The Oseen–Frank Energy Functional on Manifolds

Abstract

We observe that for a unit tangent vector field u ∈ TM on a 3-dimensional Riemannian manifold M, there is a unique unit cotangent vector field A ∈ T^∗M associated to u such that we can define the curl of u by dA. Through a unit cotangent vector field A ∈ T^∗M, we define the Oseen–Frank energy functional on 3-dimensional Riemannian manifolds. Moreover, we prove partial regularity of minimizers of the Oseen–Frank energy on 3-dimensional Riemannian manifolds.

1 Introduction

A liquid crystal is a mesomorphic phase of a material which occurs between its liquid and solid phase. The material is composed of rod like molecules which display orientational order, unlike a liquid, but lacking the lattice structure of a solid. In their pioneering works, Oseen [36] and Frank [11] established the static mathematical continuum theory on nematic liquid crystals through a director u, which is the average direction of molecules [35]. There are a lot of analytical and computational issues in study of static equilibrium configurations.

Let ${\Omega }\subset \mathbb {R}^{3}$ be an open bounded domain with smooth boundary ∂Ω. Set

$$ H^{1}({\Omega}; S^{2}) =\left\{u\in H^{1}({\Omega}; \mathbb{R}^{3}):~|u|=1\text{ a.e. on } {\Omega} \right\}. $$

The Oseen–Frank energy associated to a director u ∈ H¹(Ω;S²) is given by

$$ E(u)={\int}_{\Omega} W(u, \nabla u) dx, $$

where W(u,∇u) is the Oseen–Frank free energy density given by

$$ W(u,\nabla u)=k_{1}(\text{div} u)^{2} + k_{2} (u\cdot\text{curl} u)^{2} + k_{3} |u\times \text{curl} u|^{2} + (k_{2}+k_{4})\left[\text{tr}(\nabla u)^{2}-(\text{div} u)^{2}\right], $$

in which k₁, k₂, k₃ are the Frank constants for molecular distortion of splay, twist and bend respectively, and k₄ is the Frank constant for the surface energy (e.g. [35]).

Let γ : ∂Ω → S² be a given smooth boundary data. For any map $u\in H^{1}_{\gamma }({\Omega },S^{2})$, the integral

$$ {\Phi} (\gamma)=\frac 12 {\int}_{\Omega} [\text{tr}(\nabla u)^{2}- (\text{div} u)^{2}]dx $$

is a number depending on only γ (see [20]). Therefore, without loss of generality, we assume

$$ W(u,\nabla u) := k |\nabla u|^{2} + V(u,\nabla u), $$

(1.1)

where $k=\min \limits \{k_{1},k_{2},k_{3}\}>0$ and

$$ V(u,\nabla u) = (k_{1}-k) (\text{div} u)^{2} + (k_{2}-k ) (u \cdot \text{curl} u)^{2} + (k_{3}-k) |u\times \text{curl} u|^{2}. $$

An equilibrium configuration of liquid crystals corresponds to an extremal (critical point) of the functional E. The Euler–Lagrange equations associated with E in H¹(Ω;S²) is

$$ \left( \delta_{ik}-u^{i}u^{k}\right)\left( \nabla_{\alpha} W_{p_{\alpha}^{k}}(u,\nabla u)-W_{u^{k}}(u,\nabla u)\right)=0 $$

(1.2)

for i = 1,2,3. Here and in the sequel, we adopt the Einstein summation convention and denote by δ_ik the Kronecker delta. In the special case of k₁ = k₂ = k₃ = 1 and k₄ = 0, the equation (1.2) is

$$ \bigtriangleup u + |\nabla u|^{2} u=0~\text { in } {\Omega}, $$

which is the equation of harmonic maps from Ω into S². In 1964, Eells and Sampson [8] introduced the study of harmonic maps between two Riemannian manifolds. There are many interesting results on harmonic maps (e.g. [7, 29]). In particular, Giaquinta–Giusti [13, 14] and Schoen–Uhlenbeck [33] proved partial regularity of minimizing harmonic maps. For further developments on harmonic maps, see [18, 24].

Numerical and experimental analysis on liquid crystals has shown that equilibrium configurations of the system (1.2) expect to have point and line singularities. In physics, it is called the one-constant approximation for the special case of k₁ = k₂ = k₃ = 1 and k₄ = 0 (e.g. [35, Section 2.2.1]). For this special case, Brezis, Coron and Leib [4] investigated the local behavior of isolated singularities of energy minimizing maps. Bethuel, Brezis and Coron [3] introduced a relaxed energy for harmonic maps and proved existence of infinitely many weak solutions of harmonic maps (see also [23]). Bethuel and Brezis [2] studied the regularity problem of minimizers of modified relaxed problems for harmonic maps. Using Cartesian currents, Giaquinta, Modica and Soucek [16] proved partial regularity of minimizers of the relaxed energy for harmonic maps. In the same spirit of Sacks–Uhlenbeck [34] and Uhlenbeck [38], Giaquinta, the author and Yin [15] proposed an approximation for the relaxed energy of the Dirichlet energy and proved partial regularity of a minimizer of the relaxed energy for harmonic maps without using Cartesian currents.

In the theory of liquid crystals, the Frank elastic constants k₁, k₂, k₃ in (1.1) are unequal in general. For an example, the work of Zwetkoff in 1937 was mentioned by Stewart in [35] that the Frank elastic constants for para-Azoxyanisole (PPA) at T = 125^∘C are

$$ k_{1}=9 \times 10^{-12}N, \quad k_{2}=5.8\times 10^{-12}N, \quad k_{3}=19 \times10^{-12}N. $$

For the general case of the unequal Frank constants k₁, k₂ and k₃, Hardt, Kinderlehrer and Lin [20, 21] proved that an energy minimizer u is smooth on some open subset Ω₀ ⊂Ω and moreover $\mathcal H^{\beta }({\Omega }\backslash {\Omega }_{0})=0$ for some positive β < 1, where $\mathcal H^{\beta }$ is the Hausdorff measure. Almgren and Lieb [1] did more analysis on singularities of energy minimizing maps when k₁, k₂ and k₃ are close to k. Giaquinta, Modica and Soucek [17] studied the relaxed energy of the Oseen–Frank functional. As we pointed out before, harmonic maps have been extensively studied between two Riemannian manifolds (e.g. [7, 18, 29]), so it is interesting to generalize the Oseen–Frank energy on Riemannian manifolds.

In this paper, we investigate the Oseen–Frank energy functional on 3-dimensional Riemannian manifolds. Let (M,g) be a 3-dimensional Riemannian manifold (with possible boundary). In local coordinates around a point x ∈ M, a smooth Riemannian metric g can be represented by

$$ g = g_{ij} dx_{i}\otimes dx_{j}, $$

where (g_ij) is a positive definite symmetric n × n matrix. Let (g^ij) := (g_ij)^− 1 be the inverse matrix of (g_ij) and the volume element of (M;g) is

$$ dv_{g} = \sqrt{|g|} dx\quad \text{with }~|g|:=\det(g_{ij}). $$

For a unit tangent vector field u ∈ H¹(Ω;TM), we write $u(x)=u^{i}(x) \frac {\partial }{\partial x^{i}}$ in local coordinates with the norm

$$ |u(x)|^{2}=g_{ij} u^{i}(x) u^{j}(x)=1. $$

Although one can define curlu in the tangent space through a normal frame, there is no clear form of curlu. As pointed out in [6, Chapter 3, p. 79], curlu associates to dA through an one-form A. Motivated by this observation, we have

Theorem 1.1

For a unit tangent vector field u ∈ TM, there is a unique unit cotangent vector field A ∈ T^∗M associated to u such that

$$ |d^{\ast} A|^{2} = |\text{div} u|^{2},\quad |\langle A, \ast dA \rangle|^{2} = |u\cdot \text{curl} u|^{2},\quad |A \wedge \ast dA|^{2} = |u\times \text{curl} u|^{2}, $$

where ∗ is the Hodge star operator (e.g. [31]).

