1 Introduction

It is a classical result of continuum mechanics, known from the time of the brothers Cosserat (1896) (according to Shield [22]), that if a \(C^2\) deformation of a connected domain \(\varOmega \subset {\mathbb {R}}^3\) given by \(y: \varOmega \rightarrow {\mathbb {R}}^3\) with deformation gradient \(\nabla y =: F\) has a constant Right Cauchy-Green tensor field, i.e., \(F^TF =\) constant, then y is a homogeneous deformation, i.e., \(F=\) constant. Shield [22] gave an elegant proof (with references to other proofs by Forsyth, and Thomas) whose hypothesis was marginally weakened in[3]. An elementary proof using ideas from classical Riemannian Geometry arises from considering parametrizations of \(\varOmega \) and \(y(\varOmega )\) in a Rectangular Cartesian coordinate system. Then the condition \(F^TF =\) constant allows associating spatially constant metric tensor component fields on the two patches; a use of Christoffel’s transformation rule for the Christoffel symbols then yields \(\nabla ^2 y = 0\). This result implies that if the deformation gradient field of a deformation is known to be ‘pointwise rigid,’ i.e., \(\nabla y(x) = F(x) \in SO(3) \, \forall x \in \varOmega \), then \(F =\) constant \(\in SO(3)\), and the deformation y is globally rigid. First generalizations of this result go back to Reshetnyak who proved in [21] that if \(y_k \rightharpoonup y\) in \(W^{1,2}(\varOmega ;{\mathbb {R}}^n)\) and \({\text {dist}}(\nabla y_k,SO(n)) \rightarrow 0\) in measure then \(\nabla y\) is necessarily a constant rotation. A proof of this result using Young measures can be found in [12]. John proved in [13] that if \(y\in C^1\) and \({\text {dist}}(\nabla y,SO(n)) \le \delta \) for a sufficiently small \(\delta >0\) then \([\nabla y]_{BMO} \le C(n) \delta \). Without the assumption that \(\nabla y\) is uniformly close to SO(n), Kohn proved optimal bounds for \(\min _{R \in SO(n), b \in {\mathbb {R}}^n} \Vert y - (Rx + b) \Vert _{L^p}\) (but not for \(\Vert \nabla y - R\Vert _{L^p}\)) in [14]. Optimal bounds on \(\nabla y -R\) in \(L^2\) were derived in the celebrated work of Friesecke, James, and Müller, [9]. The authors prove that for an open, connected domain with Lipschitz boundary \(\varOmega \subseteq {\mathbb {R}}^n\) there exists \(C(\varOmega )>0\) such that for every \(y \in W^{1,2}(\varOmega ;R^n)\) there exists a rotation \(R\in SO(n)\) satisfying

$$\begin{aligned} \int _{\varOmega } |\nabla y - R|^2 \, dx \le C(\varOmega ) \int _{\varOmega } {\text {dist}}(\nabla y,SO(n))^2 \, dx. \end{aligned}$$
(1)

As pointed out in [6], \(L^p\)-versions of the above estimate also hold for \(1<p<\infty \). Generalizations to interpolation spaces were established in [4].

Regardless of the smoothness hypotheses involved, all of the above results crucially rely on the fact that the field F is the gradient of some deformation y. Going beyond the realm of deformations, it seems natural to interpret the global rigidity question in the following way: Let \(R \in C^1(\varOmega ; SO(3))\) be specified with \({\text {curl }}R = 0\) in \(\varOmega \); then \(R =\) constant. Posed in this manner, it seems natural to ask whether the hypothesis \({\text {curl }}R = 0\) is optimal or whether it can be further weakened. It is this question that is dealt with in this paper with an affirmative answer. Specifically, we show that global rigidity is obtained even for \({\text {curl }}R =\) constant on \(\varOmega \). This result, for \(\varOmega \subseteq {\mathbb {R}}^2\) and \(R \in C^2(\varOmega ;SO(2))\), was obtained in [1]. Here, we prove it for \(R: {\mathbb {R}}^3 \supseteq \varOmega \rightarrow SO(3)\) merely measurable. This three-dimensional result is based on significantly different ideas from [1], and generates also a different proof for the 2-d case.

Rigidity estimates similar to (1) for non-gradient fields were first established in the linear theory and dimension 2 in [10]. The nonlinear analogue was first proved in [17]; for a version with mixed growth see [11]. Generalizations to higher dimensions \(n\ge 3\) were then established in [15] in the Lorentz spaces \(L^{\frac{n}{n-1},\infty }\). In [5] the stronger estimate in \(L^{\frac{n}{n-1}}\) was shown for \(n\ge 3\). Precisely: For \(\varOmega \subseteq {\mathbb {R}}^n\) open and connected with Lipschitz boundary there exists \(C(\varOmega )>0\) such that for every \(F \in L^{\frac{n}{n-1}}(\varOmega ;{\mathbb {R}}^{n\times n})\) such that \({\text {curl }}F\) is a bounded measure there exists \(R \in SO(n)\) satisfying

$$\begin{aligned} {\int _{\varOmega } |F-R|^{\frac{n}{n-1}} \, dx \le C(\varOmega ) \left( \int _{\varOmega } {\text {dist}}(F,SO(n))^{\frac{n}{n-1}} \, dx + |{\text {curl }}F|(\varOmega )^{\frac{n}{n-1}} \right) }.\nonumber \\ \end{aligned}$$
(2)

Clearly, a rigidity estimate like (2) does not directly imply that rotation fields with a constant but non-zero \({\text {curl }}\) are constant as the estimate (2) applied to a field with a constant \({\text {curl }}\) does not provide more information than the same estimate applied to a field with a bounded but non-constant \({\text {curl }}\). However, there are obviously non-constant rotation fields with a bounded \({\text {curl }}\). Therefore, the proof of our result will be based on a different approach (see Sect. 2 for the idea of the proof and its connection to the gradient setting). Instead, the rigidity estimate (2) can be used to prove higher regularity for rotation fields, see Sect. 4, whereas our rigidity result is based on a PDE approach, see Sects. 2 and 3.

It turns out that the question raised above is of relevance in the theory of dislocations, as explained in detail in [1], with connections to the linear elastic theory of dislocations. Briefly, considering a nonlinear elastic material with a ‘single-well’ elastic energy density, our result shows that a traction-free body with a constant (non-vanishing) dislocation density cannot be stress-free (such a field is computed in [2, Sec. 5.3]). This is in stark contrast to the linear theory of dislocations in which the same body under identical hypotheses would necessarily be stress-free. An interesting question in this interpretation of our work is the characterization of the resulting stress field in a material with a ‘multiple-well’ energy density, in particular, whether a stress-free state can arise for a constant dislocation density.

