Abstract
The Landau-de Gennes energy in nematic liquid crystals depends on four elastic constants \(L_1\), \(L_2\), \(L_3\), \(L_4\). In the case of \(L_4\ne 0\), Ball and Majumdar (Mol. Cryst. Liq. Cryst., 2010) found an example that the original Landau-de Gennes energy functional in physics does not satisfy a coercivity condition, which causes a problem in mathematics to establish existence of energy minimizers. At first, we introduce a new Landau-de Gennes energy density with \(L_4\ne 0\), which is equivalent to the original Landau-de Gennes density for uniaxial tensors and satisfies the coercivity condition for all Q-tensors. Secondly, we prove that solutions of the Landau-de Gennes system can approach a solution of the Q-tensor Oseen-Frank system without using energy minimizers. Thirdly, we develop a new approach to generalize the Nguyen and Zarnescu (Calc. Var. PDEs, 2013) convergence result to the case of non-zero elastic constants \(L_2\), \(L_3\), \(L_4\).
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1 Introduction
A liquid crystal is a state of matter between isotropic liquid and crystalline solid. Based on the molecular positional and orientational order of liquid crystals, there are three main types: sematic, cholesterics and nematic. The nematic liquid crystal is the most common type in which the general phases are uniaxial and biaxial. In 1971, de Gennes [11] used Q-tensor order parameters to formulate the elastic energy of liquid crystals with Landau’s bulk energy. The Landau-de Gennes theory has been verified in physics as a successful theory describing both uniaxial and biaxial phases in nematic liquid crystals. Indeed, Pierre-Gilles de Gennes was awarded a Nobel prize for physics in 1991 for his discoveries in liquid crystals and polymers.
In the Landau-de Gennes framework, the space of Q-tensors in the Landau-de Gennes theory is a space of symmetric, traceless \(3\times 3\) matrices defined by
where \({\mathbb {M}}^{3\times 3}\) denotes the space of \(3\times 3\) matrices. When \(Q\in S_0\) has two equal non-zero eigenvalues, a nematic liquid crystal is said to be uniaxial. When Q has three unequal non-zero eigenvalues, a nematic liquid crystal is said to be biaxial. For material constants a, b, c, we define the constant order parameter
and denote the identity matrix by I. The subspace of uniaxial Q-tensors is given by
In this paper, we only consider the case of positive constants a, b, c, which corresponds to a lower temperature regime in liquid crystals (the constant a could also be negative; see [34, 35]).
Let \(\Omega \) be a domain in \({\mathbb {R}}^3\). For a tensor \(Q\in W^{1,2}(\Omega ; S_0)\), the Landau-de Gennes energy is defined by
where \( f_E\) is the elastic energy density with elastic constants \(L_1,\ldots ,L_4\) of the form
and \(f_B(Q)\) is the bulk energy density defined by
with positive material constants a, b and c. Here and in the sequel, we adopt the Einstein summation convention for repeated indices.
In [11], de Gennes discovered the first two terms of the elastic energy density in (1.2) with \(L_3=L_4=0\). Since both the Oseen-Frank theory and the Landau-de Gennes theory should unify for modeling uniaxial liquid crystals, Schiele and Trimper [38] pointed out that the early attempt of de Gennes’ work [11] was incomplete since it would require the splay and bend Frank constants to be equal (i.e. \(k_1=k_3\)) in the Oseen-Frank density (as defined below in (1.5)) of uniaxial tensors \(Q =s_+ (u\otimes u-\frac{1}{3} I)\). However, some experiments on liquid crystals showed that \(k_3>k_1\), so they added a third order term to original de Gennes’ elastic energy density by
with \(L_4=\frac{1}{2s_+^3}(k_3-k_1)>0\). Later, Berreman and Meiboom [6] observed that the above two groups discarded the surface energy density in the Oseen-Frank density, which correlates the blue phase theory for liquid crystals, so they proposed to recover a second order term \(\frac{L_3}{2}\frac{\partial Q_{ik}}{\partial x_j}\frac{\partial Q_{ij}}{\partial x_k}\) with four third order terms, but their density is over-determined with the Oseen-Frank density. Later, Longa et al. [31] gave an extension of the Landau-de Gennes density with 22 independent parameters. Finally, combining the work of Schiele and Trimper [38] with Berreman and Meiboom [2], Dickmann [13] found the full density (1.2), which is consistent with the Oseen-Frank density in (1.5) for uniaxial nematic liquid crystals. Since then, the general form (1.2) of the Landau-de Gennes energy density has been widely used in the study of nematic liquid crystals (e.g. [1, 33, 35]). Under the relation that
one can show (c.f. [33]) that for a uniaxial tensor \(Q =s_+ (u\otimes u-\frac{1}{3} I)\) with \(u\in S^2\),
where the Oseen-Frank density \(W(u,\nabla u)\) is defined by
for a unit director \(u\in W^{1,2} (\Omega ; S^2)\). In (1.5), \(k_1\), \(k_2\), \(k_3\) are the Frank constants for molecular distortion of splay, twist and bend respectively and \(k_4\) is the Frank constant for the surface energy (c.f. [12]). In 1937, Zvetkov established numerical values for p-azoxyanisole (PAA) at \(120^\circ C\) (with the unit \(10^{-12} \,m/J\)) as follows:
Therefore, according to physical experiments on nematic liquid crystals, the elastic constant \(L_4=\frac{1}{2s_+^3}(k_3-k_1)\) is not equal to zero in general (c.f. [12]).
A fundamental problem in mathematics on the Landau-de Gennes theory is to establish existence of a minimizer of the energy functional \(E_{LG}(Q,\Omega )\) in \(W^{1,2}_{Q_0}(\Omega ; S_0)\) with \(L_4\ne 0\). If the functional density \(f_{LG}(Q,\nabla Q)\) satisfies the coercivity condition, one can prove existence of a minimizer of the functional \(E_{LG}(Q,\Omega )\) in \(W^{1,2}(\Omega ; S_0)\). In 2010, Ball and Majumdar [2] found an example where for \(Q\in S_0\), the general Landau-de Gennes energy density (1.2) with \(L_4\ne 0\) does not satisfy the coercivity condition. Very recently, Golovaty et al. [22] emphasized that “From the standpoint of energy minimization, unfortunately, such a version of Landau-de Gennes becomes problematic, since the inclusion of the cubic term leads to an energy which is unbounded from below”. Therefore, the Landau-de Gennes density (1.2) causes a knowledge gap between mathematical and physical theories on liquid crystals, since the energy functional \(E_{LG}(Q,\Omega )\) in \(W^{1,2}(\Omega ; S_0)\) does not satisfy the coercivity condition and violates the existence theorem of minimizers (e.g. [1, 18]). In physics, concerning the third order term \(\frac{L_4}{2} Q_{\alpha \beta }\frac{\partial Q_{ij}}{\partial x_\alpha }\frac{\partial Q_{ij}}{\partial x_\beta }\) with \(L_4\ne 0\) in (1.2), Longa et al. [31] questioned that “In the presence of biaxial fluctuations the general third order theory in \(Q_{\alpha \beta }\) becomes unstable and thus is thermodynamically incorrect". In order to overcome the difficulty, they extended Landau-de Gennes densities through 22 independent second, third, fourth order terms, to preserve the stability of the free energy. Although their result is very interesting, all energy densities in [31] are complicated and have not addressed the above coercivity problem for all Q-tensors. In 2020, following similar spirit in [31], Golovaty et al. [22] proposed a new physical interpretation of the density through fourth order terms to address the above coercivity problem for all Q-tensors. We would like to point out that the new density form in [22] is completely different from the original Landau-de Gennes density (1.2) although the density in [22] can recover the Oseen-Frank density for uniaxial Q-tensors.
