1 Introduction

The estimate

$$\begin{aligned} \exists \, c > 0 \ \forall \, u\in W^{1,\,p}_{0}(\Omega ,\mathbb {R}^n): \qquad \Vert {\text {D}}u\Vert _{L^p(\Omega ,\mathbb {R}^{n\times n})}\le c\, \Vert {\text {dev}}_n{\text {sym}}{\text {D}}u\Vert _{L^{p}(\Omega ,\mathbb {R}^{n\times n})}, \end{aligned}$$
(1.1)

for \(n\ge 2\) and \(p\in (1,\infty )\) where denotes the deviatoric (trace-free) part of the square matrix X and its (compatible) generalizations of (1.1) are well known, cf. [3, 4, 14, 16, 17]. In [9] another generalization to the (incompatible) case

$$\begin{aligned} \begin{aligned} \exists \, c > 0 \ \forall \, P\in W^{1,\,p}_{0}({\text {Curl}};\Omega ,\mathbb {R}^{3\times 3})&:\\ \qquad \Vert P\Vert _{L^p(\Omega ,\mathbb {R}^{3\times 3})}&\le c\, \big (\Vert {\text {dev}}_3{\text {sym}}P\Vert _{L^{p}(\Omega ,\mathbb {R}^{3\times 3})}+ \Vert {\text {Curl}}P\Vert _{L^{p}(\Omega ,\mathbb {R}^{3\times 3})} \big ) \end{aligned} \end{aligned}$$
(1.2)

has been given. The main objective of the present paper is to extend (1.2) to the trace-free case for \(n\ge 3\) dimensions. Such a result was expected, cf. [9, Rem. 3.11], and was already proven to hold true for \(p=2\), cf. [1]. However, the latter used a Helmholtz decomposition and a Maxwell estimate and is not amenable to the \(L^p\)-case. On the contrary, the argumentation scheme using the Lions lemma resp. Nečas estimate, known from classical Korn inequalities, turned out to be also fruitful in the case of Korn inequalities for incompatible tensor fields, cf. [8,9,10,11] and also [5]. The secret of success is then to determine a linear combination of certain partial derivatives. One such expression in [9] was denoting by L a constant coefficients linear operator, for a skew-symmetric matrix field A and scalar field \(\zeta \). Here, we catch up with a corresponding linear expression in all dimensions \(n\ge 3\). For that purpose, a careful investigation of the generalized cross product, especially a corresponding matrix representation, will be given. Indeed, it is this matrix representation which allows us to obtain suitable relations which are not easily visible in index notations. Korn’s inequalities in higher dimensions for matrix-valued fields whose incompatibility is a bounded measure and corresponding rigidity estimates were obtained in the recent papers [2, 7], however, without boundary conditions. More precisely, Conti and Garroni [2] obtained as a consequence of a Hodge decomposition with critical integrability due to Bourgain and Brezis for \(P\in L^1 (\Omega ,\mathbb {R}^{n\times n})\) with \({\text {Curl}}_{n} P\in \L ^1(\Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}})\) the sharp geometric rigidity estimate

$$\begin{aligned} \inf _{R\in {\text {SO}}(n)}\Vert P-R\Vert _{L^{1^*}(\Omega )}&\le c\, \Big (\Vert {\text {dist}}(P,{\text {SO}}(n))\Vert _{L^{1^*}(\Omega )}+\Vert {\text {Curl}}_{n} P\Vert _{L^1(\Omega )}\Big ) \end{aligned}$$
(1.3)

with a constant \(c=c(n,\Omega )\), the Sobolev-conjugate exponent \(1^*:=\frac{n}{n-1}\), and where the generalized \({\text {Curl}}_{n}\) is seen without a matrix representation as \(({\text {Curl}}_{n} P)_{ijk} :=\partial _i P_{kj}-\partial _j P_{ki}\). Replacing the geometric rigidity by Korn’s inequality, they deduced from (1.3) furthermore

$$\begin{aligned} \inf _{A\in \mathfrak {so}(n)}\Vert P-A\Vert _{L^{1^*}(\Omega )}&\le c\, \Big (\Vert {\text {sym}}P\Vert _{L^{1^*}(\Omega )}+\Vert {\text {Curl}}_{n} P\Vert _{L^1(\Omega )}\Big ). \end{aligned}$$
(1.4)