Let Ω be a domain in M. For a unit tangent vector field u ∈ H¹(Ω;TM), let A ∈ H¹(Ω;T^∗M) be the unit cotangent vector field associated to u in Theorem 1.1 and denote by ∇A the covariant derivative of A (e.g. [31]). Then, we define the Oseen–Frank energy functional of A in Ω by

$$ E(A; {\Omega} )={\int}_{\Omega} W(A(x), \nabla A(x)) dv_{g}, $$

(1.3)

where W(A,∇A) is the Oseen–Frank energy density defined by

$$ W(A,\nabla A) := k |\nabla A|^{2} + V(A,d^{\ast} A, dA) $$

with $k=\min \limits \{k_{1}, k_{2}, k_{3}\}>0$, satisfying

$$ V(A,d^{\ast} A, dA) = (k_{1}-k ) |d^{\ast} A|^{2}+(k_{2}-k ) |\langle A, \ast dA \rangle|^{2}+(k_{3}-k) |A \wedge \ast dA|^{2}. $$

Denote $T^{\ast }_{M}(S^{2})=\{A\in T^{\ast } M: |A|=1\}$. Then $A\in H^{1}({\Omega }, T^{\ast }_{M}(S^{2}))$ is a weak solution to the liquid crystal system if A satisfies the Euler–Lagrange equation

$$ \begin{array}{@{}rcl@{}} &&k \left[\nabla^{\ast}\nabla A - |\nabla A|^{2} A\right] + (k_{1}-k) \left[d d^{\ast} A -\langle d d^{\ast} A, A\rangle A) \right] \\ &&+(k_{2}-k_{3})\left[d^{\ast}(\langle A, \ast dA \rangle \ast A) - \langle d^{\ast} (\langle A, \ast dA \rangle \ast A), A \rangle A)\right]\\ &&+(k_{3}-k)\left[d^{\ast} dA - \langle d^{\ast} dA, A\rangle A\right] + (k_{2}-k_{3}) \left[\langle A, \ast dA \rangle \ast dA - \langle A, \ast dA \rangle^{2}A \right]=0 \end{array} $$

(1.4)

in the sense of distribution (see details in Section 4).

Then we prove partial regularity of minimizers of the Oseen–Frank energy:

Theorem 1.2

Let A be a minimizer of the Oseen–Frank energy functional (1.3) in $H^{1}_{\gamma }({\Omega }; T^{\ast }_{M}(S^{2}))$, where γ is a given boundary value. Then, A is smooth in a set ${\Omega }_{0}\subset \bar {\Omega }$ and ${\mathscr{H}}^{\beta }(\bar {\Omega }\backslash {\Omega }_{0})=0$ for some positive β < 1, where ${\mathscr{H}}^{\beta }$ is the Hausdorff measure.

The idea of proofs of Theorem 1.2 is to modify an approach of the direct method in [22], which is based on a reverse Hölder inequality. The direct method for elliptic systems was extensively studied in [12] and [19]. Using the minimality, we prove a Caccioppoli’s inequality and a reverse Hölder inequality. Since the principal term in (1.4) is complicated, the liquid crystal system is not a standard elliptic system, which is not discussed in [12] and [19].

We would like to outline some key ideas to handle the extra terms in (1.4). We choose a normal co-frame $\{\omega _{i}\}_{i=1}^{3}$ around x₀ ∈ M such that $\omega _{3}=\frac {A_{x_{0}, R}}{|A_{x_{0}, R}|}$. Set $\tilde A= \tilde A_{1}\omega _{1}+\tilde A_{2}\omega _{2}$. Using the fact that |A| = 1, we can prove

$$ |\nabla(A-\tilde A)|^{2} \leq C\left( |A-A_{x_{0}, R} |+ R^{2}+ R^{-1}{\int}_{B_{R}(x_{0})} |\nabla A|^{2} d v_{g} \right)|\nabla A|^{2}. $$

We rewrite (1.4) into an equation on $\tilde A$ (see (4.4)). Then we can estimate the difficult terms in (1.4). Finally, we modify the freezed coefficient method from [12] to prove partial regularity.

Remark 1

Based on the static Oseen–Frank theory, Ericksen [9] and Leslie [32] proposed a hydrodynamic theory to describe the behavior of liquid crystal flows. Recently, there are a lot of progress about the Ericksen–Leslie system in $\mathbb {R}^{3}$ with unequal Frank constants k₁, k₂, k₃ (e.g. [10, 26,27,28]). Comparing with the result of Struwe [37] and Chen–Struwe [5] on the harmonic map flow between manifolds, it is interesting to study the Ericksen–Leslie system on manifolds for unequal Frank constants k₁, k₂, k₃.

Finally, I would like to dedicate this paper to Professor Jürgen Jost on the occasion of his 65th birthday. I met Jürgen first time at ETH-Zürich in 1994 when I was a postdoctoral fellow under supervision of Professor Michael Struwe. Through our collaboration [25], I learnt a lot of mathematics from Jürgen and Michael. Furthermore, I also learn a lot of knowledge on differential geometry from Jürgen’s books [30, 31].

The paper is organized as follows. In Section 2, we outline geometric setting for the Oseen–Frank energy and prove Theorem 1.1. In Section 3, we prove the Caccioppoli inequality and the reverse Hölder inequality. In Section 4, we prove Theorem 1.2.

2 Geometric Setting for the Oseen–Frank Energy

Let M be a smooth Riemannian 3-manifold M equipped with a Riemannian metric g; i.e., for each tangent space T_xM, there is an inner product 〈⋅,⋅〉. In local coordinates,

$$ g_{ij}:=\left<\frac{\partial}{\partial x^{i}}, \frac{\partial}{\partial x^{j}}\right>. $$

For $X, Y, Z\in C^{\infty }(TM)$, the connection ∇ satisfies

$$ X\langle Y, Z\rangle =\langle \nabla_{X}Y, Z\rangle + \langle Y, \nabla_{X}Z\rangle. $$

The connection, which satisfies the above identity, is called Riemannian. In local coordinates, the Christoffel symbols are defined by

$$ {\Gamma}_{ij}^{k}:=\frac 1 2 g^{kl} \left( \frac{\partial g_{lj}}{\partial x^{i}}+\frac {\partial g_{il}}{\partial x^{j}} - \frac {\partial g_{ij}}{\partial x^{l}}\right) $$

satisfying

$$ \nabla_{\frac {\partial}{\partial x^{i}}}\left( \frac {\partial}{\partial x^{j}}\right) = {\Gamma}_{ij}^{k}\frac{\partial}{\partial x^{k}},\quad \nabla_{\frac {\partial}{\partial x^{i}}}(d x^{j})=-{\Gamma}_{ik}^{j}d x^{k} . $$

We recall that the curvature tensor of Levi-Civita connection R is given by

$$ R(X,Y)Z=\nabla_{X}\nabla_{Y} Z-\nabla_{Y}\nabla_{X} Z-\nabla_{[X,Y]} Z $$

for $X, Y, Z\in C^{\infty }(TM)$. In local coordinates,

$$ R\left (\frac {\partial}{\partial x^{i}},\frac {\partial}{\partial x^{j}}\right )\frac {\partial}{\partial x^{l}} = R^{k}_{lij}\frac {\partial}{\partial x^{k}}. $$

We set

$$ R_{klij}:=g_{km} R^{m}_{lij}=\left<R\left (\frac {\partial}{\partial x^{i}},\frac {\partial}{\partial x^{j}}\right)\frac {\partial}{\partial x^{k}}, \frac {\partial}{\partial x^{l}}\right>. $$

In local coordinates, we have

$$ R^{k}_{lij}= \left( \frac {\partial {\Gamma}_{jl}^{k}}{\partial x^{i}}-\frac {\partial {\Gamma}_{il}^{k}}{\partial x^{j}} + {\Gamma}_{im}^{k}{\Gamma}_{jl}^{m} - {\Gamma}_{jm}^{k} {\Gamma}_{il}^{m} \right). $$

For each x ∈ M, let u(x) be a unit tangent vector in T_xM. In local coordinates, we write $u(x)=u^{i}(x) \frac {\partial }{\partial x^{i}}$ with the norm |u(x)|² = g_ijuⁱ(x)u^j(x) = 1. It follows from [6, Chapter 4, pp. 114–115] that the absolute differential of u is defined by

$$ \nabla u=(du^{i}+u^{j}{\omega_{j}^{i}})\otimes \frac{\partial}{\partial x^{i}}= \left( \frac{\partial u^{i}}{\partial x_{j}}+ u^{k} {\Gamma}_{kj}^{i}\right) dx^{j}\otimes \frac {\partial}{\partial x^{i}}, $$

where ${\omega _{j}^{i}}$ is defined by ${\omega _{j}^{i}}={\Gamma }^{i}_{jk} du^{k}$.