In this paper, we will propose a new Landau-de Gennes energy density to solve the above coercivity problem with \(L_4\ne 0\). At first, we observe in Lemma 2.1 that for uniaxial tensors \(Q\in S_*\), the original third order term on \(L_4\) in (1.2), proposed by Schiele and Trimper [38, p. 268] in physics, is a linear combination of a fourth order term and a second order term in the following:
In the case of \(L_4\ge 0\), we introduce a new elastic energy density
for all \(Q\in S_0\). We should point out that the fourth order term \(Q_{ln}Q_{kn}\frac{\partial Q_{ij}}{\partial x_l}\frac{\partial Q_{ij}}{\partial x_k}\) in (1.7) is a non-negative square term, so our new Landau-de Gennes density (1.7) for \(Q\in S_0\) satisfies the coercivity condition in mathematics under suitable conditions on \(L_1,\cdots , L_4\). The first three terms in (1.7) keep the original form (1.2) for \(Q\in S_0\) and the new Landau-de Gennes density (1.7) is equivalent to the original density (1.2) for \(Q\in S_*\). We also remark that our fourth order term \(Q_{ln}Q_{kn}\frac{\partial Q_{ij}}{\partial x_l}\frac{\partial Q_{ij}}{\partial x_k}\) is a linear combination of three fourth order terms \(L^{(4)}_5, L^{(4)}_6, L_7^{(4)}\) in [31]; i.e., we verify in Lemma 2.2 that
For a \(Q\in W^{1,2}(\Omega , S_0)\), we introduce a new Landau-de Gennes energy functional
where \(f_{E,1}(Q, \nabla Q)\) has the form (1.7), \(\tilde{f}_B(Q):= f_B(Q)-\min _{Q\in S_0} f_B(Q)\ge 0\) and \(L>0\) is a parameter to drive all elastic constants to zero [5, 17, 35].
Although there are many differences between the Oseen-Frank theory and the Landau-de Gennes theory, it is of great interest in mathematics whether minimizers of the Landau-de Gennes energy functional can approach a minimizer of the Oseen-Frank energy functional. When \(L_2=L_3=L_4=0\) in (1.7), Majumdar and Zarnescu [34] first proved that as \(L\rightarrow 0\), minimizers \(Q_{L}\) of \(E_{L}\) converges to \(Q_*=s_+ (u^*\otimes u^*-\frac{1}{3} I)\), where \(Q_*\) is a minimizer of the Dirichlet energy functional in \(W^{1,2}_{Q_0}(\Omega ; S_*)\). Sine then, there exist many developments on the one-constant approximation (c.f. [1]) and some special cases of unequal constants \(L_2\), \(L_3\), \(L_4\) ( [5, 29]). In theory of liquid crystals, the general expectation on the elastic constants is that \(L_4\) is not always zero (c.f. [38, p. 268], [2]). For the case of \(L_4\ne 0\), we first prove
Theorem 1
Let \(L_1\), \(L_2\), \(L_3\) and \(L_4\) be elastic constants satisfying
Then, for each \(L>0\), \(f_{LG}(Q,\nabla Q)\) in (1.8) satisfies the coercivity condition so that there exists a minimizer \(Q_L\) of the functional (1.8) in \(W^{1,2}_{Q_0}(\Omega ; S_0 )\) with boundary value \(Q_0\in W^{1,2}(\Omega ; S_* )\). As \(L\rightarrow 0\), the minimizers \(Q_L\) of \(E_L(Q;\Omega )\) converge (up to a subsequence) strongly to \(Q_*\) in \(W^{1,2}_{Q_0}(\Omega ; S_0)\) and satisfies
Furthermore, \(Q_*\) is a minimizer of the functional \(E(Q; \Omega ):=\int _{\Omega } f_{E,1}(Q, \nabla Q)\,dx\) for all uniaxial Q-tensors in \(W^{1,2}_{Q_0}(\Omega ; S_* )\).
Remark 1
In the case of \(L_4<0\), for each \(Q\in W^{1,2}(\Omega , S_0)\), we introduce an elastic energy density by
It is clear that \(|Q|^2|\nabla Q|^2-Q_{ln}\frac{\partial Q_{ij}}{\partial x_l}Q_{kn}\frac{\partial Q_{ij}}{\partial x_k}\ge 0\) for each \(Q\in W^{1,2}(\Omega , S_0)\), and that the elastic energy density \(f_E(Q,\nabla Q)\) in (1.2) is equal to \(f_{E,-}(Q,\nabla Q)\) for uniaxial tensors \(Q\in S_*\).
Next, we discuss critical points of the Landau-de Gennes energy functional (1.7) in \(W^{1,2}_{Q_0}(\Omega ; S_0)\). One can write \(f_{E}(Q,\nabla Q):=\frac{\alpha }{2} |\nabla Q|^2+V(Q, \nabla Q)\) for some \(\alpha >0\) so that \(V(Q, \nabla Q)\ge 0\) for all \(Q\in W^{1,2}_{Q_0}(\Omega ; S_0)\). Then, the Euler-Lagrange equation for the Landau-de Gennes energy functional (1.8) in \(W^{1,2}_{Q_0}(\Omega ; S_0)\cap L^{\infty }(\Omega ; S_0)\) is
in the weak sense, where \(A^T\) denotes the transpose of A,
For general elastic constants \(L_1, \cdots \), \(L_4\), we cannot find any reference having an explicit form of the Euler-Lagrange equation of \(E(Q; \Omega )\) for \(Q=s_+ (u\otimes u-\frac{1}{3} I)\in S_*\) with \(u\in S^2\), so we give an explicit form of the Euler-Lagrange equation in the following:
which is equivalent to the Oseen-Frank system for \(u\in S^2\), where \(\langle A, B\rangle = {{\,\textrm{tr}\,}}(B^TA)\) is the standard inner product of two matrices A and B. In the case of \(L_2=L_3=L_4=0\), the Euler-Lagrange Eq. (1.12) reduces to
which is equivalent to the harmonic map equation of u (c.f. [36]).
Since the Landau-de Gennes theory has been successfully used for modeling both uniaxial and biaxial states of nematic liquid crystals, it is of great interest whether the Q-tensor type of the Oseen-Frank system can be approximated by the Landau-de Gennes system (1.11) without using minimizers. In general, the problem of the convergence of solutions of the Landau-de Gennes Eq. (1.11) without using minimizers is open. Indeed, Gartland [17] pointed out that the convergence of solutions of the Landau-de Gennes Eq. (1.11) is similar to the convergence of solutions of the Ginzburg-Landau approximate equation from superconductivity theory. The Ginzburg-Landau functional was introduced in [20] to study the phase transition in superconductivity. For a parameter \(\varepsilon >0\), the Ginzburg-Landau functional of \(u:\Omega \rightarrow {\mathbb {R}}^3\) is defined by
The Euler-Lagrange equation for the Ginzburg-Landau functional is
Using the cross product of vectors, Chen [7] first proved that as \(\varepsilon \rightarrow 0\), solutions \(u_{\varepsilon }\) of the Ginzburg-Landau system (1.14) weakly converge to a harmonic map in \(W^{1,2}(\Omega ; {\mathbb {R}}^3)\). Chen and Struwe [9] proved global existence of partial regular solutions to the heat flow of harmonic maps using the Ginzburg-Landau approximation. In [3, 4], Bethuel, Brezis and Hélein obtained many results on asymptotic behavior for minimizers of \(E_{\varepsilon }\) in two dimensions as \(\varepsilon \rightarrow 0\) (see also [39]). Recently, many works ( [15, 25,26,27]) have examined the convergence of the Ginzburg-Landau approximation for the Ericksen-Leslie system with unequal Frank’s constants \(k_1, k_2, k_3\). Motivated by the above results on the Ginzburg-Landau approximation, it is natural to investigate the converging problem on solutions of the Landau-de Gennes system (1.11) as \(L\rightarrow 0\). By comparing with the result of Chen [7] (see also [8]) on the weak convergence of solutions of the Ginzburg-Landau equations, it is interesting to study whether solutions \(Q_L\) of the Landau-de Gennes equations (1.11) with uniform bound of the energy converge weakly to a solution \(Q_*\) of (1.12) in \(W^{1,2}_{Q_0}(\Omega ; S_0 )\). However, it seems that the problem is not clear when \(L_2\), \(L_3\), \(L_4\) are not zero. Under a condition, we solve this problem and prove:
Theorem 2
Let \(Q_L\) be a weak solution to the equation (1.11) with a uniform bound in L. Assume that the solution \(Q_L\) converges strongly to \(Q_*\) in \(W^{1,2}_{Q_0}(\Omega ; S_0 )\) as \(L\rightarrow 0\) and satisfies
Then, \(Q_*\) is a weak solution to (1.12).