These estimates remain true for \({\text {Curl}}_{n} P\) being a Radon measure. In that case, the \(L^1\)-norm of \({\text {Curl}}_{n} P\) has to be substituted by the total variation of the measure \({\text {Curl}}_{n} P\), cf. [2]. Lauteri and Luckhaus [7] obtained the rigidity estimate (1.3) in the Lorentz space \(L^{1^*,\infty }\). In [10] we have already established the corresponding results in the \(L^p\)-setting. Here, we focus on the trace-free case showing that the symmetric part can even be replaced by the symmetric deviatoric part.

2 Preliminaries and auxiliary results

By \(.\otimes .\) we denote the dyadic product and by \(\big \langle .,.\big \rangle \) the scalar product, \(\mathfrak {so}(n):=\{A\in \mathbb {R}^{n\times n}\mid A^T = -A\}\) is the Lie-algebra of skew-symmetric matrices and \({\text {Sym}}(n):=\{X\in \mathbb {R}^{n\times n}\mid X^T=X\}\).

Recall that usually the higher-dimensional generalization of the \({\text {curl}}\) is an operation \({\text {curl}}_{n}:\mathscr {D}'(\Omega ,\mathbb {R}^n) \rightarrow \mathscr {D}'(\Omega ,\mathbb {R}^{\frac{n(n-1)}{2}})\) given by

$$\begin{aligned} {\text {curl}}_{n}a:=\begin{pmatrix} \partial _1 a_2 - \partial _2 a_1 \\ \partial _1 a_3 - \partial _3 a_1 \\ \partial _2 a_3 - \partial _3 a_2 \\ \partial _1 a_4 - \partial _4 a_1 \\ \partial _2 a_4 - \partial _4 a_2 \\ \partial _3 a_4 - \partial _4 a_3 \\ \ldots \end{pmatrix}. \end{aligned}$$
(2.1)

Thus, in order to express this operation using the Hamiltonian formalism as a generalized cross product with \(\nabla \), we focus on the following bijection \(\mathfrak {a}_n: \mathfrak {so}(n)\rightarrow \mathbb {R}^{\frac{n(n-1)}{2}}\) given by

$$\begin{aligned} \mathfrak {a}_n (A):=(A_{12},A_{13},A_{23},\ldots ,A_{1n},\ldots ,A_{(n-1)n})^T \quad \text {for }A\in \mathfrak {so}(n) \end{aligned}$$
(2.2a)

as well as its inverse \(\mathfrak {A}_n:\mathbb {R}^{\frac{n(n-1)}{2}}\rightarrow \mathfrak {so}(n)\), so that

$$\begin{aligned} \mathfrak {A}_n(\mathfrak {a}_n (A)) = A \quad \text {for all }A\in \mathfrak {so}(n) \text { and} \quad \mathfrak {a}_n(\mathfrak {A}_n (a))= a \quad \text {for all }a\in \mathbb {R}^{\frac{n(n-1)}{2}} \end{aligned}$$
(2.2b)

and in coordinates it looks like

$$\begin{aligned} \mathfrak {A}_n(a)= \begin{pmatrix} 0 &{} a_1 &{} a_2 &{} a_4 &{} \vdots \\ -a_1 &{} 0 &{} a_3 &{} a_5 &{} \vdots \\ -a_2 &{} -a_3 &{} 0 &{} a_6&{} \vdots \\ -a_4 &{} -a_5&{} -a_6 &{} 0 &{} \vdots \\ \cdots &{} \cdots &{} \cdots &{} \cdots &{} 0 \end{pmatrix}. \end{aligned}$$
(2.3)

Moreover, we will make use of the following notations

$$\begin{aligned} b&= (\overline{b},b_n)^T \in \mathbb {R}^n \quad \text {with } \overline{b}\in \mathbb {R}^{n-1} \end{aligned}$$
(2.4a)

and

(2.4b)

2.1 A generalized cross product

Regarding our goal to express \({\text {curl}}_{n}\) by the Hamiltonian formalism, we apply the following generalization of the cross product for \(n\ge 2\) acting as \(\times _{n}:\mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathbb {R}^{\frac{n(n-1)}{2}}\) by

$$\begin{aligned} a\times _{n}b:=\mathfrak {a}_n(a\otimes b -b \otimes a) \qquad \text {for }a,b\in \mathbb {R}^n. \end{aligned}$$
(2.5a)