Additionally, we remark that recently similar questions have been studied in the context of liquid crystals. In [20, 23, 24] (see also [19] for the two–dimensional setting) the authors study the compatibility conditions for unit vector fields \(n: {\mathbb {R}}^3 \supseteq \varOmega \rightarrow S^2\), so–called director fields. In this context it is natural to distinguish four independent measures of distortion: the splay \({\text {div }}n\), the twist \(n \cdot {\text {curl }}n\), the bend \(n \times {\text {curl }} n\) and the more involved biaxial splay. It can then be shown that in Euclidean space director fields that induce a uniform (space-independent) distortion (bend, twist, splay, biaxial splay) take a very specific form, see [24]. Generalizations to curved spaces can be found in [20, 23]. However, although the rows of a matrix field \(R: \varOmega \rightarrow SO(3)\) are unit vector fields, the fact that the \({\text {curl }}\) of the different rows is constant does not translate immediately to information on the different modes of distortion above. In particular, in our setting the relations between the different rows need to be exploited to prove that R is constant, see Theorem 3.

This article is organized as follows: first, we introduce the needed notation. Then we prove that a regular rotation field with a constant \({\text {curl }}\) is constant in dimension 2 (Sect. 2) and 3 (Sect. 3). In Sect. 4 we prove regularity of rotation fields in terms of the regularity of its \({\text {curl }}\). This shows that the results proved in Sects. 2 and  3 apply more generally to measurable rotation fields with a constant \({\text {curl }}\) in the sense of distributions.

Notation Throughout the whole article we use the Einstein summation convention, i.e., we sum over indices that appear twice.

Moreover, we denote by Id the identity matrix in any dimension. For a matrix A we write \(A_i\) for its i-th row. For the set of rotations in \({\mathbb {R}}^n\) we write \(SO(n) = \{R \in {\mathbb {R}}^{n\times n}: A^T A = Id, \det (A) = 1 \}\). The trace of a matrix \(A \in {\mathbb {R}}^{n\times n}\) is given by \(\mathrm {tr}(A) = \sum _{k=1}^n A_{kk}\), the scalar product between two matrices \(A,B \in {\mathbb {R}}^{n\times n}\) is given by \(A:B = \mathrm {tr}(A^TB)\). For a matrix \(A \in {\mathbb {R}}^{n\times n}\) we write \(A_{sym} = \frac{1}{2} (A+A^T)\) and \(A_{skew} = \frac{1}{2}( A - A^T)\). The spaces of symmetric or skew-symmetric matrices are denoted by \(Sym(n) = \{A \in {\mathbb {R}}^{n\times n}: A^T = A\}\) and \(Skew(n)= \{A\in {\mathbb {R}}^{n\times n}: A^T = -A\}\), respectively. For a matrix \(A \in {\mathbb {R}}^{n\times n}\) we denote by \({\text {cof }} A\) its cofactor matrix, i.e., the \(n\times n\) matrix whose (i, j) entry is given by \((-1)^{i+j}{\text {det }}(A^{ij})\), where \(A^{ij}\) is the \((n-1)\times (n-1)\)-matrix that evolves from A by deleting the i-th row and j-th column. Cramer’s rule says that for invertible \(A \in {\mathbb {R}}^{n\times n}\) we have \(\frac{1}{{\text {det }}(A)} {\text {cof }} A = (A^{-1})^T\). For two vectors \(a,b \in {\mathbb {R}}^3\) the cross product \(a \times b \in {\mathbb {R}}^3\) is defined as usual as \((a\times b)_i = \varepsilon _{ijk} a_j b_k\). Here, \(\varepsilon _{ijk}\) is the sign of the permutation (ijk).

Let \(\varOmega \subseteq {\mathbb {R}}^n\) and connected. Throughout the whole paper we use standard notation for the n-dimensional Lebesgue measure \({\mathcal {L}}^n\), the k-dimensional Hausdorff measure \({\mathcal {H}}^k\), the space of k-times differentiable functions from \(\varOmega \) to \({\mathbb {R}}^m\), \(C^k(\varOmega ;{\mathbb {R}}^m)\), the space of p-integrable functions (more precisely, equivalence classes of these functions) on \(\varOmega \) with values in \({\mathbb {R}}^m\), \(L^p(\varOmega ;{\mathbb {R}}^m)\), Sobolev spaces, \(W^{k,p}(\varOmega ;{\mathbb {R}}^m)\), and the space of vector-valued Radon-measures, \({\mathcal {M}}(\varOmega ;{\mathbb {R}}^m)\). For a vector-valued Radon measure \(\mu \) we denote by \(|\mu |\) its total variation measure. The space of functions of bounded variation \(BV(\varOmega ;{\mathbb {R}}^m)\) consists of function \(f\in L^1(\varOmega ;{\mathbb {R}}^m)\) whose weak derivative is a vector-valued Radon measure with finite total variation i.e., there exists \(\mu \in {\mathcal {M}}(\varOmega ;{\mathbb {R}}^{n\times m})\) with \(|\mu |(\varOmega ) < \infty \) such that for all \(\varphi \in C^{\infty }_c(\varOmega ;{\mathbb {R}}^m)\) and \(i\in \{1,\dots ,n\}\) it holds that

$$\begin{aligned} \int _{\varOmega } u \cdot \partial _i \varphi \, dx = - \int _{\varOmega } \varphi \cdot d\mu _i. \end{aligned}$$

In this case we write \(Du = \mu \).

In addition we recall quickly standard notation for classical differential operators. The divergence operator for a vector field \(f=(f_1,\dots ,f_n)\) on a subset of \({\mathbb {R}}^n\) is given by \({\text {div }}(f) = \sum _{k=1}^n \partial _k f_k\). For a vector field on a subset of \({\mathbb {R}}^2\) we write \({\text {curl }}(f) = \partial _1 f_2 - \partial _2 f_1\), for a vector field f on a subset of \({\mathbb {R}}^3\) the i-th component of the vector field \({\text {curl }}(f)\) is given by \({\text {curl }}(f)_i = \varepsilon _{ijk} \partial _j f_k\). For arbitrary \(n \in {\mathbb {N}}\) we generalize this notation to \({\text {Curl}}(f) = \left( \partial _j f_k - \partial _k f_j \right) _{j,k = 1}^n\). In dimension 2 and 3 the notions \({\text {curl }}\) and \({\text {Curl}}\) can easily be identified. For matrix fields \({\text {Curl}}\), \({\text {div }}\) and \({\text {curl }}\) will always be applied rowwise.

We recall that for a function \(f \in L^1_{loc}(\varOmega ;{\mathbb {R}}^n)\) we say that \({\text {Curl}}(f) = \mu \in {\mathcal {M}}(\varOmega ; {\mathbb {R}}^{n\times n})\) in the sense of distributions if we have for all \(\varphi \in C^{\infty }_c(\varOmega ;{\mathbb {R}})\)

$$\begin{aligned} \int _{\varOmega } f_k \partial _j \varphi - f_j \partial _k \varphi _k \, dx = - \int _{\varOmega } \varphi \, d\mu _{jk}. \end{aligned}$$

Note that a function \(\alpha \in L^1_{loc}(\varOmega ;{\mathbb {R}}^m)\) can always be associated to a vector-valued Radon measure \(\mu \in {\mathcal {M}}(\varOmega ;{\mathbb {R}}^m)\) through \(\mu _{\alpha }(A) = \int _A \alpha (x) \, dx\). For \(f, \alpha \in L^1_{loc}\) we also write \({\text {Curl}} f = \alpha \) instead of \({\text {Curl}} f = \mu _{\alpha }\).