For the proof of Theorem 2, we use a concept of a projection \(\pi \) in a neighborhood \(S_{\delta }\) of the space \(S_*\), where
For a sufficiently small \(\delta >0\), there exists a smooth projection \(\pi : S_{\delta }\rightarrow S_*\) so that for any \(Q\in S_{\delta }\), \(\pi (Q)\in S_*\) (c.f. [9]). By the projection \(\pi \), we consider the modified bulk energy density
so that the Hessian of F(Q) is positive definite for each any \(Q\in S_{\delta }\) with a sufficiently small \(\delta >0\). Then, choosing a suitable test function and using (1.15), we employ Taylor’s expansion of F(Q) to cancel the limit term involving \(\frac{1}{L}\nabla _{Q_{ij}} f_B(Q_L)\). With these results, we divide the domain into three parts and then employ Egoroff’s theorem to prove Theorem 2.
When \(L_2=L_3=L_4=0\), Majumdar and Zarnescu [34] first proved that minimizers \(Q_L\) of \(E_{LG}(Q; \Omega )\) uniformly converge to \(Q_*\) away from the singular set of \(Q_*\) since there exists a monotonicity formula for minimizers \(Q_L\) of \(E_{LG}(Q; \Omega )\) in \(W^{1,2}(\Omega , S_0)\). Later, Nguyen and Zarnescu [36] improved the result by proving local smooth convergence of minimizers \(Q_L\) away from the singular set of \(Q_*\). However, in the general case of non-zero elastic constants \(L_2\), \(L_3\) and \(L_4\), there exists no such monotonicity formula for minimizers \(Q_L\) of \(E_{LG}(Q; \Omega )\) in \(W^{1,2}(\Omega , S_0)\), so it is a very interesting question whether one can improve the convergence of \(Q_L\) for general cases. For this question, we generalize Nguyen and Zarnescu’s result in [36] to the case of non-zero elastic constants \(L_2, L_3, L_4\) as follows:
Theorem 3
For each \(L>0\), let \(Q_L\) be a weak solution to the equation (1.11) and let \(Q_*\) be the limiting map of \(Q_L\) in Theorem 2. Assume that \(Q_L\) is smooth and converges to \(Q_*\) uniformly inside \(\Omega \backslash \Sigma \), where \(\Sigma \) is the singular set of \(Q_*\). Then, as \(L\rightarrow 0\) (up to a subsequence), we have
for any positive integer \(k\ge 0\).
We would like to point out that our proof of Theorem 3 is new and different from one in [36]. We outline main steps as follows:
Step I. For each \(Q \in S_\delta \), there exists a rotation \(R(Q)\in SO(3)\) such that \({{\tilde{Q}}}=R^T(Q)QR(Q)\) is diagonal. For any \(\xi \in S_0\), we prove
where \(\lambda =\min \{3a,s_+ b \}>0\). For each smooth \(Q(x)\in S_0\), \(R^T(Q(x))Q(x)R(Q(x))\) is diagonal. Then there exists a measure zero set \(\Omega _0\) such that Q(x) has a constant multiplicity of eigenvalues inside subdomains of \(\Omega \backslash \Omega _0\) and R(Q(x)) is almost differentiable in \(\Omega \). Using the geometric identity
with \(i=1, 2, 3\), we apply (1.17) to obtain
for a uniform constant C in L, where \(\phi \) is a cutoff function in \(B_{r_0}(x_0)\subset \Omega \backslash \Sigma \).
Step II. Using Step I with the technique of ‘filling hole’ on elliptic systems [18], we establish a uniform Caccioppoli inequality for solutions \(Q_L\) of (2.10) in L; i.e., there exists a uniform constant C independent of L such that
for any \(x_0\) with \(B_{r_0}(x_0)\subset \Omega \backslash \Sigma \) and any \(r\le r_0\), where 
.
Step III. Based on the uniform Caccioppoli inequality (1.18), we apply the well-known Gagliardo-Nirenberg interpolation (c.f. [28] or [15]) to obtain a control on the local \(L^3\)-estimate; i.e., there exists a uniform constant \(r_0>0\) such that
for some small \(\varepsilon _0>0\). Then, combining (1.19) with (1.17), we apply an induction method to obtain uniform estimates on higher derivatives \(\nabla ^kQ_L\) in L and prove Theorem 2.
Remark 2
When \(L_4\) is sufficiently small, the \(L^p\)-theory on the constant elliptic system (c.f. [18]) can assure that the weak solution \(Q_L\) to the equation (1.11) is smooth in \(\Omega \). In the case of \(L_4=0\), Contreras and Lamy [10] proved that the minimizers \(Q_L\) uniformly converge to \(Q_*\) in \(\Omega \backslash \Sigma \) by assuming that \(Q_L\) is uniformly bounded.
Remark 3
In a recent work [16] with Yu Mei, we can expand ideas on proofs of Theorems 2-3 to show that solutions for the Beris-Edwards system for biaxial Q-tensors converge smoothly to the solution of the Beris-Edwards system for uniaxial Q-tensors up to its maximal existence time.
Finally, we make some remarks about new forms of the Landau-de Gennes energy density through a strong Ericksen’s condition on the Oseen-Frank density. Recently, Golovaty et al. [22] proposed a novel form of the Landau-de Gennes energy density through the Oseen-Frank density. In addition to Ericksen’s inequalities that \(k_2\ge |k_4|, k_3\ge 0, 2k_1\ge k_2+k_4\), they also assumed that
However, their result is not optimal. In fact, Golovaty et al. [22] made a further remark that their assumption (1.20) could be relaxed if one includes more cubic terms in [31], but they did not do it. In Sect. 5, we improve their result to derive an implicit form of \(f_E(Q,\nabla Q)\). Assuming the strong Ericksen inequality with a weaker assumption that \(2k_3> k_2+k_4 \) instead of the condition (1.20), we write the Oseen-Frank density into a new form, which satisfies the coercivity condition for all \(u\in {\mathbb {R}}^3\). In [12], de Gennes and Prost remarked that the bending constant \(k_3\) is much larger than others \(k_1\) and \(k_2\). Therefore, the assumption \(2k_3> k_2+k_4 \) is satisfied through a strong Ericksen’s condition. We give an explicit form of \(f_E(Q,\nabla Q)\) in Proposition 5.1, which satisfies the coercivity condition for all \(Q\in S_0\).
The paper is organized as follows. In Sect. 2, we prove Theorem 1. In Sect. 3, we prove Theorem 2. In Sect. 4, we prove Theorem 3. In Sect. 5, we obtain new forms of the Landau-de Gennes energy density through a strong Ericksen’s condition.