Since for a fixed \(a\in \mathbb {R}^n\) the operation \(a\times _{n}.\) is linear, in the second component there exists a unique matrix denoted by such that

(2.5b)

The matrices can be characterized inductively, and for \(a=(\overline{a},a_n)^T\) the matrix has the form

(2.6)

so,

(2.7)

Remark 2.1

There are many possible identifications of skew-symmetric matrices with vectors. However, it is this matrix representation of the generalized cross product \(\times _{n}\) which allows us to establish the main identities needed for Lemma 2.9. Indeed, they were not easily visible to us before in index notations. Moreover, with this matrix representation in hand, the discussion of the boundary condition (see Observation 2.7) as well as the partial integration formula (2.56) is more transparent.

Remark 2.2

The entries of the generalized cross product \(a\times _3 b\), with \(a,b\in \mathbb {R}^3\), are permutations (with a sign) of the entries of the classical cross product \(a\times b\). Recall that also the operation \(a\times .\) can be identified with a multiplication with the skew-symmetric matrix

$$\begin{aligned} {\text {Anti}}(a)=\begin{pmatrix} 0 &{} -a_3 &{} a_2 \\ a_3 &{} 0 &{} -a_1 \\ -a_2 &{} a_1 &{} 0 \end{pmatrix} \end{aligned}$$
(2.8)

which differs from the expression and from \(\mathfrak {A}_3(a)\) which reads

$$\begin{aligned} \mathfrak {A}_3(a)=\begin{pmatrix} 0 &{} a_1 &{} a_2 \\ -a_1 &{} 0 &{} a_3 \\ -a_2 &{} -a_3 &{} 0 \end{pmatrix}. \end{aligned}$$
(2.9)

Moreover, we have

(2.10)

this allows us to define a generalized cross product of a vector \(b\in \mathbb {R}^n\) and a matrix \(P\in \mathbb {R}^{n\times m}\) from the left and with a matrix \(B\in \mathbb {R}^{m\times n}\) from the right via

(2.11a)

and

(2.11b)

In such a way, we obtain for all \(b\in \mathbb {R}^n\):

(2.12)

Furthermore, for \(a,b\in \mathbb {R}^n\) it holds

(2.13)

end especially for \(a=b\):

(2.14)

Hence, for all \(a,b\in \mathbb {R}^n\):

(2.15)

by induction over n, and, especially for \(a=b\):

(2.16)

The entries of are, by definition, linear combinations of \(b_i\,b_j\), the entries of \(b\otimes b\). Interestingly, for \(n\ge 3\) also the converse holds true, i.e., the entries of \(b\otimes b\) are linear combinations of the entries of which will be assertion of the subsequent lemma. Moreover, we will use this as a key observation to achieve the existence of linear combinations in     for \(n\ge 3\), so that we can follow the argumentation scheme presented in \(n=3\) dimensions, cf. [9], also in the higher-dimensional case.

Lemma 2.3

For all \(b\in \mathbb {R}^n\) with \(n\ge 3\), we have:

(2.17)

denoting by L a corresponding constant coefficients linear operator.

Remark 2.4

There are no linear combinations (2.17) in \(n=2\) dimensions. Indeed, we only have

(2.18)

so that there are no linear expressions of \({b_1}^2\), \({b_2}^2\) nor of \(b_1b_2\) in terms of the sole entry of .

Proof of Lemma 2.3by induction over \(n\ge 3\) For the base case \(n=3\), we have

(2.19)

and

(2.20)

Thus, for all \(b\in \mathbb {R}^3\)

(2.21)

and consequently from the expression (2.19) we conclude

(2.22)

Now, assume for the inductive step that for all \(\overline{b}\in \mathbb {R}^{n-1}\) with \(n\ge 4\) we have

(2.23)

For \(b\in \mathbb {R}^n\) we have

(2.24)

Surely,     by the expression (2.14). The induction hypothesis gives

(2.25)

hence, also   ,   so that finally

\(\square \)

By definition (2.11b), the entries of \(B\times _{n}b\) are linear combinations of the entries \(B_{ij}b_k\), i.e., of the entries of the matrix B multiplied with the entries of the vector b. For skew-symmetric matrices, also the converse holds true. This is the assertion of the next lemma.