2 Rigidity for Rotation Fields in Dimension \({\mathbf {2}}\)

We start by reconsidering the case \(n=2\). In [1] it was shown that a function \(R \in C^2(\varOmega ;SO(2))\) such that \({\text {curl }}R\) is constant is necessarily constant. In this section we give an alternative proof to this statement which uses the idea of the proof for gradients. A similar strategy will be used in the three-dimensional setting.

Let us quickly recall the argument for gradients in dimension n. Let \(R = \nabla u \in C^1(\varOmega ;SO(n))\) for some \(u\in C^2(\varOmega ;{\mathbb {R}}^n)\). We note that \({\text {cof }} \nabla u = \nabla u\), \({\text {div }}{\text {cof }}(\nabla u) = 0\) and \(|\nabla u|^2 = n\). Thus, \(\varDelta u = 0\) and \(0 = \varDelta |\nabla u|^2\). Then one computes \(0 = \varDelta |\nabla u|^2 = 2 \nabla (\varDelta u) : \nabla u + |\nabla ^2 u|^2 = |\nabla ^2 u|^2\). Consequently, \(\nabla u = R\) is constant.

Theorem 1

Let \(\varOmega \subseteq {\mathbb {R}}^2\) be open and connected. Let \(R \in C^2(\varOmega ;SO(2))\) and \(\alpha \in {\mathbb {R}}^2\) such that \({\text {curl }}R = \alpha \). Then R is constant.

Proof

As \(R(x) \in SO(2)\) for all \(x \in \varOmega \), there exists a \(C^2\)-vector field \(e : \varOmega \rightarrow {\mathbb {R}}^2\) such that

$$\begin{aligned} e_1(x)^2 + e_2(x)^2 = 1 \text { and } R(x) = \begin{pmatrix} e_1(x) &{}&{} e_2(x) \\ -e_2(x) &{}&{} e_1(x) \end{pmatrix} \text { for all } x\in \varOmega . \end{aligned}$$

As \({\text {curl }}R = \alpha \), we find that

$$\begin{aligned} \partial _1 e_2 - \partial _2 e_1&= \alpha _1, \\ \partial _1 e_1 + \partial _2 e_2&= \alpha _2, \end{aligned}$$

from which we derive

$$\begin{aligned} \partial _1 \partial _1 e_2 - \partial _1 \partial _2 e_1 = 0, \\ \partial _2 \partial _1 e_2 - \partial _2 \partial _2 e_1 = 0, \\ \partial _1 \partial _1 e_1 + \partial _1 \partial _2 e_2 = 0, \\ \partial _2 \partial _1 e_1 + \partial _2 \partial _2 e_2 = 0. \end{aligned}$$

Adding the fourth to the first equation and subtracting the second from the third equation we find that

$$\begin{aligned} \varDelta e_1 = \varDelta e_2 = 0. \end{aligned}$$

Using that \(e_1(x)^2 + e^2(x) = 1\) for all \(x \in \varOmega \), we obtain

$$\begin{aligned} 0 = \varDelta (e_1^2 + e_2^2) = 2 e_1 \varDelta e_1 + 2 |\nabla e_1|^2 + 2e_2 \varDelta e_2 + 2|\nabla e_2|^2 = |\nabla e_1|^2 + |\nabla e_2|^2. \end{aligned}$$

As \(\varOmega \) is connected this implies that e (and consequently R) is constant. \(\square \)

Remark 1

In the language of the literature on director fields in liquid crystals, it is shown in the proof above that \(e: \varOmega \rightarrow S^1\) has constant bend and splay. Then it is well-known that this implies in spaces without a negative Gauss curvature that e is constant, see for example [19].

In view of Theorem 1 we see that the generalized rigidity estimate (2) does not provide the optimal estimate for rotation fields with a constant \({\text {curl }}\). The naïve extension of the generalized rigidity estimate (2) incorporating the result of Theorem 1 would allow the subtraction of a constant from the \({\text {curl }}\) on the right hand side: For every open, bounded and connected set \(\varOmega \subseteq {\mathbb {R}}^2\) with Lipschitz boundary there exists \(C(\varOmega )>0\) such that for every \(F\in L^2(\varOmega ;{\mathbb {R}}^{2\times 2})\) with \({\text {curl }}F \in {\mathcal {M}}(\varOmega ;{\mathbb {R}}^2)\) and \(\alpha \in {\mathbb {R}}^2\) there exists \(R\in SO(2)\) satisfying

$$\begin{aligned} \int _{\varOmega } |F - R|^2 \, dx \le C(\varOmega ) \left( \int _{\varOmega } {\text {dist}}(F,SO(2))^2 \, dx + |{\text {curl }}(F) - \mu |(\varOmega )^2\right) , \end{aligned}$$

where \(\mu = \alpha \, {\mathcal {L}}^2\).

However, the following example shows that a statement of this type cannot be true as it does not hold true in the linearized setting, c.f. the discussion in [1].

Example 1

Let \(\varOmega = B_1(0)\). For \(\varepsilon >0\) we define \(F_{\varepsilon }: \varOmega \rightarrow {\mathbb {R}}^{2\times 2}\) by

$$\begin{aligned} F_{\varepsilon }(x) = Id + \varepsilon \begin{pmatrix} 0 &{}&{} x_1 \\ -x_1 &{}&{} 0 \end{pmatrix}. \end{aligned}$$

First we notice that \({\text {curl }}F_{\varepsilon } = \varepsilon \begin{pmatrix} 1\\ 0\end{pmatrix}\). Next, we observe that and therefore \(\int _{\varOmega } |F_{\varepsilon } - Id|^2 \, dx \le \int _{\varOmega } |F_{\varepsilon } - R|^2 \, dx\) for all \(R\in SO(2)\). Now, we compute \(\int _{\varOmega } |F_{\varepsilon } - Id|^2 \, dx = \int _{\varOmega } 2\varepsilon ^2 x_1^2 \, dx = \frac{\pi }{2} \varepsilon ^2\). On the other hand, a second order Taylor expansion at Id shows that

$$\begin{aligned} {\text {dist}}(F_{\varepsilon }(x),SO(2))^2 \le |(F_{\varepsilon }(x) - Id)_{sym}|^2 + C |F_{\varepsilon } - Id|^3 \le C \varepsilon ^3. \end{aligned}$$

Consequently, \(\int _{\varOmega } {\text {dist}}(F_{\varepsilon }(x),SO(2))^2 \, dx \le C \varepsilon ^3\). In particular we see that there cannot exist a constant \(C(\varOmega ) > 0\) such that for every \(\varepsilon >0\) there exists \(R_{\varepsilon } \in SO(2)\) satisfying

$$\begin{aligned} \int _{\varOmega } |F_{\varepsilon } - R_{\varepsilon }|^2 \, dx \le C(\varOmega ) \left( \int _{\varOmega } {\text {dist}}(F_{\varepsilon }, SO(2))^2 \, dx + \left( \left| {\text {curl }}(F) - \varepsilon \begin{pmatrix} 1 \\ 0 \end{pmatrix}\right| (\varOmega )\right) ^2 \right) . \end{aligned}$$

3 Rigidity for Rotation Fields in Dimension \({\mathbf {3}}\)

This section is devoted to prove that in three dimensions a rotation field whose \({\text {curl }}\) is constant has to be locally constant.