The divergence of the vector $u(x)=u^{i}(x) \frac {\partial }{\partial x^{i}}$ (e.g. [31]) is defined as

$$ \text{div} u=\frac 1{\sqrt{|g|}}\frac{\partial}{\partial x^{j}}\left( \sqrt{|g|} u^{j}\right). $$

(2.1)

For a unit vector $u(x)=u^{i}(x) \frac {\partial }{\partial x^{i}}\in TM$, there is a unique corresponding cotangent vector A in T^∗M defined by

$$ A(x)=A^{i}(x) dx^{i}=g_{ij}u^{j}dx^{i} $$

(2.2)

satisfying

$$ |A|^{2}=\left<g_{ij}u^{j}dx^{i}, g_{kl}u^{l}dx^{k} \right> = g_{ij}g_{kl}g^{ik}u^{l}u^{j} =g_{ij}u^{i}u^{j}=1. $$

Then the absolute differential of A (e.g. [6, Chapter 4]) is given by

$$ \nabla A= (d A^{i}-A^{j} {\omega_{i}^{j}})\otimes du^{i}=\left( \frac {\partial A^{i}}{\partial x^{j}}-A^{k} {\Gamma}_{ij}^{k}\right)du^{j}\otimes du^{i}. $$

Moreover,

$$ dA =dA^{j}\wedge dx^{j}=\frac 12\left( \frac {\partial A^{j}}{\partial x^{i}}- \frac{\partial A^{i}}{\partial x_{j}}\right)dx^{i}\wedge dx^{j}={\sum}_{i<j}\left( \frac {\partial A^{j}}{\partial x_{i}}- \frac{\partial A^{i}}{\partial x_{j}}\right)dx^{i}\wedge dx^{j}. $$

It implies

$$ |dA|^{2} =\frac 1 4 g^{ik}g^{jl}\left( \frac {\partial A^{j}}{\partial x^{i}}- \frac{\partial A^{i}}{\partial x^{j}}\right)\left( \frac {\partial A^{l}}{\partial x^{k}}- \frac{\partial A^{k}}{\partial x^{l}}\right). $$

Let d be the exterior derivative given by

$$ d : {\Omega}^{k}(TM) \to {\Omega}^{k+1}(TM) \quad \text{for an integer}~k \geq 0 $$

and the adjoint operator

$$ d^{\ast}: {\Omega}^{k+1}(TM )\to {\Omega}^{k}(TM) $$

satisfying the property

$$ {\int}_{M} \langle da, b\rangle dv={\int}_{M} \langle a, d^{\ast} b\rangle dv. $$

Let ∗ be the Hoge star operator (e.g. [31]) by

$$ \ast: {\Lambda}^{k} (T^{\ast}_{x} M) \to {\Lambda}^{3-k} (T^{\ast}_{x}M). $$

Using the star operator ∗, we have

$$ \ast dA = {\sum}_{i<j}\left( \frac{\partial A^{j}}{\partial x^{i}}- \frac{\partial A^{i}}{\partial x^{j}}\right) \ast(dx^{i}\wedge dx^{j}) \in {\Lambda}^{1} (T^{\ast}_{x}M). $$

Then

$$ \begin{array}{@{}rcl@{}} \langle A, \ast dA \rangle & = &A^{1}\!\left( \!\frac {\partial A^{3}}{\partial x_{2}} - \frac{\partial A^{2}}{\partial x_{3}}\!\right) \!\langle dx_{1},\ast(dx^{2}\wedge dx^{3}) \rangle + A^{2} \!\left( \!\frac{\partial A^{3}}{\partial x_{2}} - \frac{\partial A^{2}}{\partial x_{3}}\!\right)\langle dx_{2}, \ast(dx^{i}\!\wedge\! dx^{j})\rangle\\ &&+ A^{3} \left( \frac{\partial A^{2}}{\partial x^{1}} - \frac{\partial A^{1}}{\partial x^{2}}\right) \langle dx^{3},\ast(dx^{1}\wedge dx^{2})\rangle. \end{array} $$

Now we complete a proof of Theorem 1.1.

Proof

Through the dual operator d^∗, we have

$$ {\int}_{M} \langle d^{\ast} A, f \rangle d v_{g} = {\int}_{M} \langle A, d f \rangle d v_{g} $$

for a smooth function f with compact support in M. Then we have in local coordinates

$$ d^{\ast} A = -\frac 1{\sqrt{|g|}}\frac{\partial}{\partial x^{j}}\left( \sqrt {|g|} g^{ij} A^{i}\right). $$

It can be checked that d^∗A = −divu through (2.1)–(2.2).

In normal coordinates at each fixed x₀ ∈ M (e.g. [31, p. 21]), we have

$$ g_{ij}(x_{0})=\delta_{ij}, \quad {\Gamma}_{jk}^{i} (x_{0})=0,\quad \frac {\partial g_{ij}}{\partial x_{k}} (x_{0})=0 $$

for i,j,k. At each x₀ ∈ M, we can write $u(x_{0})=u^{i}(x_{0}) e_{i}\in S^{2}\subset T_{x_{0}}M=\mathbb {R}^{3}$ with a normal frame $\{e_{i}=\frac {\partial }{\partial x_{i}}\}$. Let {ω_i = dx_i} be a normal co-frame at x₀. Then at x₀, we have

$$ A=A^{1}dx_{1}+A^{2}dx_{2}+A^{3}dx_{3}=u^{1} \omega_{1} + u^{2} \omega_{2}+u^{3} \omega_{3}. $$

At each x₀, we have

$$ dA = \left( \frac{\partial A^{3}}{\partial x^{2}} - \frac{\partial A^{2}}{\partial x^{3}}\right) \omega_{2}\wedge \omega_{3} + \left( \frac{\partial A^{3}}{\partial x_{1}} - \frac{\partial A^{1}}{\partial x_{3}}\right)\omega_{1}\wedge \omega_{3} +\left( \frac{\partial A^{2}}{\partial x_{1}}- \frac{\partial A^{1}}{\partial x_{2}}\right) \omega_{1}\wedge \omega_{2} $$

and

$$ \ast dA = \left( \frac {\partial A^{3}}{\partial x_{2}} - \frac{\partial A^{2}}{\partial x_{3}}\right) \omega_{1} + \left( \frac {\partial A^{1}}{\partial x_{3}}- \frac{\partial A^{3}}{\partial x_{1}}\right) \omega_{2} + \left( \frac {\partial A^{2}}{\partial x_{1}}- \frac{\partial A^{1}}{\partial x_{2}}\right) \omega_{3}. $$

Using the formula (2.2.1) of Chapter 2 in [31] with |A| = 1, we have

$$ \begin{array}{@{}rcl@{}} |\ast dA |^{2} &=& |\text{curl} u|^{2}, \quad |\langle A, \ast dA \rangle |^{2} = |u\cdot \text{curl} u|^{2},\\ |A \wedge \ast dA|^{2} &=& |A|^{2}|\ast dA|^{2} - |\langle A, \ast dA \rangle|^{2} = |u\times \text{curl} u|^{2}. \end{array} $$

This proves our claim. □

3 Caccioppoli’s Inequality and the Reverse Hölder Inequality

In the section, we will follow the approach in [22] for proving Caccioppoli’s inequality and a reverse Hölder inequality of energy minimizers.