Lemma 2.5

For all \(A\in \mathfrak {so}(n)\) and \(b\in \mathbb {R}^n\) with \(n\ge 2\), we have

  1. (i)

    \(A\times _{n}b = L(\mathfrak {a}_n (A) \otimes b)\)

  2. (ii)

    \(\mathfrak {a}_n (A) \otimes b = L(A\times _{n}b)\)

denoting by L a corresponding constant coefficients linear operator which can differ in both cases.

Proof

For \(A\in \mathfrak {so}(n)\) and \(b\in \mathbb {R}^n\), we have

(2.26)

and on the other hand

(2.27)

Thus, the conclusions of the lemma follow by induction over the dimension n. Indeed, for the base case \(n=2\) we have

$$\begin{aligned} \begin{pmatrix} 0 &{} \alpha \\ -\alpha &{} 0 \end{pmatrix}\times _{2}\begin{pmatrix}b_1 \\ b_2\end{pmatrix} = \begin{pmatrix} 0 &{} \alpha \\ -\alpha &{} 0 \end{pmatrix}\begin{pmatrix}-b_2 \\ b_1\end{pmatrix} =\begin{pmatrix} -\alpha \,b_1 \\ -\alpha \,b_2\end{pmatrix} = -\left( \alpha \otimes b\right) ^T \end{aligned}$$
(2.28)

which establishes (i) and (ii) of the lemma for \(n=2\).

For the inductive step, let us assume that the statement of the lemma holds for all \(\overline{A}\in \mathfrak {so}(n-1)\) and all \(\overline{b}\in \mathbb {R}^{n-1}\), i.e., it holds:

  1. (i)

    \(\overline{A}\times _{n-1}\overline{b} = L(\mathfrak {a}_{n-1}(\overline{A}) \otimes \overline{b})\)       and

  2. (ii)

    \(\mathfrak {a}_{n-1} (\overline{A}) \otimes \overline{b} = L(\overline{A}\times _{n-1}\overline{b})\).

Thus, returning to \(A\in \mathfrak {so}(n)\) and \(b\in \mathbb {R}^n\) we have by the expressions (2.26) and (2.27), respectively,

  1. (i)

    \(\overline{A}\times _{n-1}\overline{b} = L(\mathfrak {a}_{n} (A) \otimes b)\)       and

  2. (ii)

    \(\mathfrak {a}_{n-1}(\overline{A}) \otimes \overline{b} = L(A\times _{n} b)\)

and the conclusion of part (i) of the lemma follows then from the definition of the generalized cross product, indeed,

$$\begin{aligned} -\left( \,\overline{A\,e_n}\times _{n-1}\overline{b}\right) ^T=\left( \mathfrak {a}_{n-1}(\overline{b} \otimes \overline{A\,e_n}- \overline{A\,e_n}\otimes \overline{b} )\right) ^T \overset{(2.26)}{=}L(\mathfrak {a}_{n}(A) \otimes b). \end{aligned}$$
(2.29)

On the other hand, we have

$$\begin{aligned} b_n\cdot \overline{A}-\overline{A\,e_n}\otimes \overline{b}-\left( b_n\cdot \overline{A}-\overline{A\,e_n}\otimes \overline{b}\right) ^T + \mathfrak {A}_{n-1}\left( \overline{A\,e_n}\times _{n-1}\overline{b}\right) = 2\,b_n\cdot \overline{A}, \end{aligned}$$
(2.30)

so that,   \(b_n\cdot \overline{A} = L(A\times _{n} b)\)   and also   \(b_n\cdot \mathfrak {a}_{n-1}(\overline{A}) = L(A\times _{n} b)\) which by (2.27) implies that \(\overline{A\,e_n}\otimes \overline{b}= L(A\times _{n} b)\).   This finishes the proof of (ii) since we have shown that all the entries of \(\mathfrak {a}_n (A) \otimes b\) can be written as linear combinations of the entries of \(A\times _{n}b\). \(\square \)