2 The coercivity condition and existence of minimizers
At first, we note
Lemma 2.1
For a uniaxial \(Q= s_+(u\otimes u-\frac{1}{3} I)\in S_*\) with \(u\in S^2\), we have
Proof
Using the fact that \(|u|=1\), we have
Through the identity (2.2), we obtain
\(\square \)
Recall from Longa et al. [31] that
Then we have
Lemma 2.2
For a uniaxial \(Q\in S_*\), we obtain
Proof
Let \(Q= s_+(u\otimes u-\frac{1}{3} I)\) for \(u\in S^2\). Noting that \(u_i\nabla u_i =0\), we calculate
Similarly, we can calculate other terms to obtain
Moreover, we calculate
It follows from using (2.1) and (2.4) that
We can verify from [31] that
Substituting (2.6) into (2.5), we have
\(\square \)
Under the condition (1.9), one can verify from Lemma 1.2 in [30] that there are two uniform constants \(\alpha >0\) and \(C>0\) such that the form \(f_{E,1}(Q,p)\) also satisfies
for any \(Q\in {\mathbb {M}}^{3\times 3}\) and \(p \in {\mathbb {M}}^{3\times 3}\times {\mathbb {R}}^3\). Since \( f_{E,1}(Q,p)\) is quadratic in p and satisfies (2.7), it can be checked (c.f. [28]) that \( f_{E,1}(Q,p)\) is uniformly convex in p; that is
Now we give a proof of Theorem 1.
Proof
Under the condition on \(L_1, \cdots , L_4\) in Theorem 1, it is clear from using Lemma 1.2 in [30] that
By the standard theory in the calculus of variations (c.f. [17]), there exists a minimizer \(Q_{L}\) of \(E_L\) in \(W_{Q_0}^{1,2}(\Omega ; S_0)\). For each \(Q\in W^{1,2}_{Q_0}(\Omega ; S_0)\), we set
It implies that
for any \(Q\in W^{1,2}_{Q_0}(\Omega ; S_*)\) with the fact that \({{\tilde{f}}}_B(Q)=f_B(Q)- \inf _{S_0} f_B=0\).
As \(L\rightarrow 0\), minimizers \(Q_{L}\) converge (possible passing subsequence) weakly to a tensor \(Q_*\in W^{1,2}(\Omega ; S_0)\) with that \({{\tilde{f}}}_B( Q_*)=0\), which implies that \(Q_*\in S_*\) a.e. in \(\Omega \). Then, for any \(Q\in W_{Q_0}^{1,2}(\Omega ; S_*)\), we have
Therefore \(Q_*\) is also a minimizer of E in \(W^{1,2}_{Q_0}(\Omega ; S_*)\). Choosing \(Q=Q_*\) in the above inequality, it implies that
Moreover, it is known that
It implies that \(\int _{\Omega } |\nabla Q_*|^2\,dx= \liminf _{L\rightarrow 0}\int _{\Omega } |\nabla Q_L|^2\,dx\). Otherwise, there is a subsequence \(L_k\rightarrow 0\) such that
Then
This is impossible. Therefore, minimizers \(Q_{L_k}\) converge strongly, up to a subsequence, to a minimizer \(Q_*=s_+(u_*\otimes u_*-\frac{1}{3} I)\) of E in \(W^{1,2}_{Q_0}(\Omega ; S_0)\). Following from Lemma 3.1, \(Q_*\) satisfies (1.12) and \(u_*\) is a minimizer of the Oseen-Frank energy in \(W^{1,2}(\Omega ; S^2)\). Due to the well-known result of Hardt, Kinderlehrer and Lin [23], \(u_*\) is partially regular in \(\Omega \) (see also [24]). Thus \(Q_*\) is partially regular. \(\square \)
Lemma 2.3
If Q is a minimizer of \(E_{L}(Q;\Omega )\) from (1.8) in \(W^{1,2}_{Q_0}(\Omega ; S_0)\), it satisfies
in the weak sense, where \(V_{p^k_{ij}}:=V_{p^k_{ij}}(Q,p)\) with \(p=(\nabla _k Q_{ij})\).
Proof
For any test function \(\phi \in C^\infty _0(\Omega ; S_0)\), consider \(Q_t:=Q+t\phi \) for \(t\in {\mathbb {R}}\). Then for all \(\phi \in C^\infty _0(\Omega ; S_0)\), we calculate
where we used the fact that \(\phi \) is traceless. This proves our claim. \(\square \)
In the case of \(L_2=L_3=L_4=0\), Majumdar and Zarnescu [34] proved that the weak solution of (1.11) is bounded by using a maximum principle. However, when \(L_2\), \(L_3\), \(L_4\) are non-zero, the system (1.11) is a nonlinear elliptic system, so there exists no such maximum principle for it (e.g. [17, 21]). Therefore, it is not clear whether each minimizer \(Q_L\) of \(E_{L}(Q; \Omega )\) in \(W_{Q_0}^{1,2}(\Omega , S_0)\) is bounded and the energy density \(f_{E,1}(Q, \nabla Q)\) in (1.8) can be bounded above by \(C|\nabla Q|^2 +C\). Without this above growth condition on the density, it is a well-known fact that a minimizer \(Q_L\) of the Landau-de Gennes energy functional in \(W^{1,2}_{Q_0}(\Omega ; S_0 )\) may not satisfy the Euler-Lagrange equation in \(W^{1,2}(\Omega , S_0)\). To overcome this difficulty, we can introduce a smooth cutoff function \(\eta (r)\) in \([0, \infty )\) so that \(\eta (r)=1\) for \(r\le M\) with a very large constant \(M>0\) and \(\eta (r)=0\) for \(r\ge M+1\). Then for each \(Q\in W^{1,2}(\Omega , S_0)\), one can modify the Landau-de Gennes density by
with the property that
For a large \(M>0\) in (2.9), we consider a modified Landau-de Gennes functional
Then we obtain
Lemma 2.4
Let \(Q_L\) be a weak solution to the equation (1.11) with the boundary value \(Q_0\in W^{1,2}(\Omega ; S_*)\) associated to the functional \({{\tilde{f}}}_E(Q,\nabla Q)\) in (2.9). Then, \(|Q_L|\le M+1\) for a sufficient large M.
Proof
Recall from the definition of \({{\tilde{f}}}_E(Q,\nabla Q)\) in (2.9) that for a \(Q\in S_0\) with \(|Q|\ge M+1\),
Similarly to one in [8], choose a test function \(\phi =Q (1-\min \{1, \frac{M+1}{|Q|}\})\). Multiplying (1.11) by the test function \(\phi \), we have
Note the fact that \( \nabla _k |Q|^2=2Q_{ij} \nabla _k Q_{ij} \). The above second term is non-negative. For a sufficiently large \(M>0\), the third term is positive. This implies that the set \(\{|Q|\ge M+1\}\) is empty; i.e., \(|Q|\le M+1\) a.e. in \(\Omega \). \(\square \)
The following result is a variant result of Giaquinta-Giusti [19] (see more details in page 206 of [18]):
Proposition 2.1
For each \(L>0\), let \(Q_L\) be a bounded minimizer of (2.10) in \(W^{1,2}_{Q_L}(\Omega ; S_0 )\). Then there exists an open set \(\Omega _L\subset \Omega \) such that \(Q_L\in C_{loc}^{1, \alpha }(\Omega \backslash \Omega _L)\) for each \(\alpha <1\). Moreover, there is a small constant \(\varepsilon _0\) independent of \(Q_L\) such that
and the Hausdorff measure \({\mathcal {H}}^{q}(\Sigma _L )=0\) with \(0<q<1\).
3 Proof of Theorem 2
At first, let us recall that for a uniaxial tensor \(Q\in W^{1,2}_{Q_0}(\Omega ; S_* )\), its energy is given by
where \(f_{E}(Q, \nabla Q)=\frac{\alpha }{2} |\nabla Q|^2+V(Q, \nabla Q)\). Then we have
Lemma 3.1
If Q is a minimizer of \(E(Q; \Omega )\) in \(W^{1,2}_{Q_0}(\Omega ; S_*)\), it satisfies
in the weak sense.