At first, we generalize Hardt–Lin’s extension Lemma in [21]:

Lemma 3.1

Let B_R(x₀) be a geodesic ball in M for all R ≤ R₀ with some R₀ > 0. For any v ∈ H¹(B_R(x₀);T^∗M) with |v| = 1 on ∂B_R(x₀), there exists an one form $w\in H^{1} (B_{R}(x_{0}); T^{\ast }_{M}(S^{2}))$ such that

$$ w=v\quad \text{on }\partial B_{R}(x_{0}) $$

and

$$ {\int}_{B_{R}(x_{0})} |\nabla w|^{2} d v_{g}\leq C{\int}_{B_{R}(x_{0})}(|\nabla v|^{2}+|x-x_{0}|^{2}) dv_{g} $$

for a constant C independent of v, w and R.

Proof

We modify a proof in the Appendix of [21]. At a fixed x₀ ∈ M, there is normal coordinates (e.g. [31, p. 21]) in $B_{R_{0}}(x_{0})$ such that

$$ g_{ij}(x_{0})=\delta_{ij}, \quad \left|\frac{\partial g_{ij}}{\partial x_{k}} (x)\right|\leq C|x-x_{0}| $$

for i,j,k. In the coordinate at x₀, we write

$$ v= v^{i}(x) dx^{i}. $$

Let $\tilde a=a^{i} dx^{i}\in T^{\ast } M$ be the one form corresponding to the constant $a=(a^{1}, a^{2}, a^{3})\in \mathbb {R}^{3}$ with $|\tilde a|\leq \frac 12$. Then we consider a one form

$$ w_{a}(x)=\frac{v(x)-\tilde a}{|v(x)-\tilde a|},\quad x\in {\Omega}. $$

Then at x = x₀

$$ \nabla w_{a}=\frac {\nabla (v-\tilde a)} {| v- \tilde a|}- \frac{(v - \tilde a) \langle v- \tilde a, \nabla v\rangle}{| v - \tilde a|^{3}}. $$

Integrating over Ω with respect to x and over B_1/2 with respect to a, we obtain

$$ \begin{array}{@{}rcl@{}} {\int}_{B_{1/2}} {\int}_{B_{R}(x_{0})} |\nabla w_{a}|^{2} d v_{g} d\tilde a &=& {\int}_{B_{R}(x_{0}) }{\int}_{B_{1/2}} | \nabla w_{a}|^{2} d\tilde a d v_{g}\\ &\leq& C{\int}_{B_{R}(x_{0})} |\nabla (v -\tilde a)|^{2} d v_{g}{\leq} C{\int}_{B_{R}(x_{0})} |\nabla v|^{2}{+}|x-x_{0}|^{2} d v_{g} \end{array} $$

due to the fact that

$$ {\int}_{B_{1/2}} | v -\tilde a|^{-2} d\tilde a\leq K, $$

where K is a positive constant. Hence there exists a point a₀ with $| a_{0}|\leq \frac 12$ such that

$$ \label {3.1} {\int}_{B_{R}(x_{0})} | \nabla w_{a_{0}}|^{2} dx \leq C{\int}_{B_{R}(x_{0})} |\nabla B|^{2} +|x-x_{0}|^{2} d v_{g}. $$

(3.1)

For any a ∈ T^∗M with $|a|\leq \frac 12$, we define π_a to be a C¹-bilipshitz diffeomorphism of $T^{\ast }_{M}(S^{2})$ onto itself by

$$ {\Pi}_{a}(\xi)=\frac {\xi -a}{|\xi -a|}. $$

Indeed,

$$ {\Pi}^{-1}_{a}(\eta)=a+\left( [(a\cdot\eta )^{2}+(1-|a|^{2})]^{1/2}-a\cdot \eta \right)\eta $$

and

$$ |\nabla {\Pi}_{a}^{-1} (\eta)|\leq {\Lambda} $$

for a uniform constant Λ independently of a with $|a|\leq \frac 12$.

Therefore, taking

$$ w={\Pi}^{-1}_{a_{0}}\circ w_{a_{0}}, $$

we have

$$ \label {3.2} |\nabla w|\leq C({\Lambda} ) |\nabla w_{a_{0}}|. $$

(3.2)

Our claim follows from (3.1) and (3.2). □

We recall Lemma 3.1 in Chapter V of [12]:

Lemma 3.2

Let f(t) be a nonnegative bounded function defined in [r₀,r₁], r₀ ≥ 0. Suppose that for any two t, s with r₀ ≤ t < s ≤ r₁ we have

$$ f(t) \leq \left[C(s-t)^{\alpha} +B \right]+\theta f(s), $$

(3.3)

where C, B, α, 𝜃 are nonnegative constants with 0 ≤ 𝜃 < 1. Then all ρ, R with r₀ ≤ ρ < R ≤ r₁ we have

$$ f(\rho)\leq C\left[(R-\rho)^{\alpha} + B \right]. $$

Using Lemma 3.1 and Lemma 3.2, we prove

Lemma 3.3 (Caccioppoli’s inequality)

Let x₀ ∈Ω and R₀ > 0 such that $B_{R_{0}}(x_{0})\subset {\Omega }$. Let A be a minimizer of E in $H^{1}_{\gamma } ({\Omega }, T^{\ast }_{M}(S^{2}))$. Then for any R ≤ R₀, we have

$$ {\int}_{B_{R/2}(x_{0})}|\nabla A|^{2} d v_{g} \leq C R^{-2}{\int}_{B_{R}(x_{0})}|A-A_{x_{0}, R}|^{2} d v_{g}+CR^{5}, $$

(3.4)

where $A_{x_{0}, R}:={\sum }_{i=1}^{3} A^{i}_{x_{0}, R} dx^{i}$ is denoted by

$$ \label {AV} A^{i}_{x_{0}, R}=\frac 1 {|B_{R}(x_{0})|}{\int}_{B_{R}(x_{0})} A^{i} d v_{g}. $$

(3.5)

Proof

Let $v\in H^{1}_{\gamma }(B_{R_{0}}(x_{0}),T^{\ast } M)$ with v = A on $\partial B_{R_{0}}(x_{0})$ for a R₀ > 0. By Lemma 3.1, there is a $w\in H^{1}(B_{R_{0}}(x_{0}); T_{M}^{\ast } (S^{2}))$ with w = A on $\partial B_{R_{0}}(x_{0})$ such that

$$ {\int}_{B_{R_{0}}(x_{0})} |\nabla w|^{2} dv_{g} \leq C{\int}_{B_{R_{0}}(x_{0})}|\nabla v|^{2} +|x-x_{0}|^{2} d v_{g}. $$

Since A is a minimizer of E in $H^{1}_{\gamma } ({\Omega }, T_{M}^{\ast } (S^{2}))$ and w = A on $\partial B_{R_{0}}(x_{0})$, we have

$$ \begin{array}{@{}rcl@{}} k {\int}_{B_{R_{0}}(x_{0})}|\nabla A|^{2} dv_{g} &\leq& {\int}_{B_{R_{0}}(x_{0})}W(A,\nabla A) dx\\ &\leq& {\int}_{B_{R_{0}}(x_{0})} W(w,\nabla w) dv_{g}\leq C{\int}_{B_{R_{0}}(x_{0})} |\nabla v|^{2} +|x-x_{0}|^{2} dv_{g} \end{array} $$

for any $v\in {H^{1}_{A}}(B_{R_{0}}(x_{0}))$.

