R be a simply-connected bounded open set with smooth boundary. Let Q L k) W, Ω,S 0 ) be a sequence of global minimizers for the energy F LG [Q] in the admissible space A Q. Assume that as L k 0 we have Q L k) Q 0) in W, Ω,S 0 ). 9

22 Let K Ω be a compact set which contains no singularity of Q 0). Then lim k QL k) x) = Q 0) x), uniformly for x K 76) Proof. From the hypothesis and Proposition 4 we have that f B Q Lk) x)) 0 uniformly in K. Thus for any ε 0 > 0 there exists a L 0 > 0 such that for L k < L 0 we have that Q Lk) x) s + nx) nx) Id) < ε 0 for all x K and for each x K, we have nx) S ). Thus we can apply Lemmas 5, 6 and 7. In order to show the uniform convergence it suffices to show that we have uniform independent of L k ) Lipschitz bounds on Q L k) x) for x K. We reason similarily to the proof in Proposition 4 see also [6]). We first claim that there exists an ε > 0 so that ε 0,ε ), there exists r 0 ε) depending only on ε, Ω,K, and boundary data Q b so that QL k) x) + f B Q Lk) x)) dx ε, x K, provided that L k < L ε,r 0 ε)) 77) L k r 0 K B r0 x) In order to prove the claim let us first recall that Q 0) has no singularities on the compact set K. Thus there exists a larger compact set K with K K and a constant C > 0 so that Q 0) x) C, x K. We choose ε > 0 so that Bx,ε ) K K hence for an arbitrary ε 0, ε ) there exists r 0 ε) > 0 so that K B r0 x) r 0 K B r0 x) Q0) x) dx < ε provided that x K and r 0 ε) is chosen small enough. We also have, from the W, Ω,S 0 ) convergence of Q Lk) to Q 0), that there exists Lε) so that r 0 QL k) x) x)dx r 0 Q0) x) dx + ε, L k < Lε) r 0 Ω Recall from the proof of Lemma that lim Lk 0 f B Q L k ) x)) L k K B r0 x) Ω f B Q L k ) x)) L k dx = 0. Hence there exists Lε) so that dx < ε, L < Lε). Letting L ε,r 0 ε)) = min{ L, L} and combining the two relations above we obtain the claim 77). Choosing ε > 0 smaller than the constant C from Lemma 7, we apply Lemma 7 to conclude that Q L k) x) can be bounded, independently of L k, on the set K. The uniform convergence result now follows. 4. The analysis near the boundary In this section we consider the behaviour of a global minimizer Q L) near the boundary, Ω, in the limit L 0. For x 0 Ω we define the region Ω r to be: Ω r def = Ω B r x 0 ), r > 0. 78) Lemma 8 Let Ω be a simply-connected, bounded open set with Lipschitz boundary. There exists a constant D > 0, depending only on Ω, and a constant r 0 > 0 such that for all r < r 0 and for any x 0 Ω, we have: H Ω B r x 0 )) Dr. 79) 0

23 Proof. Since Ω has Lipschitz boundary, we have that for any x 0 Ω, there exists a λx 0 ) > 0 and an orthonormal coordinate system X = x,x,x ) such that x 0 = 0, 0,0) and there exists a Lipschitz function, f x 0 : R R, with the property U x 0 def = {x Ω, x i < λx 0 ),i =,,} = {x R,x < f x 0x,x ), x i < λx 0 ),i =,,}. As Ω is bounded, it is necessarily uniformly Lipschitz see for instance [0]). Hence, for each x 0 Ω, we can choose the system of coordinates as before such that there exists a constant c > 0, independent of x 0, so that f x 0 c, x 0 Ω. Letting r 0 def = λ we have: H Ω B r x 0 )) + fx 0x,x ) dx dx + c dx dx = 4 + c r [ r,r] [ r,r] r < r 0. We have a boundary analogue of the interior mononicity lemma, Lemma, namely : Lemma 9 boundary monotonicity) Let Ω be a simply-connected bounded open set with smooth boundary. Let Q L) be a global minimizer of F LG [Q] in the admissible class A Q. Let E r = r Ωr Q L) + f B Q L) ) L dv 80) Then there exists r 0 > 0 so that where the positive constant C is independent of L. d dr E r C a, b, c,q b,r 0,Ω ), 0 < r < r 0 8) Proof. Step We assume that the domain Ω is star-shaped. Then the proof of 8) closely follows the arguments in [0] combined with an idea from []. Recall that Q L) satisfies the equation: Q L) ij = L [ f B Q L) ) Q L) ij In what follows, we drop the superscript L for convenience. + b δ ij trql) ) We multiply both sides of 8) by x p x 0 p)q ij,p and integrate over Ω r. Then Q ij,kk x p x 0 p)q ij,p dx = Ω r Q ij,k Q ij,p x p x 0 p)n k dσ Ω r Q + Q ij,k Q ij,kp x p x 0 p) dx Ω r 8) where n is the unit outward normal to Ω r and dσ is the area element on Ω r. The integral Ω r Q ij,k Q ij,p x p x 0 p)n k dσ is evaluated by considering the contributions from Ω B r x 0 ) and Ω B r x 0 ) separately. On Ω B r, nx) = x x0 x x 0 so that Q ij,k Q ij,p x p x 0 p)n k dσ = Ω B r Ω B r r ] Q n dσ. 8)