3.1 A simple argument for \(\mathbf {\varOmega = {\mathbb {R}}^3}\)

We start with a simple argument for \(\varOmega = {\mathbb {R}}^3\) which is based on Stokes’ theorem.

Theorem 2

Let \(R \in C^1({\mathbb {R}}^3;SO(3))\) such that \({\text {curl }}R = \alpha \) for some \(\alpha \in {\mathbb {R}}^{3\times 3}\). Then \(\alpha = 0\) and R is constant.

Proof

If \(\alpha = 0\) then the result follows by the classical rigidity result for gradients. So we assume that \(\alpha \ne 0\). Hence, there exists \(v \in {\mathbb {R}}^3\) such that \(\alpha v \ne 0\). Up to a rotation we may assume that \(v = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\). Now, we define for \(\rho > 0\) the two-dimensional disk and circle with radius \(\rho \) as

$$\begin{aligned}&D^{(2)}_{\rho } = \left\{ (x_1,x_2,x_3) \in {\mathbb {R}}^3: x_1^2 + x_2^2 < \rho ^2, x_3=0 \right\} \end{aligned}$$
(3)
$$\begin{aligned} \text { and }&S_{\rho }^{(2)} = \left\{ (x_1,x_2,x_3) \in {\mathbb {R}}^3: x_1^2 + x_2^2 = \rho ^2, x_3 = 0 \right\} . \end{aligned}$$
(4)

We choose v to be the normal to \(D_{\rho }^{(2)}\) and denote by \(\tau \in S^2\) the corresponding positively oriented tangent to \(S^{(2)}_{\rho }\). Using Stokes’ theorem we compute

$$\begin{aligned} \pi \rho ^2 \, \left\| \alpha v\right\| = \left\| \int _{D^{(2)}_{\rho }} {\text {curl }}R \cdot \nu \, {\mathcal {H}}^2\right\| = \left\| \int _{S^{(2)}_{\rho }} R \tau \, d{\mathcal {H}}^1\right\| \le 2\pi \rho . \end{aligned}$$
(5)

For the last inequality we used that \(\Vert R \tau \Vert = 1\) since \(R \in SO(3)\). This yields a contradiction for every \(\rho > \frac{2}{\Vert \alpha v\Vert }\). \(\square \)

Remark 2

The proof shows that there cannot be \(R \in C^1(\varOmega ;SO(3))\) with \({\text {curl }}R = \alpha \) and \(B_{2\Vert \alpha \Vert _{op}+\delta }(x) \subseteq \varOmega \) for some \(x \in \varOmega \), \(\delta > 0\) and \(\Vert \alpha \Vert _{op} = \sup \{ \alpha v: \Vert v\Vert = 1\}\).

3.2 The general result

In this section we prove our main result, namely that on any open and connected set \(\varOmega \subseteq {\mathbb {R}}^3\) every sufficiently regular function \(R: \varOmega \rightarrow SO(n)\) with a constant \({\text {curl }}\) is constant.

Our approach is quite similar to the proof of Theorem 1, namely we first show that a field of rotations \(R:\varOmega \rightarrow {\mathbb {R}}^3\) satisfies a linear elliptic PDE. Together with the assumption that \({\text {curl }}R\) is constant this will yield an equality for \(|\nabla R|^2\) in terms of R and \({\text {curl }}R\).

Before we prove the main result we collect a few results that will be needed later.

Proposition 1

Let \(\varOmega \subseteq {\mathbb {R}}^3\) be open and \(R \in C^2(\varOmega ; SO(3))\) with \({\text {curl }}R = \alpha \) for some constant matrix \(\alpha \in {\mathbb {R}}^{3\times 3}\). Then the following hold:

  1. (i)

    \({\text {div }}R_i = \varepsilon _{ijk} \, \alpha _j \cdot R_k\) for \(i\in \{1,2,3\}\).

  2. (ii)

    \(\varDelta R_i = \varepsilon _{ijk} \nabla (\alpha _j \cdot R_k)\).

  3. (iii)

    \(|\nabla R|^2 =- \mathrm {tr} (R^T \alpha R^T \alpha )\).

  4. (iv)

    \(\mathrm {tr}(R^T \alpha R^T \alpha ) = |(R^T \alpha )_{sym}|^2 - |(R^T \alpha )_{skew}|^2\).

  5. (v)

    If \(R(x_0) = Id\) then \(|{\text {div }}(R)(x_0)|^2 = 2 |\alpha _{skew}|^2\).

  6. (vi)

    \(\sum _{i=1}^3 |(\nabla R_i)_{sym}|^2 \ge \frac{1}{3} |{\text {div }}(R)|^2\).

  7. (v)

    \(\sum _{i=1}^3 |(\nabla R_i)_{skew}|^2 = \frac{1}{2} |\alpha |^2\).

Proof

As R takes values in SO(3) we note that the rows of R form an orthonormal frame. Hence, for \(i\in \{1,2,3\}\) we have

$$\begin{aligned} 2 R_i = \varepsilon _{ijk}\, R_j \times R_k. \end{aligned}$$

Consequently, we can compute \({\text {div }}R_i\) as follows that

$$\begin{aligned} 2\, {\text {div }}(R_i) = \varepsilon _{ijk} {\text {div }}\left( R_j \times R_k \right)&= \varepsilon _{ijk} \left( {\text {curl }}(R_j) \cdot R_k - R_j \cdot {\text {curl }}(R_k) \right) \end{aligned}$$
(6)
$$\begin{aligned}&= \varepsilon _{ijk} \,(\alpha _j \cdot R_k - R_j \cdot \alpha _k) \end{aligned}$$
(7)
$$\begin{aligned}&= 2 \varepsilon _{ijk} \, \alpha _j \cdot R_k. \end{aligned}$$
(8)

This shows (i). Now we recall the well-known identity \({\text {curl }}{\text {curl }}= -\varDelta + \nabla {\text {div }}\). As \({\text {curl }}R\) is constant, this yields

$$\begin{aligned} 0 = -\varDelta R_i + \nabla {\text {div }}R_i, \end{aligned}$$
(9)

which shows in combination with (i) claim (ii). For (iii) we first observe for \(i \in \{1,2,3\}\) that

$$\begin{aligned} 0 = \varDelta (|R_i|^2) = 2 \varDelta (R_i) \cdot R_i + 2 |\nabla R_i|^2. \end{aligned}$$