Proof
Let \(\phi \in C^\infty _0(\Omega ; {\mathbb {R}}^3)\) be a test function. For each \(u_t = \frac{u+t\phi }{|u+t\phi |}\) with \(t\in {\mathbb {R}}\), we define
For any \(\eta \in C^\infty _0(\Omega ; S_0)\), we choose a test function \(\phi \) such that \(\phi _{i}:=u_k\eta _{ik} \). If Q is a minimizer, the first variation of the energy of Q is zero; that is
Note that
where we used the fact that \(|u|=1\) and \(\phi _{i}=u_l\eta _{il} \). Then we have
Using the fact that \(\nabla _k |u+t\phi |^2 =0\) at \(t=0\) and substituting \(\phi _{i}:=u_l\eta _{il} \), a simple calculation shows
In the special case of \(\frac{1}{2} \int _{\Omega } {|\nabla Q|^2}\,dx\), it follows from using (3.3) and \(\langle Q, \nabla Q\rangle =0\) that
for all \(\eta \in C^\infty _0(\Omega ; S_0)\).
For the term \(V(Q,\nabla Q)\), using (3.2)-(3.3) and integrating by parts, we have
Combining above two identities (3.4) and (3.5), we prove Lemma 3.1. \(\square \)
Corollary 1
Assume that \(Q=s_+(u\otimes u-\frac{1}{3} I)\). Then \(Q=(Q_{ij})\) is a solution of equation
if and only if u is a harmonic map from \(\Omega \) into \(S^2\); i.e., \(-\Delta u=|\nabla u|^2 u\).
Now we give a proof of Theorem 2.
Proof
For each \(L>0\), let \(Q_L\) be a weak solution to the equation (1.11) with boundary value \(Q_0\in W^{1,2}(\Omega , S_*)\) and assume that \(Q_L\) is uniformly bounded in \(\Omega \).
For each \(\delta >0\), define a set
For each \(Q\in \Sigma _{\delta }\), we have \(\pi (Q)\in S_*\); i.e., \( \pi (Q) =s_+\left( u\otimes u-\frac{1}{3} I\right) \) with \(u\in S^2\).
For a \( \pi (Q_L) =s_+\left( u_L\otimes u_L-\frac{1}{3} I\right) \) with \(u_L\in S^2\) and a test function \(\phi \in C^\infty _0(\Omega ; {\mathbb {R}}^3)\) with a small \(t\in {\mathbb {R}}\), we set \(u_{L, t}:= \frac{u_L+t\phi }{|u_L+t\phi |}\). Then we have
For any \(Q\in S_{\delta }\), set
Using the Taylor expansion of \(F\left( (\pi (Q_L))_t\right) \) at \(Q_L\in S_{\delta }\), we derive
where \(Q_{\tau _1}:=(1-\tau _1) (\pi (Q_L))_t+\tau _1 Q_L\) for some \(\tau _1\in [0,1]\). Note that
Since \((\pi (Q_L))_t\in S_*\), it implies that \(F((\pi (Q_L))_t)=0\). Note that the function F(Q) is smooth in Q. For sufficiently small t, we have \(|Q_{\tau _1}-(\pi (Q_L))_t|\le 3\delta \) for \(Q_L\in S_{2\delta }\). For each \(Q\in S^*\), it is known that the Hessian of \({{\tilde{f}}}_B(Q)\) is semi-positive definite at \(Q=Q^*\). Therefore each \(Q\in S_{\delta }\), the Hessian of \(F_B(Q)\) is positive definite with sufficiently small \(\delta >0\); i.e., for any \(Q_L\in S_{2\delta }\), we have
with sufficiently small t and \(\delta \). Then it follows from (3.8)-(3.9) that
provided \(\Omega _{L,2\delta }=\{x\in \Omega : Q_L(x)\in S_{2\delta }\}\) for \(\delta >0\).
By using Young’s inequality, we have
where we used a result in [36] or Corollary 2 in Sect. 3.
In order to extend (3.11) to \(\Omega \), we define
It can be checked that \({\hat{Q}}_{L,t}\in W^{1,2}_{Q_0}(\Omega ; S_0)\). Then
On the other hand, there exists a uniform bound \(C(\delta )>0\) such that for all \(x\in \Omega \backslash \Omega _{L,\delta }\), \(\tilde{f}_B(Q_L(x)) \ge C(\delta )\). Using Lemma 2.4, we observe that
By the assumption (1.15) in Theorem 2, we deduce from (3.11) and (3.14) that
Multiplying (1.11) by \(({{\hat{Q}}}_{L,t}-Q_L)\) and using (3.15) yield
Here we used the fact that \({{\hat{Q}}}_{L,t}-Q_L\) is symmetric and traceless.
In order to pass a limit, we claim that \({\hat{Q}}_{L,t}\rightarrow Q_{*,t}\) strongly in \(W^{1,2}_{Q_0}(\Omega ; S_0)\).
In fact, it follows from (3.13) that
Note that
When \(Q_L\) approaches to \(Q_*\), \(\nabla _{Q}\pi (Q_{\xi })\) is close to the identity map I and \(\nabla _{Q}\pi (Q_{\xi })_t\) for small t. Therefore
As \(Q_L\rightarrow Q_*\), the term \(\pi (Q_L)_t\) is close to \(\pi (Q_*)_t\) and \(\nabla _Q\pi (Q_\xi )_t\) is close to the identity map for small t. Note that \(\nabla ^2_{QQ}\pi (Q_\xi )_t\) is bounded. Then
Then the inequality (3.17) reads as
Here we employ the Egoroff theorem; i.e., for all \(\varepsilon >0\), there exists a measurable subset \(\Sigma _{\varepsilon } \subset \Omega \) such that
As \(\varepsilon \rightarrow 0\) and \(L\rightarrow 0\), we prove the claim that \({{\hat{Q}}}_{L,t}\rightarrow Q_{*,t}\) strongly in \(W^{1,2}_{Q_0}(\Omega ; S_0)\).
We observe that
and
Using the uniform convergence of \(Q_L\) in \(\Omega \backslash \Sigma _\varepsilon \) and strong convergence of \({{\hat{Q}}}_{L,t}, Q_L\) in \(W^{1,2}_{Q_0}(\Omega ,S_0)\), we derive
As \(L\rightarrow 0\), the estimate (3.16) yields
For each \(\eta \in C_0^{\infty }(\Omega ,S_0)\), we define
In view of (3.2) and (3.3), we have
For the estimate (3.19), the limit in t exists. Dividing (3.19) by t then as \(t\rightarrow 0^+\) and \(t\rightarrow 0^-\), we have
Repeating the same steps in (3.4) and (3.5), we prove that \(Q_*\) satisfies (1.12). \(\square \)
4 Smooth convergence of solutions
In this section, we will prove Theorem 3. At first, we derive some key lemmas.
For any tensor \(Q \in S_0\), there exists a rotation \(R(Q)\in SO(3)\) such that \({{\tilde{Q}}}:=R^T(Q) Q R(Q)\) is diagonal. Moreover, the space \(S^*\) has only three diagonal tensors so for each \(Q\in S_*\), we assume that
Lemma 4.1
For any \(Q\in S_\delta \) and \(\xi \in S_0\) with a sufficiently small \(\delta >0\), the Hessian of the bulk density \(f_B(Q)\) satisfies the following estimate
where \(\lambda =\min \{3a,s_+ b \}>0\) and \({{\tilde{Q}}}=R^T(Q) Q R(Q)\) is diagonal.