For any two positive t, s with $\frac R 2\leq t<s\leq R$, we choose a cut-off function $\eta \in C_{0}^{\infty } (B_{s})$ such that 0 ≤ η ≤ 1 with η ≡ 1 in B_t and $|\nabla \eta |\leq \frac C{s-t}$. Taking $v=A-\eta (A-A_{x_{0}, R})$, we see

$$ \nabla \text{v}=(1-\eta)\nabla (A-A_{x_{0}, R})-\nabla \eta (A-A_{x_{0}, R}). $$

It follows from (3.3) that

$$ {\int}_{B_{s}}|\nabla A|^{2} dx \leq C{\int}_{B_{s}} |\nabla v|^{2}+|x-x_{0}|^{2} dv_{g}. $$

Then

$$ {\int}_{B_{s}} |\nabla A|^{2} d v_{g} \leq C_{1} {\int}_{B_{s}\backslash B_{t}} |\nabla A|^{2} d v_{g}+C_{1} R^{5}+C_{1}(s-t)^{-2}{\int}_{B_{R}}|A-A_{x_{0}, R}|^{2} dv_{g}. $$

By the standard filling hole trick, there exists a positive $\theta = \frac {C_{1}}{1+C_{1}} <1$ such that

$$ {\int}_{B_{t}} |\nabla A|^{2} d v_{g} \leq \theta {\int}_{B_{s}} |\nabla A|^{2} d v_{g}+CR^{5} + C(s-t)^{-2}{\int}_{B_{R}}|A-A_{x_{0}, R}|^{2} dv_{g} $$

for all t, s with $\frac R 2\leq t<s\leq R$. It implies from using Lemma 3.2 that

$$ {\int}_{B_{R/2}}|\nabla A|^{2} dv_{g} \leq C R^{-2}{\int}_{B_{R}}|A-A_{x_{0}, R}|^{2} d v_{g}+C R^{5}. $$

This proves our claim. □

By applying the Sobolev–Poincare inequality to (3.4), we have

$$ \begin{array}{@{}rcl@{}} \left( -{\int}_{B_{R/2}(x_{0})} |\nabla A|^{2} dv_{g}\right)^{1/2} &\leq& \frac C R\left( -{\int}_{B_{R} (x_{0})} |A-A_{R}|^{2} d v_{g}\right)^{1/2}+CR\\ &\leq& C \left( -{\int}_{B_{R} (x_{0})} |\nabla A|^{\frac {6}{5}} dv_{g}\right)^{\frac{5}{6}}+CR \end{array} $$

for any x₀ ∈Ω and any R > 0 with $B_{R}(x_{0})\subset B_{\frac {R_{0}}2 }(x_{0})\subset {\Omega }$ for some R₀ > 0. Then

$$ \left( -{\int}_{B_{R/2}(x_{0})} (|\nabla A|+R)^{2} dv_{g}\right)^{1/2} \leq C \left( -{\int}_{B_{R} (x_{0})} (|\nabla A|+R)^{\frac {6}{5}} d v_{g}\right)^{\frac {5}{6}}. $$

Using the above result, we use the standard trick (see Proposition 1.1 of [12]; pp. 122–123) to obtain that there exists an exponent q > 2 such that for all x₀ ∈Ω and R ≤ R₀, we have

$$ \left( -{\int}_{B_{R/2}(x_{0})} (|\nabla A|+R)^{q} dx\right)^{1/q} \leq C \left (-{\int}_{B_{R} (x_{0})} (|\nabla A|+R)^{2} dx\right )^{1/2}, $$

(3.6)

where C is a constant independent of A. Equation (3.6) is called a reverse Hölder inequality.

4 Partial Regularity of Weal Solution of Liquid Crystal Systems

In the section, we will modify an approach in [22] to prove partial regularity of weak solutions having Caccioppoli’s inequality (see Lemma 3.3).

For a smooth one-form $\phi \in C_{0}^{\infty }({\Omega })$, we consider a variation

$$ A_{t}(x)=\frac {A+t\phi}{|A(x)+t\phi(x)|} = \frac{A+t\phi}{(1+2t\langle A, \phi\rangle + t^{2}|\phi|^{2})^{1/2}}. $$

We calculate

$$ \frac{dA_{t}}{dt} = \frac {\phi} {|A(x)+t\phi(x) |}-\frac {(A+t\phi)(\langle A, \phi\rangle + t|\phi|^{2})} {(1+2t \langle A,\phi\rangle + t^{2}|\phi|^{2})^{3/2}}. $$

Note |A| = 1. Then

$$ \frac{dA_{t}}{dt}\Big|_{t=0} = \phi-A \langle A, \phi\rangle,\quad \frac{d \nabla A_{t}}{dt}\Big|_{t=0} = \nabla \phi - \nabla (A \langle A, \phi\rangle). $$

Note that

$$ d^{\ast} A_{t} = -\frac1{\sqrt{|g|}}\frac{\partial}{\partial x^{j}}\left( \sqrt{|g|} g^{ij} {A_{t}^{i}}\right). $$

Using the fact that |A| = 1, we have

$$ \frac{d}{dt}d^{\ast} A_{t}\Big|_{t=0} = d^{\ast} (\phi - A\langle A, \phi\rangle) $$

and

$$ \frac{d}{dt}d A_{t}\Big|_{t=0} = d(\phi -A \langle A, \phi\rangle). $$

To derive the Euler–Lagrange equation, we compute

$$ \frac{d}{dt}{\int}_{\Omega} W(A_{t},\nabla A_{t}) dx \Big|_{t=0}=0. $$

Using |A|² = 1, we have

$$ \begin{array}{@{}rcl@{}} &&{}{\int}_{\Omega}k \langle \nabla A, \nabla \phi - \nabla A \langle A, \phi\rangle\rangle + (k_{1}-k) \langle d^{\ast} A, d^{\ast} [\phi - A \langle A, \phi\rangle] \rangle\\ &&\quad+(k_{2}-k) \langle A, \ast dA \rangle \langle \ast A, d (\phi - A \langle A, \phi\rangle) \rangle \\ &&\quad +(k_{2}-k) \langle A, \ast dA \rangle \langle \ast dA, \phi-A \langle A, \phi\rangle \rangle\\ &&\quad + (k_{3}-k) \langle A \wedge \ast dA, A \wedge \ast [d(\phi - A \langle A, \phi\rangle)] \rangle\\ &&\quad+(k_{3}-k) \langle A \wedge \ast dA, (\phi-A \langle A, \phi\rangle) \wedge \ast dA \rangle d v_{g} =0 \end{array} $$

for any one-form $\phi \in C_{0}^{\infty }({\Omega })$.

Using the formula (2.2.1) of Chapter 2 in [31] and that |A| = 1, we have

$$ \begin{array}{@{}rcl@{}} &&{}\langle A \wedge \ast dA, A \wedge \ast [d \phi - d(A \langle A, \phi\rangle)] \rangle\\ &&=|A|^{2}\langle \ast dA, \ast [d \phi - d(A \langle A, \phi\rangle)]\rangle - \langle A, \ast [d \phi - d(A \langle A, \phi\rangle)]\rangle\langle \ast dA, A \rangle\\ &&= \langle dA, d[\phi - A \langle A, \phi\rangle]\rangle - \langle \langle \ast dA, A \rangle \ast A, d(\phi - A\langle A, \phi\rangle)\rangle \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} && {}\langle A \wedge \ast dA, (\phi-A \langle A, \phi\rangle) \wedge \ast dA \rangle\\ && =\langle A, \phi-A \langle A, \phi\rangle \rangle |\ast dA|^{2} - \langle A, \ast dA \rangle \langle \ast dA, \phi-A \langle A, \phi\rangle \rangle\\ &&=-\langle A, \ast dA \rangle \langle \ast dA, \phi-A \langle A, \phi\rangle \rangle. \end{array} $$

Therefore, $A\in H^{1}({\Omega }, T^{\ast }_{M}(S^{2}))$ is said to be a weak solution to the liquid crystal system if A satisfies (1.4) in weak sense, i.e.,

$$ \begin{array}{@{}rcl@{}} &&{}{\int}_{\Omega}k \langle \nabla A, \nabla \phi - \nabla A \langle A, \phi\rangle \rangle + (k_{1}-k)\langle d^{\ast} A, d^{\ast} [\phi - A \langle A, \phi\rangle ] \rangle \\ &&\quad + (k_{2}-k_{3}) \langle A, \ast dA \rangle \langle \ast A, d(\phi - A \langle A, \phi\rangle) \rangle \\ &&\quad + (k_{2}-k_{3}) \langle A, \ast dA \rangle \langle \ast dA, \phi-A \langle A, \phi\rangle \rangle \\ &&\quad +(k_{3}-k) \langle dA, d[\phi - A\langle A, \phi\rangle]\rangle dv_{g} =0 \end{array} $$

for any one-form $\phi \in C_{0}^{\infty }({\Omega })$.