In combination with (ii) this implies

$$\begin{aligned} -|\nabla R|^2 = \varepsilon _{ijk} \nabla (\alpha _j \cdot R_k) \cdot R_i = \varepsilon _{ijk} \,\alpha _{jl} \, \left( \partial _m R_{kl}\right) R_{im}. \end{aligned}$$
(10)

Next, we use (23), i.e., we have for \(m,k,l \in \{1,2,3\}\) that

$$\begin{aligned} 2 \left( \partial _m R\right) _{kl} = \varepsilon _{rml} \alpha _{kr} + \varepsilon _{rsl} R_{ks} \left( R^T \alpha \right) _{mr} + \varepsilon _{rsm} R_{ks} \left( R^T \alpha \right) _{lr}.\end{aligned}$$

Plugging this identity into (10) yields

$$\begin{aligned} 2 \varepsilon _{ijk} \,\alpha _{jl} \, \left( \partial _m R_{kl}\right) R_{im} =&\varepsilon _{ijk} \,\alpha _{jl} \, R_{im} \, \varepsilon _{rml} \alpha _{kr} \\&+ \varepsilon _{ijk} \,\alpha _{jl} \, R_{im} \, \varepsilon _{rsl} R_{ks} \left( R(x)^T \alpha \right) _{mr} \\&+ \varepsilon _{ijk} \,\alpha _{jl} \, R_{im} \varepsilon _{rsm} R_{sr} \left( R^T \alpha \right) _{lr} \\ =:&(I) + (II) + (III). \end{aligned}$$

No we compute

$$\begin{aligned} (I)&= \varepsilon _{ijk} \,\alpha _{jl} \, R_{im} \, \varepsilon _{rmq} \alpha _{kr} = \varepsilon _{ijk} \,\alpha _{jl} \, (\alpha _k \times R_i)_{l} = \varepsilon _{ijk} \, (\alpha _k \times R_i) \cdot \alpha _j, \\ (II)&= \varepsilon _{ijk} \,\alpha _{jl} \, \varepsilon _{rsl} R_{ks} \, \alpha _{ir} \\&= \varepsilon _{ijk} \, (R_k \times \alpha _j)_r \alpha _{ir} =\varepsilon _{ijk} \, (R_k \times \alpha _j) \cdot \alpha _i = -\varepsilon _{ijk} (\alpha _j \times R_k) \cdot \alpha _i = -(I), \\ (III)&= \varepsilon _{ijk} \,\alpha _{jl} \, R_{im} \varepsilon _{rsm} R_{ks} \left( R^T \alpha \right) _{lr} \\&= \varepsilon _{ijk} \, \alpha _{jl} \, (R_k \times R_i)_{r} \, \left( R^T \alpha \right) _{lr} \\&= \varepsilon _{ijk} (R_k \times R_i)_r\, (\alpha R^T \alpha )_{jr} \\&= 2 R_{jr} \, (\alpha R^T \alpha )_{jr} = 2 (R^T \alpha R^T \alpha )_{rr} = 2 \mathrm {tr}(R^T\alpha R^T \alpha ). \end{aligned}$$

Combining (10), (I), (II) and (III) yields (iii).

For (iv) we simply compute

$$\begin{aligned} \mathrm {tr}(R^T\alpha R^T\alpha ) =&(R^T\alpha )^T : (R^T\alpha ) \\ =&\left( (R^T\alpha )_{sym} - (R^T\alpha )_{skew} \right) : \left( (R^T\alpha )_{sym} + (R^T\alpha )_{skew} \right) \\ =&\left| (R^T \alpha )_{sym} \right| ^2 - \left| (R^T \alpha )_{skew} \right| ^2. \end{aligned}$$

Next, we assume that \(R(x_0) = Id\). By (i) we have that \({\text {div }}(R_i)(x_0) = \varepsilon _{ijk} \, \alpha _j \cdot R_k(x_0) = \varepsilon _{ijk} \alpha _{jk}\). Consequently,

$$\begin{aligned} \alpha _{skew} = \frac{1}{2} \begin{pmatrix} 0 &{}&{} {\text {div }}(R_3)(x_0) &{}&{} -{\text {div }}(R_2)(x_0) \\ -{\text {div }}(R_3)(x_0) &{}&{} 0 &{}&{} {\text {div }}(R_1)(x_0) \\ {\text {div }}(R_2)(x_0) &{}&{} -{\text {div }}(R_1)(x_0) &{}&{} 0 \end{pmatrix} \end{aligned}$$

and therefore \(|\alpha _{skew}|^2 = \frac{1}{2} |{\text{ div } }(R)(x_0)|^2\), which is (v).

For (vi), we estimate

$$\begin{aligned} \sum _{i=1}^3 |(\nabla R_i)_{sym}|^2&\ge \sum _{i=1}^3 \left( (\nabla R_i)_{11}^2 + (\nabla R_i)_{22}^2 + (\nabla R_i)_{33}^2 \right) \\&\ge \sum _{i=1}^3 \frac{1}{3} \left( \mathrm {tr}(\nabla R_i)\right) ^2 \\&= \frac{1}{3} \sum _{i=1}^3 \left( {\text {div }}(R_i) \right) ^2 = \frac{1}{3} |{\text {div }}(R)|^2. \end{aligned}$$

Eventually, we prove (vii). We observe for \(i \in \{1,2,3\}\) that

$$\begin{aligned} (\nabla R_i)_{skew}&= \frac{1}{2} \begin{pmatrix} 0 &{}&{} \partial _2 R_{i1} - \partial _1 R_{i2} &{}&{} \partial _3 R_{i1} - \partial _1 R_{i3} \\ \partial _1 R_{i2} - \partial _2 R_{i1} &{}&{} 0 &{}&{} \partial _3 R_{i2} - \partial _2 R_{i3} \\ \partial _1 R_{i3} - \partial _3 R_{i1} &{}&{} \partial _2 R_{i3} - \partial _3 R_{i2} &{}&{} 0 \end{pmatrix} \\&= \frac{1}{2} \begin{pmatrix} 0 &{}&{} -\alpha _{i3} &{}&{} \alpha _{i2} \\ \alpha _{i3} &{}&{} 0 &{}&{} -\alpha _{i1} \\ -\alpha _{i2} &{}&{} \alpha _{i1} &{}&{} 0 \end{pmatrix}. \end{aligned}$$

Therefore,

$$\begin{aligned} \sum _{i=1}^3 \left| (\nabla R_i)_{skew} \right| ^2 = \sum _{i=1}^3 \frac{1}{2} |\alpha _i|^2 = \frac{1}{2} |\alpha |^2. \end{aligned}$$
(11)

\(\square \)

Armed with the results from Proposition 1 we can now show that every field of rotations with a constant \({\text {curl }}\) has to be locally constant.