Proof
For a fixed \(\pi (Q_0)\in S_*\), there exists a rotation \(R(\pi (Q_0))\in SO(3)\) in (4.1) such that \(R^T(\pi (Q_0))\pi (Q_0) R(\pi (Q_0))=Q^+\). For \(i=1, 2, 3\), we calculate the first derivative of \(f_B( {\tilde{Q}})\) by
Then the second derivative of \(f_B({\tilde{Q}})\) with \(i,j=1, 2, 3\) is
For the case of \(i=j\) at \(Q=Q_0\), \({\tilde{Q}}=Q^+\), From the equality \(\frac{2}{3} cs^2_+=\frac{1}{3}bs_++a \) (c.f. [34]), we find
Then, at \({\tilde{Q}}=Q^+\), we have
For the case of \(i\ne j\), at \({\tilde{Q}}=Q^+\), we observe that
In conclusion, using the fact that \(\xi _{33}=-(\xi _{11}+\xi _{22})\), we have at \({\tilde{Q}}=Q^+\)
with \(\lambda =\min \{3a,s_+b\}>0\). Then
Due to the fact that \(|{{\tilde{Q}}}-Q^+| = |Q-\pi (Q)|\), we prove (4.2) for a sufficiently small \(\delta >0\). \(\square \)
Corollary 2
For any \(Q\in S_{\delta }\) with a sufficiently small \(\delta >0\), there exists constants \(C_1,C_2,C_3>0\) such that
Proof
It follows from the Taylor expansion of \({{\tilde{f}}}_B(Q)\) at \(\pi (Q)\) that
where \(Q_\tau \) is an intermediate point between \(Q_L\) and \(\pi (Q)\).
Since Q commutes with \(\pi (Q)\) (c.f. [36]), they can be simultaneously diagonalized. Note that \({{\tilde{f}}}_B(\pi (Q))=0\), \(\nabla _{Q_{ij}}f_B(\pi (Q))=0\) and \(Q_\tau \) is sufficiently close to \(\pi (Q)\). Using Lemma 4.1 with the fact that
we have
Then we obtain
The left-hand side of (4.10) is a direct consequence of (4.12) by using Young’s inequality. Taking Taylor expansion of \( g_B(Q)\) at \(\pi (Q)\) yields
Multiplying both side by \((Q-\pi (Q))_{ij}\) and using (4.13), we obtain (4.11). \(\square \)
From now on, for each \(L>0\), let \(Q_L\) be a solution to the equation (1.11) and assume that \(Q_L\) is smooth and converges to \(Q_*\) uniformly inside \(\Omega \backslash \Sigma \), where \(\Sigma \) is the singular set of \(Q_*\). For a sufficiently small L, \({{\,\textrm{dist}\,}}(Q_L;S_*)\le \delta \) inside \(\Omega \backslash \Sigma \).
Set
and
Due to the fact that \({{\tilde{Q}}}:=R^T(Q) Q R(Q)\) is diagonal, \(g_B({{\tilde{Q}}})=R^T(Q)\, g_B(Q)R(Q)\) is also diagonal for a rotation \(R(Q)\in SO(3)\).
Let Q be differentiable in \(\Omega \). Then there exists a set \(\Sigma _Q\), which has measure zero, such that R(Q) is differentiable in \(\Omega \backslash \Sigma _Q\) (c.f. Corollary 2 [34, 37]). Therefore, we have the following geometric identity of rotations:
Lemma 4.2
Assume that for any \(x\in \Omega \backslash \Sigma _Q\), there exists a differentiable rotation R(Q) such that both \(R^T(Q)QR(Q)\) and \(R^T(Q)h(Q)R(Q)\) are diagonal. Then, for each i, we have
Proof
Let \(x_0\) be a fixed point in \(\Omega \backslash \Sigma _Q\) and fix \(i=1,2,3\). For \(Q_0=Q(x_0)\in S_0\), there exists \(R_0:=R(Q_0)\in SO(3)\) such that \(R_0^TQ_0R_0\) is diagonal. Denote \(\tilde{R}(Q)=R_0^TR(Q)\) with \({{\tilde{R}}}(Q_0)=I\). Fix \(Q_0\in S_0\), there is \(R_0:=R(Q_0)\in SO(3)\) such that \(R_0^TQ_0R_0\) and \(R_0^Th(Q_0)R_0\) diagonal. Denote \({{\tilde{R}}}(Q)=R_0^TR(Q)\), so \({{\tilde{R}}}(Q_0)=I\). Since \(R(Q)\in SO(3)\), \(R_{ki}(Q)R_{kj}(Q)=\delta _{ij}\). Then, for each i, we have at \(x_0\)
Note that \(R^T_0 h(Q_0) R_0\) is diagonal, \(\tilde{R}_{ik}(Q_0)=\delta _{ik}\) and \(\nabla {{\tilde{R}}}_{ii}(Q_0)=0\). It can be seen that the term \(\nabla {{\tilde{R}}}_{ki}(Q)(R^T_0Q_0 R_0)_{kl}{{\tilde{R}}}_{li}(Q)\) at \(Q=Q_0\) is zero. Therefore
Since \(x_0\) is any point, we prove (4.16). \(\square \)
Denote the inner product by \(\langle A,B\rangle =A_{ij}B_{ij}\) for \(A,B\in {\mathbb {M}}^{3\times 3}\). Using the above geometric identity, we have
Lemma 4.3
Let k and l be two integers, for \(x\in \Omega \backslash \Sigma _Q\), \(Q= Q(x)\) and any smooth scalar function \(\phi \), we have
Proof
For \(x_0\in \Omega \backslash \Sigma _Q\), R(x) is differentiable in the neighborhood of \(x_0\). Fixing \(Q_0=Q(x_0)\in S_0\), there exists \(R_0=R(Q(x_0))\) such that \(R_0^TQ_0R_0\) is diagonal. Recall that \({{\tilde{R}}}(Q)=R_0^TR(Q)\), \({{\tilde{R}}}_{ik}(Q_0)=\delta _{ik}\) and \(\nabla {{\tilde{R}}}_{ii}(Q_0)=0\). For any matrix A, let \(A_D\) be the diagonal part of A and \(A_N\) the non-diagonal part of A such that \(A=A_D+A_N\). Recall that \(\nabla ^k g_B({{\tilde{Q}}})\) and \(\nabla ^{k+1} g_B({{\tilde{Q}}})\) are diagonal. By employing an analogous argument in the proof of Lemma 4.2, we obtain
Here we used that
Similarly, using (4.19), at \(Q=Q_0\), we find
Since \(x_0\in \Omega \backslash \Sigma _Q\) is arbitrary, this completes the proof. \(\square \)
Let \(R_L=R(Q_L)\) be a rotation such that \(R^T_LQ_LR\) is diagonal. Then we have
Lemma 4.4
Let \(x_0\in \Omega \) with some \(B_{r_0}(x_0)\subset \Omega \backslash \Sigma \) for a sufficiently small \(r_0\). Then, for any \(\phi \in C^\infty _0 ( B_{r_0}(x_0))\) and \(Q_L\in S_\delta \) with sufficiently small \(\delta \), we have
where C is a constant independent of L.
Proof
Let \(\varphi _\varepsilon \) be a cutoff function such that \(\varphi _\varepsilon (x)=0\) for \({{\,\textrm{dist}\,}}(x,\Sigma _{Q_L})\le \varepsilon \) and \(\varphi _\varepsilon (x)=1\) for \({{\,\textrm{dist}\,}}(x,\Sigma _{Q_L})\ge 2\varepsilon \). Multiplying (1.11) by \(\nabla (\phi ^2 \varphi _{\varepsilon }^2 \nabla Q_L)\) yields
Utilizing Lemma 4.1 with a sufficiently small \(\delta >0\), we derive
Using Lemma 4.2 and (4.23), we have
As \(\varepsilon \) tends to zero, we observe that
It follows from using integrating by parts, (2.8) and Young’s inequality that
Combining (4.24) with (4.25) yields
Integrating by parts and using Young’s inequality, we deduce
Here 
. Note that
and for \(x\in B_r(x_0)\subset \Omega \backslash \Sigma \), \(Q_L(x)\) uniformly converges to \(Q_*(x)\). For a sufficiently small \(r_0\) and L, we see that
Then we conclude that
\(\square \)
As an application of Lemma 4.4, we obtain a uniform Caccioppoli inequality for minimizer \(Q_L\) as follows.