Now we prove

Theorem 4.1

Let $A\in H^{1}({\Omega }, T_{M}^{\ast } (S^{2}))$ be any weak solution of (1.4) and assume that A has the Caccioppoli inequality. Then A is smooth in an open set Ω₀ ⊂Ω and $\mathcal H^{\beta } ({\Omega }\backslash {\Omega }_{0})=0$ for some positive β < 1.

Proof

Let x₀ be a point in Ω with $B_{R_{0}}(x_{0})\subset {\Omega }$ with $R_{0}\leq \frac 12 \text {dist}(x_{0}, \partial {\Omega })$. For any R, we denote $A_{x_{0}, R}:={\sum }_{i=1}^{3} A^{i}_{x_{0}, R} dx^{i}$ as in (3.5). In normal coordinates around a point x₀ ∈ M (e.g. [31, p. 21]), we have

$$ g_{ij}(x_{0})=\delta_{ij},\quad \left |\frac {\partial g_{ij}}{\partial x_{k}} (x)\right |\leq C|x-x_{0}|,\quad |g_{ij}(x)-\delta_{ij}|\leq C|x-x_{0}|^{2}. $$

By the Sobolev–Poincare inequality, we have

$$ \begin{array}{@{}rcl@{}} |1-|A_{x_{0}, R}|^{2} | &\leq& -{\int}_{B_{R}(x_{0})} |A-A_{x_{0}, R}|^{2} d v_{g}+ C|x-x_{0}|^{2}\\ &\leq& CR^{-1}{\int}_{B_{R}(x_{0})} |\nabla A|^{2} d v_{g}+ CR^{2}. \end{array} $$

(4.1)

If $R^{-1}{\int \limits }_{B_{R}(x_{0})} |\nabla A|^{2} d v_{g}$ and R are sufficiently small, it can be seen from (4.1) that $|A_{x_{0}, R}|\neq 0$. We set $\omega _{3}=\frac {A_{x_{0}, R}}{|A_{x_{0}, R}|}$. Then it follows from (4.1) that

$$ \begin{array}{@{}rcl@{}} |\omega_{3}-A_{x_{0}, R}| &=& |1-|A_{x_{0}, R}|| \leq |1-|A_{x_{0}, R}|^{2}|\\ &\leq& C R^{-1}{\int}_{B_{R}(x_{0})} |\nabla A|^{2} d v_{g}+ CR^{2}. \end{array} $$

Then we have

$$ \begin{array}{@{}rcl@{}}\label {4.3} |A-\omega_{3}| &\leq& |A-A_{x_{0}, R} |+|A_{x_{0}, R}-\omega_{3}|\\ &\leq& |A-A_{x_{0}, R}| + C R^{-1}{\int}_{B_{R}(x_{0})} |\nabla A|^{2} d v_{g}+ CR^{2}. \end{array} $$

(4.2)

At each point $x\in B_{R_{0}}(x_{0})$ with a sufficiently small R₀ > 0, there exists an normal frame field $e_{j}=\frac {\partial }{\partial x^{j}}$ in T_M (e.g. [6, 31]). Assume that $\{\omega _{i}\}_{i=1}^{3}$ is a normal co-frame field in T^∗M in $B_{R_{0}}(x_{0})$, where $\omega _{3}=\frac {A_{x_{0}, R}}{|A_{x_{0}, R}|}$. Then we write

$$ \begin{array}{@{}rcl@{}} A(x)&=& \langle A,\omega_{1}\rangle \omega_{1} + \langle A, \omega_{2}\rangle \omega_{2} + \langle A, \omega_{3}\rangle \omega_{3} \\ &:=& \tilde A_{1}\omega_{1}+\tilde A_{2}\omega_{2}+\tilde A_{3}\omega_{3}. \end{array} $$

Using |A(x)|² = 1, we have

$$ \nabla\tilde A_{3} =-\tilde A_{1} \nabla \tilde A_{1}-\tilde A_{2}\nabla \tilde A_{2}+(1-\tilde A_{3})\nabla \tilde A_{3}. $$

It follows from (4.2) and using Cauchy’s inequality that

$$ \begin{array}{@{}rcl@{}} |\nabla \tilde A_{3}|^{2} &\leq& (1-|\tilde A_{3}|^{2}+(1-\tilde A_{3})^{2})|\nabla A|^{2}= 2 (1-\tilde A_{3}) |\nabla A|^{2}\\ &=&2 (|A|^{2} - \langle A, \omega_{3}\rangle) |\nabla A|^{2} \leq 2|A-\omega_{3}||\nabla A|^{2} \\ &\leq& C\left( |A-A_{x_{0}, R}|+ R^{2} + R^{-1}{\int}_{B_{R}(x_{0})} |\nabla A|^{2} d v_{g} \right)|\nabla A|^{2}. \end{array} $$

(4.3)

Set $\tilde A= \tilde A_{1}\omega _{1}+\tilde A_{2}\omega _{2}$. Then we re-write (1.4) into the following equation:

$$ \begin{array}{@{}rcl@{}} &&{}k[\nabla^{\ast}\nabla \tilde A] + (k_{1}-k)dd^{\ast}\tilde A + (k_{2}-k_{3}) d^{\ast} (\langle A, \ast d\tilde A \rangle \ast A) + (k_{3}-k)d^{\ast} d\tilde A \\ && = -k \big[\nabla^{\ast} \nabla(A-\tilde A) - | \nabla A|^{2} A\big] - (k_{1}-k) \big[dd^{\ast} (A-\tilde A) - \langle dd^{\ast} A, A\rangle A\big] \\ &&\quad -(k_{2}-k_{3})\big[d^{\ast}(\langle A, \ast dA \rangle \ast A) - \langle d^{\ast} (\langle A, \ast dA \rangle \ast A), A \rangle A\big]\\ &&\quad -(k_{3}-k)\big[d^{\ast} d(A-\tilde A) - \langle d^{\ast} dA, A\rangle A\big]\\ &&\quad -(k_{2}-k_{3})\big[\langle A, \ast dA \rangle \ast dA - \langle A, \ast dA\rangle^{2}A \big]. \end{array} $$

(4.4)

For each R ≤ R₀, let $\tilde v=\tilde v_{1} \omega _{1}+ \tilde v_{2} \omega _{2}\in H^{1}(B_{R}(x_{0}))$ be the solution of

$$ \begin{array}{@{}rcl@{}} &&k \nabla^{\ast} \nabla v+(k_{1}-k) \big[d d^{\ast} v - \langle dd^{\ast} v, \omega_{3}\rangle \omega_{3}\big]\\ &&+ (k_{2}-k_{3})\big[d^{\ast}(\langle \omega_{3}, \ast d v \rangle \ast\omega_{3}) - \langle d^{\ast} (\langle\omega_{3}, \ast d v \rangle \ast \omega_{3}), \omega_{3} \rangle \omega_{3}\big]\\ &&+(k_{3}-k)\big[d^{\ast} d v - \langle d^{\ast} d v, \omega_{3}\rangle \omega_{3}\big]=0 \end{array} $$

(4.5)

with boundary value $v|_{\partial B_{R}(x_{0})} = \tilde A_{1}\omega _{1}+\tilde A_{2}\omega _{2} |_{\partial B_{R}(x_{0})}$. Note that (4.5) is a strong elliptic linear system on (v₁,v₂). Then for every ρ < R, we have (e.g. [12])

$$ {\int}_{B_{\rho}(x_{0})}|\nabla \tilde v|^{2} dx \leq C \left( \frac {\rho}R\right)^{3}{\int}_{B_{R}(x_{0})}|\nabla \tilde v|^{2} dx. $$

(4.6)

Moreover, using the maximum principle of a linear elliptic system (e.g. Proposition 2.3 of Chapter III in [12]) and |A| = 1, $|\tilde v| \leq C$ in B_R(x₀) for some positive constant C.