Theorem 3

Let \(\varOmega \subseteq {\mathbb {R}}^3\) open and connected, and \(R \in C^2(\varOmega ;SO(3))\) such that \({\text {curl }}R = \alpha \) for some \(\alpha \in {{\mathbb {R}}^{3\times 3}}\). Then R is constant.

Proof

We assume first that \(\varOmega \) is simply-connected. For \(\alpha = 0\) the result is the well-known result for gradients. Hence, it suffices to prove that \(\alpha = 0\). Now, let \(x_0 \in \varOmega \). We may assume that \(R(x_0) = Id\). Otherwise consider \({\tilde{R}}(x) = R(x_0)^T R(x)\) and \({\tilde{\alpha }} = R(x_0)^T \alpha \). By Proposition 1 (iii) and (iv) we have

$$\begin{aligned} |\nabla R|^2 = |(R^T \alpha )_{skew}|^2 - |(R^T \alpha )_{sym}|^2. \end{aligned}$$
(12)

On the other hand, combining Proposition 1 (vi) and (vii) yields

$$\begin{aligned} |\nabla R|^2 = \sum _{i=1}^3 |(\nabla R_i)_{sym}|^2 + |(\nabla R_i)_{skew}|^2 \ge \frac{1}{3} |{\text {div }}(R)|^2 + \frac{1}{2} |\alpha |^2. \end{aligned}$$
(13)

Using Proposition 1 (v) we find from combining (12) and (13) at the point \(x_0\)

$$\begin{aligned} |\alpha _{skew}|^2 - |\alpha _{sym}|^2 \ge \frac{2}{3} |\alpha _{skew}|^2 + \frac{1}{2} |\alpha |^2 = \frac{7}{6} |\alpha _{skew}|^2 + \frac{1}{2}|\alpha _{sym}|^2. \end{aligned}$$
(14)

This implies that \(\alpha _{skew} = \alpha _{sym} = 0\) i.e, \(\alpha =0\). This completes the proof if \(\varOmega \) is simply-connected.

Eventually we notice that around every point there exists a simply-connected neighborhood which is included in \(\varOmega \). Then we proved that R is constant in this neighborhood i.e., R is locally constant. As \(\varOmega \) is connected this implies that R is constant. \(\square \)

In combination with Corollary 1 in Sect. 4, Theorem 3 shows our main result.

Theorem 4

Let \(\varOmega \subseteq {\mathbb {R}}^3\) be open and bounded. Then every measurable \(R:\varOmega \rightarrow SO(3)\) with a constant \({\text {curl }}\) in the sense of distributions is constant.

4 Regularity of Rotation Fields is Dominated by Regularity of Their Curl

In this section \(\varOmega \subseteq {\mathbb {R}}^n\) denotes an open set. We will show that the regularity of a measurable field \(R: \varOmega \rightarrow SO(n)\) is determined by the regularity of its \({\text {Curl}}\). Precisely, we will show that if \({\text {Curl}}(R) \in C^k(\varOmega ;{\mathbb {R}}^{n\times n \times n})\) for some \(k\in {\mathbb {N}}\) then \(R \in C^{k+1}(\varOmega ; {\mathbb {R}}^{n\times n})\). In particular, if \({\text {Curl}}(R)\) is constant then R is smooth.

As a first step we recall a statement from [15, 16]. It states that a field of rotations R whose \({\text {Curl}}\) is a finite vector-valued Radon measure is already a function of bounded variation. For the convencience of the reader we present a slight variation of the argument from [15, 16] which can be simplified using the recently obtained generalized rigidity estimates in \(L^{\frac{n}{n-1}}\) from [5] replacing the rigidity estimates in the weak spaces \(L^{\frac{n}{n-1},\infty }\) from [16]. The argument implies local estimates which we will use to derive that DR is absolutely continuous with respect to the measure \({\text {Curl}}(R)\).

Proposition 2

Let \(n\ge 2\) and \(\varOmega \subseteq {\mathbb {R}}^n\) open and bounded. Then there exists a constant \(C>0\) such that for every measurable function \(R: \varOmega \rightarrow SO(n)\) such that \({\text {Curl}}(R) \in {\mathcal {M}}(\varOmega ;{\mathbb {R}}^{n\times n\times n})\) and \(|{\text {Curl}}R|(\varOmega ) < \infty \) it holds for every Borel set \(A \subseteq \varOmega \) that

$$\begin{aligned} |DR|(A) \le C |{\text {Curl }} R|(A). \end{aligned}$$
(15)

In particular, \(R \in BV(\varOmega ;{\mathbb {R}}^{n\times n})\).

Proof

First, let \(A \subseteq \varOmega \) be open. For this let \(\varOmega ' \subseteq A\) be open such that \(\varOmega ' \subset \subset \varOmega \). For \(\delta > 0\) we define

$$\begin{aligned} I_{\delta } = \left\{ i \in \delta {\mathbb {Z}}^n \,|\, i + (-\delta ,\delta )^n \subseteq A \right\} \end{aligned}$$

and for \(i \in I_{\delta }\)

$$\begin{aligned} q_i^{\delta } = i + (-\delta /2,\delta /2)^n \text { and } Q_i^{\delta } = i + (-\delta ,\delta )^n. \end{aligned}$$

Then it holds for \(\delta > 0\) small enough that \(\varOmega ' \subseteq \bigcup _{i \in I_{\delta }} q_i^{\delta } \cup N \subseteq A\), where \(N \subseteq \varOmega \) is a set of Lebesgue measure 0, see Fig. 1.

Fig. 1
figure 1

Sketch of the situation in Proposition 2. The open set \(A \subseteq \varOmega \) is colored in gray, the set \(\varOmega ' \subset \subset A\) is colored in blue. The points in \(I_{\delta }\) are indicated by black dots. The corresponding cubes \(q_i^{\delta }\) are sketched with black boundaries. One specific of the larger cubes \(Q_i^{\delta }\) is sketched in red. Note that they cover \(\varOmega '\) for \(\delta >0\) small enough. The function \(R_{\delta }\) is constant on each of the cubes \(q_i^{\delta }\). Hence, \(DR_{\delta }\) is concentrated on the faces of \(\partial q_i^{\delta }\)

Full size image

Now, fix \(i \in I_{\delta }\). By the generalized rigidity estimate from [5] (if \(n\ge 3\)) or from [17] (if \(n=2\)) there exists \(R_i \in SO(n)\) such that

$$\begin{aligned} { \int _{ Q_i^{\delta }} |R - R_i|^{\frac{n}{n-1}} \, dx \le C |{\text {Curl }} R|(Q_i^{\delta })^{\frac{n}{n-1}} }. \end{aligned}$$
(16)

Note that by a scaling argument it can be shown that for all \(\delta >0\) and \(i\in I_{\delta }\) for \(C>0\) one can use the constant for the domain \((0,1)^n\). In particular, C in the inequality above does not depend on \(\delta \) nor i.