Lemma 4.5
Let \(x_0\in \Omega \) with \(B_{r_0}(x_0)\subset \Omega \backslash \Sigma \) for a sufficiently small \(r_0>0\). Then for any \(r\le r_0\), we have
where 
and C is a constant independent of L.
Proof
For two s, t such that \(\frac{r}{2}\le t< s\le r\), choose a cutoff function \(\phi \in C_0^{\infty } (B_s(x_0))\) such that \(0\le \phi \le 1\), \(\phi =1\) on \(B_t\) and \(|\nabla \phi |\le C/(s-t)\).
Integrating by parts and using Young’s inequality, we have
Then, by Young’s inequality and Lemma 4.4, we obtain
Through the standard technique of ’filling hole’, we have
for \(\theta = \frac{C_1}{1+C_1} <1\) and two s, t such that \(r/2\le t< s\le r\). In view of Lemma 3.1 in Chapter V of [18], the relation (4.21) follows. \(\square \)
Using Lemma 4.5, we have local uniform estimates on higher derivatives.
Lemma 4.6
Let \(x_0\in \Omega \) with \(B_{r_0}(x_0)\subset \Omega \backslash \Sigma \). Assume that there exists a constant \(\varepsilon _0>0\) such that
Then, for any integer \(k\ge 1\), there exist a constant \(r_k\ge r_0/2\) and a positive constant \(C_k\) independent of L such that
Proof
For simplicity of notations, we denote \(Q=Q_L\) and \(R=R_L\). The claim (4.28) is true for \(k=1\). At first, we show the case of \(k=2\).
Assume that there exists a constant \(C_1>0\) such that
Let \(\phi \) be a cutoff function in \(C_0^{\infty }(B_{r_1}(x_0))\), where \(r_2\) satisfies \(\frac{r_0}{2}<r_2<r_1<r_0\) and \(\phi =1\) in \(B_{r_2}(x_0)\). Let \(\varphi _\varepsilon \) be another cutoff function such that \(\varphi _\varepsilon (x)=0\) for \({{\,\textrm{dist}\,}}(x,\Sigma _Q)\le \varepsilon \) and \(\varphi _\varepsilon (x)=1\) for \({{\,\textrm{dist}\,}}(x,\Sigma _Q)\ge 2\varepsilon \). We differentiate (1.11) twice and multiply by \(\nabla ^2 Q\phi ^2\varphi _\varepsilon ^2\) to get
Applying Lemma 4.3 and Lemma 4.1 to the right-hand side of (4.29), we find
where we used that
Observer that, for any fixed \(\varepsilon >0\), R(x) in (4.30) is differentiable. Then it follows from (4.1) that \(Q^+=R^T\pi (Q) R\) and
Letting \(\varepsilon \rightarrow 0\) in (4.32), using (1.11) and the fact that \(|Q-\pi (Q)|\le C| g_B(Q)|\) in (4.11), we find
In view of (4.32)-(4.32), we deduce (4.30) to
Applying (2.8) and Young’s inequality to the left-hand side of (4.29), we obtain
Using Hölder’s inequality, we have
Combining (4.34) with (4.36) and choosing \(\varepsilon _0\) sufficiently small, we obtain
Set \(r_k:=(1-\sum _{i=1}^k2^{-(i+1)})r_0>\frac{r_0}{2}\). For any \(k\le l\), we can assume that there is a constant \(C_k\) such that
As a consequence of the Sobolev inequality, we have
for any \(k\le l-1\).
Next, we prove it for \(k=l+1\). Let \(r_{l+1}\) be the constant satisfying
Let \(\phi \) be a cutoff function in \(C_0^{\infty }(B_{r_{l}}(x_0))\) with \(\phi =1\) in \(B_{r_{l+1}}(x_0)\). We apply \(\nabla ^{l+1}\) to (1.11) and multiply by \( \nabla ^{l+1} Q\phi ^2\varphi _\varepsilon ^2 \) to have
It follows from Lemma 4.3 that
Using a similar argument in (4.31), for \(i\ge 3\), one can check that
for some scalar function Z(x). Observe that \(\partial ^j_{\tilde{Q}}f_B({{\tilde{Q}}})=0\) for \(j\ge 5\). Applying Lemma 4.1 with a sufficiently small \(\delta \) and (4.41) to (4.40), we obtain
As \(\varepsilon \) tends to zero, we obtain from (4.38) and (4.42) that
Using Young’s inequality and integration by parts, we have
In view of (4.37)-(4.38), we have
Combining (4.44) with (4.45), our claim (4.28) follows for \(k=l+1\). \(\square \)
Now we give a proof of Theorem 3.
Proof
For any \(x_0\in \Omega \backslash \Sigma \), let \(B_{2R_0}(x_0)\) be a ball such that \(B_{2r_0}(x_0)\subset \Omega \backslash \Sigma \). From Lemma 4.5, we deduce the following estimate
It follows from (4.21) that
As a consequence of the Gagliardo-Nirenberg interpolation (c.f. [15]), we have
Using Lemma 4.5 with any \(k\ge 1\), we obtain
Then, \(Q_L\) converges smoothly to \(Q_*\) in \(\Omega \backslash \Sigma \). \(\square \)
5 The Landau-de Gennes density through the Oseen-Frank density
In this section, we will obtain a new form of the Landau-de Gennes energy density through the Oseen-Frank density. Under the condition (1.9), it was shown in [33] that for each \(Q =s_+ (u\otimes u-\frac{1}{3} I)\in S_*\), one has
Assuming the strong Ericksen condition
it was pointed out in [28] (see also [1, 14, 15]) that there are positive constants \(\lambda \) and C such that the density \(W(u,\nabla u)\) is equivalent to a new form that \({{\widetilde{W}}}(u,p)\) satisfies
for any \(u\in {\mathbb {R}}^3\) and any \(p\in {\mathbb {M}}^{3\times 3}\). However, there seems no reference for an explicit form of \(\widetilde{W}(u,\nabla u)\), so we give an explicit form \({{\widetilde{W}}}(u,\nabla u)\) here. For \(u\in {\mathbb {R}}^3\), it can be checked that
Lemma 5.1
Assume the Frank constants \(k_1, \cdots ,k_4\) satisfy
Then the density \(W(u, \nabla u)\) of the form (1.5) for each \(u\in S^2\) is equivalent to the new form
where \( \alpha =\min \{k_2-|k_4|,2k_1- k_2-k_4,2k_3- k_2-k_4\}>0\).
Proof
Note that \(W(u, \nabla u)\) is rotational invariant (c.f. [24]); i.e., for each \(R\in SO(3)\), \({{\tilde{x}}}=R(x-x_0)\) and \({{\tilde{u}}} = R u(x)=R u\). Then we have
Then for any \(u\in S^2\), we can find some \(R=R(u(x_0))\in SO(3)\) at each point \(x_0\in \Omega \) such that
Using the relation
for all \(i=1,2,3\), we evaluate four terms of the Oseen-Frank energy density at \({{\tilde{x}}}_0\)
Substituting above identities into the density, we have
where \( {{\tilde{\alpha }}}\) is a positive constant due to the strong Ericksen condition (5.1). Using (5.5), we find
If we further assume that \(2k_3>k_2+k_4\), then we can rewrite (5.6) into
From (5.2), we prove (5.4). \(\square \)
It is clear that the new form \({{\widetilde{W}}}(u,p)\) in (5.4) with \(p=\nabla u\) satisfies
for all \(u\in {\mathbb {R}}^3\) and \(p\in {\mathbb {M}}^{3\times 3}\).