Choosing $\phi =\tilde w= v-\tilde A= \tilde v_{1} \omega _{1}+ \tilde v_{2} \omega _{2}-\tilde A_{1}\omega _{1}-\tilde A_{2}\omega _{2}$ as test function in (4.5), we have

$$ \begin{array}{@{}rcl@{}} &&{\int}_{B_{R}(x_{0})} k \langle \nabla v, \nabla \tilde w \rangle + (k_{1}-k)\langle d^{\ast} v, d^{\ast} \tilde w \rangle + (k_{2}-k) \langle\omega_{3}, \ast dv \rangle \langle \ast\omega_{3}, d \tilde w \rangle\\ &&+(k_{3}-k) \langle\omega_{3} \wedge \ast dv, \omega_{3} \wedge \ast d \tilde w \rangle dv_{g} =0. \end{array} $$

(4.7)

Here we use the fact that 〈ϕ,ω₃〉 = 0 and

$$ \langle\omega_{3} \wedge \ast dv, \omega_{3} \wedge \ast d \tilde w \rangle = \langle dv, d\tilde w \rangle - \langle\omega_{3}, \ast dv \rangle\langle\omega_{3}, \ast d \tilde w \rangle. $$

Multiplying (4.4) by $\tilde w= v-\tilde A= \tilde v_{1} \omega _{1}+ \tilde v_{2} \omega _{2}-\tilde A_{1}\omega _{1}-\tilde A_{2}\omega _{2}$, we have

$$ \begin{array}{@{}rcl@{}} &&{}{\int}_{B_{R}(x_{0})}k \langle \nabla \tilde A, \nabla\tilde w \rangle + (k_{1}-k) \langle d^{\ast}\tilde A, d^{\ast} \tilde w \rangle + (k_{2}-k)\langle A, \ast d\tilde A\rangle \langle \ast A, d\tilde w\rangle \\ && + (k_{3}-k)\langle A \wedge \ast d\tilde A, A \wedge \ast d \tilde w \rangle dv_{g} \\ &&= {\int}_{B_{R}(x_{0})}k \langle \nabla (A- \tilde A),\nabla \tilde w \rangle - k |\nabla A|^{2} \langle A, \tilde w\rangle + (k_{1}-k) \langle d^{\ast} (A- \tilde A), d^{\ast} \tilde w \rangle\\ &&\quad-(k_{1}-k) \langle d^{\ast} A, d^{\ast} [A\langle A,\tilde w\rangle] \rangle + (k_{2}-k_{3}) \langle A, \ast d(A-\tilde A)\rangle \langle \ast A, d \tilde w \rangle\\ &&\quad -(k_{2}-k_{3}) \langle A, \ast dA \rangle \langle \ast A, d (A\langle A, \tilde w\rangle) \rangle + (k_{3}-k) \langle d(A-\tilde A), d\tilde w \rangle \\ &&\quad+(k_{3}-k) \langle dA, d[A\langle A,\tilde w\rangle]\rangle dv_{g}. \end{array} $$

(4.8)

Combining (4.8) with (4.7), we obtain

$$ \begin{array}{@{}rcl@{}} &&{}{\int}_{B_{R}(x_{0})}k |\nabla \tilde w|^{2} + (k_{1}-k) |d^{\ast}\tilde w|^{2} + (k_{2}-k)| \langle\omega_{3}, \ast d \tilde w \rangle |^{2} +(k_{3}-k) |\omega_{3} \wedge \ast d \tilde w|^{2} dv_{g}\\ &&={\int}_{B_{R}(x_{0})} (k_{2}-k) \big[\langle\omega_{3}, \ast d\tilde A \rangle \langle \ast\omega_{3}, d\tilde w \rangle - \langle A, \ast d\tilde A\rangle \langle \ast A, d\tilde w\rangle\big] \\ &&\quad+(k_{3}-k)\big[\langle A \wedge \ast d\tilde A, A \wedge \ast d \tilde w \rangle - (k_{3}-k)\langle A \wedge \ast d\tilde A, A \wedge \ast d \tilde w \rangle \big] \\ &&\quad+k \langle \nabla (A- \tilde A), \nabla \tilde w \rangle - k |\nabla A|^{2} \langle A, \tilde w\rangle + (k_{1}-k) \langle d^{\ast} (A- \tilde A), d^{\ast} \tilde w \rangle \\ &&\quad-(k_{1}-k) \langle d^{\ast} A, d^{\ast} [A\langle A, \tilde w\rangle] \rangle + (k_{2}-k_{3}) \langle A, \ast d(A-\tilde A)\rangle \langle \ast A, d \tilde w \rangle\\ &&\quad - (k_{2}-k_{3}) \langle A, \ast dA \rangle \langle \ast A, d (A\langle A, \tilde w\rangle) \rangle + (k_{3}-k) \langle d(A-\tilde A), d \tilde w \rangle \\ &&\quad + (k_{3}-k) \langle dA, d[A\langle A,\tilde w\rangle]\rangle dv_{g}. \end{array} $$

(4.9)

Since $\tilde w=\tilde v_{1} \omega _{1}+ \tilde v_{2} \omega _{2}-\tilde A_{1}\omega _{1}-\tilde A_{2}\omega _{2}$, then $\langle \tilde w, \omega _{3}\rangle =0$, which implies

$$ \langle A, \tilde w\rangle = \langle A-\omega_{3}, w\rangle. $$

To hand those difficult terms on the right-hand side of (4.9), we have

$$ \begin{array}{@{}rcl@{}} d[\langle A, \tilde w\rangle A] &=& d\langle A-\omega_{3}, \tilde w\rangle \wedge A + \langle A-\omega_{3}, \tilde w\rangle dA\\ &&=(\langle \nabla A, \tilde w\rangle + \langle A-\omega_{3}, \nabla \tilde w\rangle) \wedge A + \langle A - \omega_{3}, \tilde w\rangle dA \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} d^{\ast}[\langle A, \tilde w\rangle A] &=& \ast d\ast[\langle A, \tilde w\rangle A]\\ &=&\ast(\langle \nabla A,\tilde w\rangle + \langle A-\omega_{3}, \nabla \tilde w\rangle) \wedge A + \langle A- \omega_{3}, \tilde w\rangle d^{\ast} A, \end{array} $$

where we used the fact that ∇ω₃ = 0.