We define a function \(R_{\delta }: \varOmega ' \rightarrow SO(n)\) by \(R_{\delta }(x) = R_i\) if \(x \in q_i^{\delta }\) where \(i \in I_{\delta }\) (note that while each \(R_i\) is defined on \(Q_i^\delta \) which overlap for neighboring indices, the smaller cubes \(q_i^{\delta }\) are mutually disjoint). It follows that \(R_{\delta } \in BV(\varOmega ';SO(n))\) and the distributional derivative of \(R_{\delta }\) is concentrated on the boundaries of neighboring cubes \(q_i\), namely

$$\begin{aligned} |D R_{\delta }|(\varOmega ') = \sum _{i,j \in I_{\delta }, |i - j| = \delta } |R_i - R_j| \, {\mathcal {H}}^{n-1}( \partial q_i^{\delta } \cap \partial q_j^{\delta } \cap \varOmega ') . \end{aligned}$$
(17)

Next, we fix two neighboring indices \(i,j \in I_{\delta }\), i.e., \(|i-j| = \delta \). Then we use (16) to find

$$\begin{aligned} 2^{n-1} \cdot \delta ^n |R_i - R_j|^{\frac{n}{n-1}}&= \int _{Q_i^{\delta } \cap Q_j^{\delta }} |R_i - R_j|^{\frac{n}{n-1}} \, dx \\&\le \int _{Q_i^{\delta }} |R - R_i|^{\frac{n}{n-1}} \, dx + \int _{Q_j^{\delta }} |R - R_j|^{\frac{n}{n-1}} \, dx \\&\le C \left( |{\text {Curl }} R|(Q^{\delta }_i)^{\frac{n}{n-1}} + |{\text {Curl }} R|(Q^{\delta }_j)^{\frac{n}{n-1}} \right) . \end{aligned}$$

In particular, we obtain

$$\begin{aligned} { |R_i - R_j| \le C' \delta ^{-(n-1)} \left( |{\text {Curl }} R|(Q^{\delta }_i) + |{\text {Curl }} R|(Q^{\delta }_j) \right) }. \end{aligned}$$

By the finite overlap of the cubes \(Q_i^{\delta }\) we derive from (17) that

$$\begin{aligned} |D R_{\delta }|(\varOmega ')&\le C' \sum _{\begin{array}{c} i,j \in I_{\delta }, |i - j| = \delta \end{array}} \delta ^{-(n-1)} \delta ^{n-1} ( |{\text {Curl }} R|(Q_i) + |{\text {Curl }} R|(Q_j)) \nonumber \\&\le C'' |{\text {Curl }} R|\left( \bigcup _{i\in I_{\delta }} Q_i \right) \le C'' \, |{\text {Curl }} R|(A). \end{aligned}$$
(18)

Additionally, Hölder’s inequality then yields

$$\begin{aligned} \int _{\varOmega '} |R_{\delta } - R| \, dx&= \sum _{i \in I_{\delta }} \int _{\varOmega ' \cap q_i^{\delta }} |R_i - R| \, dx \\&\le \sum _{i \in I_{\delta }} \int _{Q_i^{\delta }} |R_i - R| \, dx \\&\le \sum _{i \in I_{\delta }} 2\delta \Vert R_{i} - R\Vert _{L^{\frac{n}{n-1}}(Q_i^{\delta })} \\&\le 2C \delta \, \sum _{i \in I_{\delta }} |{\text {Curl }}R|(Q_i^{\delta }) \\&\le C' \delta |{\text {Curl }}R|(A). \end{aligned}$$

For the last inequality we used again the finite overlap of the cubes \(Q_i^{\delta }\). It follows that \(R_{\delta } \rightarrow R\) in \(L^1(\varOmega ';{\mathbb {R}}^{n\times n})\) as \(\delta \rightarrow 0\). By the lower-semicontinuity of the total variation we find from (18) that \(R \in BV(\varOmega ';{\mathbb {R}}^{n\times n})\) and

$$\begin{aligned} |D R|(\varOmega ')| \le \liminf _{\delta \rightarrow 0} |DR_{\delta }|(\varOmega ') \le C'' |{\text {Curl }} R|(A). \end{aligned}$$
(19)

Note that the constant \(C''\) can be chosen independently from \(\varOmega '\).

Now we exhaust A by compactly contained open sets. Precisely, we find a sequence of open sets \(\varOmega '_k \subset \subset A\) such that \(\varOmega _k' \subseteq \varOmega '_{k+1}\) and \(\bigcup _{k \in {\mathbb {N}}} \varOmega _k' = A\). Then DR is a vector-valued Radon measure on A and (19) yields

$$\begin{aligned} |DR|(A) = \lim _{k \rightarrow \infty } |DR|(\varOmega _k') \le C'' |{\text {Curl }} R|(A). \end{aligned}$$

For \(A = \varOmega \) it follows immediately that \(R \in BV(\varOmega ;{\mathbb {R}}^{n\times n})\).

For an arbitrary Borel set \(A \subseteq \varOmega \), we can find for \(\varepsilon > 0\) an open set \(A \subseteq O \subseteq \varOmega \) such that \(|{\text {Curl}} R|(O) \le |{\text {Curl}} R|(A) + \varepsilon \). It follows

$$\begin{aligned} |DR|(A) \le |DR|(O) \le C'' |{\text {Curl}}R|(O) \le C'' \left( |{\text {Curl}} R|(A) + \varepsilon \right) . \end{aligned}$$

Sending \(\varepsilon \rightarrow 0\) yields (15). \(\square \)

Remark 3

We note that (15) shows that the vector-valued Radon measure DR is absolutely continuous with respect to the Radon measure \(|{\text {Curl }}R|\). In particular, if \({\text {Curl }} R \in L^1(\varOmega ;{\mathbb {R}}^{n\times n \times n})\) then DR is absolutely continuous with respect to the Lebesgue measure. In this case by the Radon-Nikodym Theorem (see [7, Section 1.6]) we may write \(D R = g \, {\mathcal {L}}^n\) for some \(g \in L^1(\varOmega ;{\mathbb {R}}^{n\times n \times n})\) and obtain for almost every \(x \in \varOmega \)

In particular, it follows that \(\Vert g\Vert _{L^1} \le C \Vert {\text {Curl }} R\Vert _{L^1}\) which implies that \(R \in W^{1,1}(\varOmega ;{\mathbb {R}}^n)\). In addition, if \({\text {Curl }} R \in L^{\infty }\) then \(R \in W^{1,\infty }\) and (c.f. [18])

$$\begin{aligned} \Vert DR \Vert _{L^{\infty }} \le C \Vert {\text {Curl }} R\Vert _{L^{\infty }}. \end{aligned}$$

In this case, by the Sobolev embedding theorem (see, for example, [25, Theorem 2.4.4]) R can be identified with a function which is locally Lipschitz continuous.

In light of Remark 3 we recall here Rademacher’s theorem (see [8, Theorem 3.1.6]) which states that every Lipschitz function is differentiable at almost every point. Next, we show that for a differentiable function \(R: \varOmega \subseteq {\mathbb {R}}^n \rightarrow SO(n)\) the derivative DR can be expressed in terms of the functions R and \({\text {Curl }} R\).