Through the relation (5.4), we can have the new Landau-de Gennes energy density satisfying the coercivity in the following:
Proposition 5.1
Assume that \({{\hat{L}}}_1\), \({{\hat{L}}}_2\), \({{\hat{L}}}_3\) and \({{\hat{L}}}_4\) satisfy the condition
Then for each \(Q\in S_*\), we obtain
where \(Q_i\) is the i-th column of the Q matrix and \({\tilde{\alpha }}\) is given by
Proof
Due to the fact that \(|u|^2=1\), a direct calculation yields
One can verify that
Here we used the fact that \(|Q|=\sqrt{\frac{2}{3}}s_+\). Then we can derive \(f_E(Q,\nabla Q)\) from (5.4) that
It then follows from (5.2) and (5.10) that
Substituting the identities (5.11)-(5.14) into the equation (5.4), we complete a proof. \(\square \)
Data availability.
No datasets were generated or analyzed during the current study.
Change history
21 January 2023
Missing Open Access funding information has been added in the Funding Note.
References
Ball, J.M.: Mathematics and liquid crystals. Mol. Cryst. Liq. Cryst. 647, 1–27 (2017)
Ball, J.M., Majumdar, A.: Nematic liquid crystals: from maier-saupe to a continuum theory. Mol. Cryst. Liq. Cryst. 525, 1–11 (2010)
Bethuel, F., Brezis, H., Hélein, F.: Asymptotics for the minimization of a ginzburg-landau functional. Calc. Var. PDEs 1, 123–148 (1993)
Bethuel, F., Brezis, H., Hélein, F.: Ginzburg-Landau Vortices, Progress in Nonlinear Differential Equations and their Applications 13. Birkha user Boston Inc, Boston, MA (1994)
Bauman, P., Park, J., Phillips, D.: Analysis of nematic liquid crystals with disclination lines. Arch. Ration. Mech. Anal. 205, 795–826 (2012)
Berreman, D.W., Meiboom, S.: Tensor representation of oseen-frank strain energy in uniaxial cholesterics. Phys. Rev. A 30, 1955–1959 (1984)
Chen, Y.: Weak solutions to the evolution problem for harmonic maps into spheres. Math. Z. 201, 69–74 (1989)
Chen, Y., Hong, M.-C., Hungerbühler, N.: Heat flow of \(p\)-harmonic maps with values into spheres. Math. Z. 215, 25–35 (1994)
Chen, Y., Struwe, M.: Existence and partial regular results for the heat flow for harmonic maps. Math. Z. 201, 83–103 (1989)
Contreras, A., Lamy, X.: Singular perturbation of manifold-valued maps with anisotropic energy. Preprint, (2018). arXiv:1809.05170
de Gennes, P.G.: Short range order effects in the isotropic phase of nematics and cholesterics. Mol. Cryst. Liq. Cryst. 12, 193–214 (1971)
de Gennes, P.G., Prost, J.: The Physics of Liquid Crystals, 2nd edn. Oxford University Press, Oxford (1993)
Dickmann, S.: Numerische berechnung von feld und molekülausrichtung in flüssigkristallan-zeigen. PhD thesis, University of Karlsruhe, (1995)
Ericksen, J.L.: Inequalities in liquid crystals theory. Phys. Fluid 9, 1205–1207 (1966)
Feng, Z., Hong, M.-C., Mei, Y.: Convergence of the ginzburg-landau approximation for the ericksen-leslie system. SIAM J. Math. Anal. 52, 481–523 (2020)
Feng, Z., Hong, M.-C., Mei, Y.: Existence and convergence of the Beris-Edwards system with general Landau-de Gennes energy. Preprint, (2021). arXiv preprint arXiv:2112.04074
Gartland, E.C., Jr.: Scalings and limits of landau-de gennes models for liquid crystals: a comment on some recent analytical papers. Math. Model Anal. 23, 414–432 (2018)
Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, New Jersey (1983)
Giaquinta, M., Giusti, E.: On the regularity of the minima of variational integrals. Acta Math. 148, 31–46 (1982)
Ginzburg, V., Landau, L.: On theory of superconductivity. Zh. Eksp. Teor. Fiz. 20, 1064–1082 (1950)
Giusti, E.: Direct Methods in the Calculus of Variations, p. 38. World Scientific, Singapore (2003)
Golovaty, D., Novack, M., Sternberg, P.: A novel landau-de gennes model with quartic elastic terms. Eur. J. Appl. Math. 32, 1–22 (2020)
Hardt, R., Kinderlehrer, D., Lin, F.-H.: Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105, 547–570 (1986)
Hong, M.-C.: Partial regularity of weak solutions of the liquid crystal equilibrium system. Indiana Univ. Math. J. 53, 1401–1414 (2004)
Hong, M.-C.: Global existence of solutions of the simplified ericksen-leslie system in dimension two. Calc. Var. PDEs 40, 15–36 (2011)
Hong, M.-C., Li, J., Xin, Z.: Blow-up criteria of strong solutions to the ericksen-leslie system in \({\mathbb{R} }^3\). Commun. Partial Differ. Equ. 39, 1284–1328 (2014)
Hong, M.-C., Xin, Z.: Global existence of solutions of the liquid crystal flow for the oseen-frank model in \({\mathbb{R} }^2\). Adv. Math. 231, 1364–1400 (2012)
Hong, M.-C., Mei, Y.: Well-posedness of the ericksen-leslie system with the oseen-frank energy in \(L^3_{uloc}({\mathbb{R} }^3)\). Calc. Var. PDEs 58, 1–38 (2019)
Iyer, G., Xu, X., Zarnescu, A.: Dynamic cubic instability in a 2D Q-tensor model for liquid crystals. Math. Model Method Appl. Sci. 25, 1477–1517 (2015)
Kitavtsev, G., Robbins, J.M., Slastikov, V., Zarnescu, A.: Liquid crystal defects in the Landau-de gennes theory in two dimensions - beyond the one-constant approximation. Math. Model Method Appl. Sci. 26, 2769–2808 (2016)
Longa, L., Monselesan, D., Trebin, H.-R.: An extension of the Landau-ginzburg-de gennes theory for liquid crystals. Liq. Cryst. 2, 769–796 (1987)
Madsen, L.A., Dingemans, T.J., Nakata, M., Samulski, E.T.: Thermotropic biaxial nematic liquid crystals. Phys. Rev. Lett. 92, 145505 (2004)
Mori, H., Gartland, E.C., Kelly, J.R., Bos, P.J.: Multidimensional director modeling using the Q tensor representation in a liquid crystal cell and its application to the \(\pi \) cell with patterned electrodes. Jpn. J. Appl. Phys. 38, 135–146 (1999)
Majumdar, A., Zarnescu, A.: Landau-de gennes theory of nematic liquid crystals: the qseen-frank limit and beyond. Arch. Ration. Mech. Anal. 196, 227–280 (2010)
Mottram, N.J., Newton, C.J.P.: Introduction to Q-tensor theory. Preprint (2014). arXiv:1409.3542v2
Nguyen, L., Zarnescu, A.: Refined approximation for minimizers of a landau-de gennes energy functional. Calc. Var. PDEs 47, 383–432 (2013)
Nomizu, K.: Characteristic roots and vectors of a differentiable family of symmetric matrices. Linear Multilinear Algebra 1, 159–162 (1973)
Schiele, K., Trimper, S.: Elastic constants of a nematic liquid crystal. Phys. Stat. Sol. B 118, 267–274 (1983)
Struwe, M.: On the asymptotic behavior of minimizers of the ginzburg-landau model in 2 dimensions. Differ. Int. Eqs. 7, 1613–1624 (1994)
Acknowledgements
We would like to thank Professors John Ball and Arghir Zarnescu for their useful comments on our early version (arXiv:2007.11144). This article supersedes that version.
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Feng, Z., Hong, MC. Existence of minimizers and convergence of critical points for a new Landau-de Gennes energy functional in nematic liquid crystals. Calc. Var. 61, 219 (2022). https://doi.org/10.1007/s00526-022-02321-5
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DOI: https://doi.org/10.1007/s00526-022-02321-5