Then, using Young’s inequality, we have

$$ \begin{array}{@{}rcl@{}} \!\!\!&&{}{\int}_{B_{R}(x_{0})}k |\nabla \tilde w|^{2} +(k_{1}-k) |d^{\ast}\tilde w|^{2}+(k_{2} - k)| \langle\omega_{3}, \ast d \tilde w \rangle |^{2} + (k_{3} - k) |\omega_{3} \wedge \ast d \tilde w|^{2} dv_{g}\qquad\\ \!\!\!\!\!&&\leq \frac k2 {\int}_{B_{R}(x_{0})}|\nabla w|^{2} dv_{g} + C{\int}_{B_{R}(x_{0})}|\nabla A^{3}|^{2} + |\nabla A|^{2} (|A - \omega_{3}|^{2} + |w|^{2} + R^{2}+|w|) dv_{g}. \end{array} $$

(4.10)

Applying (4.2) and (4.3) to (4.10), we have

$$ \begin{array}{@{}rcl@{}} {\int}_{B_{R}(x_{0})} |\nabla \tilde w|^{2} dx &\leq & C{\int}_{B_{R}(x_{0})} |\nabla A|^{2} (|A-A_{x_{0}, R}|^{2}+ R^{2}+|w|) dv_{g}\\ && +C R^{-1}{\int}_{B_{R}(x_{0})} |\nabla A|^{2} d v_{g} {\int}_{B_{R}(x_{0})}|\nabla A|^{2} dv_{g}. \end{array} $$

(4.11)

By (4.5), we have

$$ {\int}_{B_{R}(x_{0})} |\nabla \tilde v|^{2} dx\leq C{\int}_{B_{R}(x_{0})} |\nabla A|^{2} dx. $$

Then it follows from (4.6), (4.3) and (4.11) that

$$ \begin{array}{@{}rcl@{}} {\int}_{B_{\rho}(x_{0})} |\nabla A|^{2}dx &\leq& C\left( \frac {\rho}R\right)^{3}{\int}_{B_{R}(x_{0})} |\nabla A|^{2} dx + C{\int}_{B_{R}(x_{0})} |\nabla \tilde w|^{2} dx + C{\int}_{B_{R}(x_{0})} |\nabla A^{3}|^{2} dx\\ &\leq& C \left[\left( \frac {\rho}R\right)^{3} + R^{2} + \frac 1R {\int}_{B_{R}(x_{0})} |\nabla A|^{2} dx\right]{\int}_{B_{R}(x_{0})} |\nabla A|^{2} dx\\ &&+ C{\int}_{B_{R}(x_{0})} (|A-A_{x_{0}, R}|+ |\tilde w|) |\nabla A|^{2} dx. \end{array} $$

By the reverse Hölder inequality (3.6) and the Sobolev inequality, we obtain

$$ \begin{array}{@{}rcl@{}} {\int}_{B_{R}(x_{0})} |\tilde w| |\nabla A|^{2} dx &\leq& \left( {\int}_{B_{R}(x_{0})} |\nabla A|^{q} \right)^{\frac 2q}\left( {\int}_{B_{R}(x_{0})} |\tilde w|^{\frac q{q-2}} dx \right )^{\frac {q-2}q}\\ &\leq& CR^{\frac {3(2-q)}q}{\int}_{B_{2R}(x_{0})} (|\nabla A|^{2}+R^{2}) dx \left ({\int}_{B_{R} (x_{0})}|\tilde w|^{2} dx\right )^{\frac {q-2}q}\\ &\leq& C \left (R^{-1} {\int}_{B_{R}(x_{0})} |\nabla A|^{2} \right )^{\frac{q-2}q}{\int}_{B_{2R}(x_{0})}(|\nabla A|^{2}+R^{2}) dx \end{array} $$

for some q > 2.

By a similar argument, it follows from using the Hölder inequality and the Sobolev–Poincare inequality that

$$ {\int}_{B_{R}(x_{0})} |A-A_{x_{0}, R}| |\nabla A|^{2} dx \leq C\left( R^{-1} {\int}_{B_{R}(x_{0})} |\nabla A|^{2} \right)^{\frac {q-2}q}{\int}_{B_{2R}(x_{0})}(|\nabla A|^{2}+R^{2}) dx. $$

Then for every ρ and R with $0<\rho <R\leq R_{0}<\frac 12 \text {dist}(x_{0}, \partial {\Omega })$, we have

$$ \begin{array}{@{}rcl@{}}\label {3.10} {\int}_{B_{\rho}(x_{0})} |\nabla A|^{2} &\leq& C\left[\left( \frac {\rho} R\right)^{3} + R^{2}+\frac 1R {\int}_{B_{R}(x_{0})} |\nabla A|^{2} dx\right]{\int}_{B_{2R}(x_{0})}|\nabla A|^{2} dx\\ &&+C\left( R^{-1} {\int}_{B_{R}(x_{0})} |\nabla A|^{2} \right)^{\frac{q-2}q}{\int}_{B_{2R}(x_{0})}(|\nabla A|^{2}+R^{2}) dx. \end{array} $$

(4.12)

By (4.12), it implies from the standard method [12] that A is Hölder continuous in α < 1 inside Ω∖Σ, where

$$ {\Sigma} = \left\{x\in{\Omega}:~\liminf_{R\to 0^{+}} R^{-1}{\int}_{B_{R}(x)} |\nabla A|^{2} dx>0\right\}. $$

For completeness, we give a detailed proof here. For any x₀ ∈Ω∖Σ, there is a sufficiently small R₀ such that $B_{R_{0}}(x_{0})\subset {\Omega } \backslash {\Sigma }$. For each R ≤ R₀, set

$$ \begin{array}{@{}rcl@{}}\label {3.11} \phi (x_{0}, R) &=& \frac1R {\int}_{B_{R}(x_{0})}(|\nabla A|^{2} +R^{2}) dx = \frac1R{\int}_{B_{R}(x_{0})} |\nabla A|^{2} dv_{g}+ R|B_{R}(x_{0})|,\\ \xi (x_{0}, R) &=& \frac 1R {\int}_{B_{R}(x_{0})} |\nabla A|^{2} dx + \left( R^{-1} {\int}_{B_{R}(x_{0})} |\nabla A|^{2} \right)^{\frac{q-2}q}. \end{array} $$

Choosing ρ = 2rR in (4.12) with 0 < r < 1, we have

$$ \label {3.12} \phi (x_{0},r 2R)\leq C_{1}\left[r^{2}(1+\xi (x_{0},R) r^{-3} )+R^{2}\right]\phi (x_{0}, 2R)+4r^{2} R^{2}|B_{2rR}(x_{0})|. $$

(4.13)

For some α with 0 < α < 1, we choose r such that

$$ \label {3.13} 2(C_{1}+1)r^{2-2\alpha}=1. $$

(4.14)

For the above r, there are a sufficiently small ε₁ and R with 2R ≤ R₀ such that

$$ \xi (x_{0}, R)=\frac 1{R} {\int}_{B_{R}(x_{0})} |\nabla A|^{2} dx+\left (\frac 1{R} {\int}_{B_{R}(x_{0})} |\nabla A|^{2} \right )^{\frac{q-2}q}<\varepsilon_{1}, $$

(4.15)

which implies

$$ \label {3.13} \xi (x_{0},R) r^{-3}\leq 2\varepsilon_{1} r^{-3}\leq 1. $$

(4.16)

Assume that ϕ(x₀,2R) < ε₀ for 2R ≤ R₀. It implies that ξ(x₀,R) < ε₁ for a sufficiently small ε₀. Then it follows from (4.13)–(4.16) that

$$ \phi (x_{0},r 2R)\leq r^{2\alpha}\phi (x_{0}, 2R) $$

implying

$$\label {3.14} \phi (x_{0},r^{l} 2R)\leq r^{2l\alpha}\phi (x_{0}, 2R)<\varepsilon_{0} $$

for all number l.

We conclude that if ϕ(x₀,2R) < ε₀ for some 2R < R₀, then

$$ \phi (x_{0},r^{l} 2R)\leq r^{2l\alpha}\varepsilon_{0}. $$

This implies that for any ρ < 2R ≤ R₀, we have

$$ \phi (x_{0},\rho )\leq C \left( \frac {\rho}{2R}\right )^{2\alpha}. $$

Hence, A belongs to $C^{\alpha }_{loc}({\Omega }\backslash {\Sigma })$ for some α < 1. In fact, using (4.12), A belongs to $C^{\alpha }_{loc}({\Omega }\backslash {\Sigma })$ for any α < 1. Repeating the same argument in Theorem 1.5 of Chapter IV of [12], we can prove that $\nabla \tilde A$ are Hölder continuous inside Ω∖Σ, so is ∇A. Then it can be proved by the standard theory that A is smooth in Ω∖Σ. By the reverse Hölder inequality, $A\in W^{1,q}_{loc}({\Omega })$ for some q > 2. Therefore $\mathcal H_{loc}^{\beta } ({\Sigma } )=0$ for some positive β < 1. □

Theorem 1.2 is a consequence of Theorem 4.1 by using Lemma 3.3.