Proposition 3

Let \(n\in {\mathbb {N}}\), \(\varOmega \subseteq {\mathbb {R}}^n\). Assume that \(R: \varOmega \rightarrow SO(n)\) is differentiable at a point \(x\in \varOmega \). Then we have for \(i,k,l,p \in \{1,\dots ,n\}\)

$$\begin{aligned} 2 \left( R(x)^T (\partial _i R)(x)\right) _{kl} =&R(x)_{mk} \left( {\text {Curl}}R(x)\right) _{mil} + R(x)_{mi} \left( {\text {Curl}}R(x)\right) _{mkl} \\&+ R(x)_{ml} \left( {\text {Curl}}R(x)\right) _{mki} \nonumber \end{aligned}$$
(20)

and

$$\begin{aligned} 2 \left( (\partial _i R)(x)\right) _{pl} =&\left( {\text {Curl}}R(x)\right) _{pil} + R(x)_{pk} R(x)_{mi} \left( {\text {Curl}}R(x)\right) _{mkl} \\&+ R(x)_{pk} R(x)_{ml} \left( {\text {Curl}}R(x)\right) _{mki}. \nonumber \end{aligned}$$
(21)

In particular, we have for \(n=3\)

$$\begin{aligned} 2 \left( R(x)^T (\partial _i R)(x)\right) _{kl}&= \varepsilon _{nil} \left( R(x)^T ({\text {curl }}R)(x)\right) _{kn} + \varepsilon _{nkl} \left( R(x)^T ({\text {curl }}R)(x)\right) _{in} \nonumber \\&\quad + \varepsilon _{nki} \left( R(x)^T ({\text {curl }}R)(x)\right) _{ln}. \end{aligned}$$
(22)

and

$$\begin{aligned} 2 \left( (\partial _i R)(x)\right) _{pl} =&\varepsilon _{nil} \left( ({\text {curl }}R)(x)\right) _{pn} + \varepsilon _{nkl} R(x)_{pk} \left( R(x)^T ({\text {curl }}R)(x)\right) _{in} \\&+ \varepsilon _{nki} R(x)_{pk} \left( R(x)^T ({\text {curl }}R)(x)\right) _{ln}. \nonumber \end{aligned}$$
(23)

Proof

Since \(R(x) \in SO(n)\) for all \(x \in \varOmega \) it follows for all \(i\in \{1,\dots ,n\}\) that \(R(x)^T (\partial _i R)(x)\) is skew-symmetric. Consequently we find that

$$\begin{aligned} 2 \left( R(x)^T (\partial _i R)(x)\right) _{kl} =&\left( R(x)^T (\partial _i R)(x)\right) _{kl} - \left( R(x)^T (\partial _i R)(x)\right) _{lk} \\ =&\left[ \left( R(x)^T (\partial _i R)(x)\right) _{kl} - \left( R(x)^T (\partial _i R)(x)\right) _{lk} \right] \\&+ \left[ \left( R(x)^T (\partial _k R)(x)\right) _{il} + \left( R(x)^T (\partial _k R)(x)\right) _{li} \right] \\&- \left[ \left( R(x)^T (\partial _l R)(x)\right) _{ki} + \left( R(x)^T (\partial _l R)(x)\right) _{ik} \right] \\ =&\left[ \left( R(x)^T (\partial _i R)(x)\right) _{kl} - \left( R(x)^T (\partial _l R)(x)\right) _{ki} \right] \\&+ \left[ \left( R(x)^T (\partial _k R)(x)\right) _{il} - \left( R(x)^T (\partial _l R)(x)\right) _{ik} \right] \\&+ \left[ \left( R(x)^T (\partial _k R)(x)\right) _{li} - \left( R(x)^T (\partial _i R)(x)\right) _{lk} \right] \\ =&R(x)_{mk} \left( {\text {Curl}}R(x)\right) _{mil} + R(x)_{mi} \left( {\text {Curl}}R(x)\right) _{mkl} \\&+ R(x)_{ml} \left( {\text {Curl}}R(x)\right) _{mki}. \end{aligned}$$

This shows (20). Then (21) follows immediately by multiplication from the left with R. Now, we notice that for \(n=3\), it holds \(({\text {Curl}}R)_{qrs} = \varepsilon _{nrs} ({\text {curl }}R)_{qn}\). Plugging this identity into (20) and (21) yields immediately (22) and (23). \(\square \)

Combining Proposition 2, Remark 3 and Proposition 3 allows us to prove regularity of rotation fields with a regular \({\text {curl }}\).

Theorem 5

Let \(n,k \in {\mathbb {N}}\), \(\varOmega \subseteq {\mathbb {R}}^n\) open and \(R: \varOmega \rightarrow SO(n)\) measurable. Assume that \({\text {Curl}} R = f\) in the sense of distributions for \(f \in C^k(\varOmega ;{\mathbb {R}}^{n\times n\times n})\). Then \(R \in C^{k+1}(\varOmega ;{\mathbb {R}}^{n\times n})\).

Proof

First, let \(k=0\). As differentiability is a local property we may assume that \(\varOmega \) is bounded and \({\text {Curl }} R\) is bounded. By Remark 3 it follows that \(R \in W^{1,\infty }(\varOmega ;{\mathbb {R}}^{n\times n})\). By the Sobolev-embedding theorem it can hence be identified with a function which is locally Lipschitz-continuous. Then Rademacher’s theoerem (see, for example, [8, Theorem 3.1.6]) yields that R is differentiable almost everywhere and that at almost every point the classical and the weak derivative coincide. Then Proposition 3 implies that the weak derivative DR is for almost every point the sum of terms which are products of components of R and \({\text {curl }}R\). Thus DR can be represented through a continuous function. This implies that \(R \in C^1(\varOmega ;SO(n))\) which is the statement for \(k=0\). If \(k > 0\) we can bootstrap this argument. We see now that DR is the sum of products of terms which are \(C^1\) (components of R) or \(C^k\) (components of \({\text {curl }}R\)). Hence, by the product rule \(R \in C^2\) and second derivatives are sums of products of R, DR, \({\text {curl }}R\), or \(D{\text {curl }}R\). This is the statement for \(k=1\). All appearing terms of DR are again \(C^1\) if \(k\ge 2\). Inductively, one can show that derivatives of order \(k+1\) exist and are given by sums of products which consist of components of the first k derivatives of R and \({\text {curl }}R\). \(\square \)

An immediate consequence of this is that rotation fields with a constant \({\text {Curl}}\) in the sense of distributions are necessarily smooth.

Corollary 1

Let \(n \in {\mathbb {N}}\), \(\varOmega \subseteq {\mathbb {R}}^n\) open and \(R: \varOmega \rightarrow SO(n)\) be measurable. Assume that the distributional \({\text {Curl}} R\) is locally constant. Then \(R \in C^{\infty }(\varOmega ;{\mathbb {R}}^{n\times n})\).