Remark 2.6

The identity (2.30) is not a new result, and usually it is expressed using coordinates:

$$\begin{aligned} (A\times _{n}b)_{kij}-(A\times _{n}b)_{kji}+(A\times _{n}b)_{jik} = 2\,A_{ij}\,b_k \qquad \forall \ i,j,k=1,\ldots ,n. \end{aligned}$$
(2.31)

However, we included the statement as well as the proof not only for the sake of completeness, but also since the use of the matrix representation of the generalized cross product allows us to give a coordinate-free proof and provides a deeper insight into the algebra needed in the present paper. \(\square \)

For a square matrix \(P\in \mathbb {R}^{n\times n}\), we can take the generalized cross product with a vector \(b\in \mathbb {R}^n\) both left and right, and simultaneously we obtain for the identity matrix

(2.32)

and for a general matrix \(P\in \mathbb {R}^{n\times n}\)

(2.33)

Thus, especially for a symmetric matrix \(S\in {\text {Sym}}(n)\) and a skew-symmetric matrix \(A\in \mathfrak {so}(n)\) we obtain

$$\begin{aligned} b\times _{n}S\times _{n} b\in {\text {Sym}}\left( \frac{n(n-1)}{2}\right) \qquad \text {and}\qquad b\times _{n}A\times _{n} b\in \mathfrak {so}\left( \frac{n(n-1)}{2}\right) . \end{aligned}$$
(2.34)

Observation 2.7

Let \(A\in \mathfrak {so}(n)\) and \(\alpha \in \mathbb {R}\). Then,   for \(b\in \mathbb {R}^n\backslash \{0\}\)   implies \(A=0\) and \(\alpha =0\).

Proof

Taking the generalized cross product from the left on both sides of gives

(2.35)

so that taking the trace of the symmetric part on both sides we obtain

(2.36)

which implies   \(\alpha =0\). Consequently, we moreover have  \(A\times _{n}b=0\)  which by Lemma 2.5 (ii) yields   \(\mathfrak {a}_n(A)\otimes b =0\), and thus \(\mathfrak {a}_n(A)=0\) and \(A=0\). \(\square \)

2.2 Considerations from vector calculus

Subsequently, we make use of the algebraic behavior of the vector differential operator \(\nabla \) as a vector for formal calculations. So, the derivative and the divergence of a vector field \(a\in \mathscr {D}'(\Omega ,\mathbb {R}^n)\) can be seen as

(2.37)

In a similar way, the generalized curl is related to the generalized cross product

(2.38)

The latter expression gives a generalized row-wise matrix \({\text {Curl}}_{n}\) operator for \(B\in \mathscr {D}'(\Omega ,\mathbb {R}^{m\times n})\) via

(2.39)

This differential operator kills the Jacobian matrix of a vector field (a compatible field), indeed

(2.40)

since \(b\times _{n}b=0\) for all \(b\in \mathbb {R}^n\). Furthermore, for a scalar field \(\zeta \in \mathscr {D}'(\Omega ,\mathbb {R})\) we obtain

(2.41)

For \(P\in \mathscr {D}'(\Omega ,\mathbb {R}^{n\times m})\), we consider also the column-wise differential operator of first order coming from a cross product from the left, namely

(2.42)

which kills the transposed Jacobian \(({\text {D}}a)^T\).

It is clear that   \({\text {Curl}}_{n} B=L({\text {D}}B)\),   i.e., the entries of \({\text {Curl}}_{n} B\) are linear combinations of the entries of \({\text {D}}B\) for all \(B\in \mathscr {D}'(\Omega ,\mathbb {R}^{m\times n})\). However, for skew-symmetric matrix fields also the converse holds true:

Corollary 2.8

For all \(A\in \mathscr {D}'(\Omega ,\mathfrak {so}(n))\) with \(n\ge 2\) it holds: \({\text {D}}A=L({\text {Curl}}_{n}A)\).

It is a well-known result and follows from the linear expression (ii) in Lemma 2.5 replacing b by \(-\nabla \) as well as from its analogous statement written out in coordinates (2.31).

We now catch up with the crucial linear relation needed in our argumentation scheme.

Lemma 2.9

Let \(n\ge 3\), \(A\in \mathscr {D}'(\Omega ,\mathfrak {so}(n))\) and \(\zeta \in \mathscr {D}'(\Omega ,\mathbb {R})\). Then, the entries of are linear combinations of the entries of .