24 Similarly Q ij,k Q ij,p x p x 0 p)n k dσ = x x 0 ) n Q Ω B r Ω B r n + x x 0 ) τ Q b τ Q n dσ where τx) S is the tangential direction to the boundary at x Ω. In order to estimate Ω r Q ij,k Q ij,kp x p x 0 p) dx we note that Q ij,k Q ij,kp x p x 0 p) = [x p x 0p) ] x Q p Q and therefore Q + Q ij,k Q ij,kp x p x 0 p) dx = x p x 0 p) Ω r Ω r Q n p dσ Ω r Q dx. The surface integral over Ω r can again be expressed in terms of separate contributions from Ω B r x 0 ) and Ω B r x 0 ). Combining the above, we have ) Q ij,kk x p x 0 Q p)q ij,p dx = Ω r Ωr dx + r Q Ω B r n Q dσ + [ + x x 0 ) n Q Ω B r n Q b ] dσ + x x 0 ) τ Q b τ Ω B r τ Q dσ. 84) n In 84), we use the fact that Q = Q n + Q b τ on Ω. Using the same sort of arguments as above, we compute L Ω r f B x p x 0 Q p)q ij,p dx = ij L Ω r [ ] fb Q)x p x 0 x p) f B Q) dx 85) p where f B Q) = f B Q b ) = 0 on Ω from our choice of the boundary condition Q b in )). Equating 84) and 85) we obtain ) Q Ωr dx + r Q Ω B r n Q dσ + [ + x x 0 ) n Q Ω B r n Q b ] dσ + x x 0 ) τ Q b τ Ω B r τ Q n dσ = = r f B Q) fb Q) dσ dx. 86) Ω B r L L Ω r We multiply both sides of 86) by r and after some re-arrangement, obtain Q r + f B Q) dx + Q Ω r L r Ω Br + f B Q) dσ = L = Q r n dσ + fb Q) r dx + L + r Ω B r Ω r [ x x 0 Q ) n Ω B r n Q b τ ] dσ + r x x 0 ) τ Q b Ω B r τ Q n dσ. 87)

25 For a star-shaped domain x x 0 ) n 0 on Ω. Therefore, the negative contributions to the right hand side of 87) are r Ω B r x x 0 ) n Q b τ dσ and potentially r Ω B r x x 0 ) τ Q b τ Q n dσ. The first integral can be easily estimated since Q b is known. Using the fact that Q b τ Cs + for some C > 0 as Q b C Ω) by hypothesis) where s + is defined in 0), we have that r x x 0 ) n Q b Ω B r τ dσ Crs +. 88) Here we have used x x 0 ) n r and Lemma 8. Using Cauchy-Schwarz, we have r x x 0 ) τ Q b Ω B r τ Q n dσ r Ω B r Q b τ dσ ) / Ω B r Q n dσ) /. 89) The first integral on the right hand side is easily dealt with i.e. Q b Ω B r τ dσ Cs +r, from Lemma 8. The second integral involving Q is estimated using Lemma 0: n Ω B r where G > 0 is a constant independent of L. Combining the above we have that Q r + f B Q) dx + Ω r L r Q n Ω Br Q dσ G Q b, Ω) + f B Q) L where C and G are positive constants independent of L. We note that d dr E r = Q r Ωr + f B Q) L dx + r dσ Crs + G a,b,c,ω ) 90) Ω Br Q + f B Q) L and the above holds for any 0 < r < r 0 where r 0 is the constant from Lemma 8. Therefore where G > 0 is independent of L. Step : General domain Ω. d dr E r G a, b,c,q b,r 0,Ω ) dσ 9) We do not assume that the domain Ω is star-shaped and take into account the perturbation terms induced by omitting this assumption. As in [0], the boundary regularity of the domain implies that x x 0 ) n x x 0 ) n cr where c > 0 is independent of r or x 0 Ω. Then r x x 0 ) n Q Ω B r n dσ r x x 0 ) n Q Ω B r n The inequality 8) now follows from Lemma 0. dσ c Ω B r Q n 9) dσ. 9) Lemma 0 Let Q L) be a minimizer of F LG [Q] in A Q see )) for a fixed L > 0. Then Q L) n dσ GQ b, Ω) 94) where G > 0 only depends on the boundary condition Q b and Ω. Ω