Remark 2.10

The statement is false in \(n=2\) dimensions. Indeed, with \(\alpha ,\zeta \in \mathscr {D}'(\Omega ,\mathbb {R})\) we have

$$\begin{aligned} {\text {Curl}}_{2}\begin{pmatrix} \zeta &{} \alpha \\ -\alpha &{} \zeta \end{pmatrix} = \begin{pmatrix} \zeta &{} \alpha \\ -\alpha &{} \zeta \end{pmatrix} \begin{pmatrix} -\partial _2 \\ \partial _1\end{pmatrix} =\begin{pmatrix} \partial _1\alpha - \partial _2\zeta \\ \partial _2\alpha + \partial _1\zeta \end{pmatrix} \end{aligned}$$
(2.43)

so that

$$\begin{aligned} {\text {D}}{\text {Curl}}_{2}\begin{pmatrix} \zeta &{} \alpha \\ -\alpha &{} \zeta \end{pmatrix} = \begin{pmatrix} \partial _1\partial _1\alpha -\partial _1\partial _2\zeta &{} \,\,\partial _1\partial _2\alpha -\partial _2\partial _2\zeta \\ \partial _1\partial _2\alpha + \partial _1\partial _1\zeta &{} \,\,\partial _2\partial _2\alpha +\partial _1\partial _2\zeta \end{pmatrix} \end{aligned}$$
(2.44)

and we cannot extract \(\partial _1\partial _1\, \alpha \) from the components of (2.44).

Proof of Lemma 2.9

The proof is divided into the two observations

  1. 1.

    ,

  2. 2.

denoting by L a corresponding constant coefficients linear operator which can differ in both cases. To show that the entries of the Hessian \({\text {D}}^2 \zeta \) can be written as linear combinations of the entries of , we introduce the following second-order derivative operator for square matrix fields \(P\in \mathscr {D}'(\Omega ,\mathbb {R}^{n\times n})\):

$$\begin{aligned} \mathbf{inc }_n P:=\nabla \times _{n}P\times _{n}\nabla = - \nabla \times _{n}({\text {Curl}}_{n}P) \end{aligned}$$
(2.45)

in the style of the incompatibility operator known from the three-dimensional case. In regard of (2.32), we see

(2.46)

so that substituting b by \(-\nabla \) in Lemma 2.3 we obtain

(2.47)

Moreover, with regard to (2.34)\(_2\) we have for a skew-symmetric matrix field \(A\in \mathscr {D}'(\Omega ,\mathfrak {so}(n))\):

$$\begin{aligned} \mathbf{inc }_n A = \nabla \times _{n}A\times _{n}\nabla \in \mathfrak {so}\left( \frac{n(n-1)}{2}\right) , \end{aligned}$$
(2.48)

concluding for the 1. part that

(2.49)

where in the last step we have used that the entries of \(\mathbf{inc }_n P = - \nabla \times _{n}({\text {Curl}}_{n}P)\) are, of course, linear combinations of the entries from \({\text {D}}{\text {Curl}}_{n} P\).

To establish part 2, recall that the entries of \({\text {D}}A\) for a skew-symmetric matrix field are linear combinations of the entries of \({\text {Curl}}_{n}A\), giving

(2.50)

The conclusion follows by taking the \(\partial _j\)-derivative of (2.50) together with the observation from the 1. part:

\(\square \)

In the last result of this section, we focus on the kernel of \({\text {dev}}_n{\text {sym}}\) and \({\text {Curl}}_{n}\):

Lemma 2.11

Let \(n\ge 3\), \(A\in L^p(\Omega ,\mathfrak {so}(n))\) and \(\zeta \in L^p(\Omega ,\mathbb {R})\). Then,   in the distributional sense if and only if there exists constant \(b\in \mathbb {R}^n\), \(d\in \mathbb {R}^{\frac{n(n-1)}{2}}\), \(\beta \in \mathbb {R}\) such that   almost everywhere in \(\Omega \).

Remark 2.12

The “only if”-part is false in \(n=2\) dimensions. To see this, take in (2.43) \(\alpha \) and \(\zeta \) to be the real and imaginary part of a holomorphic function.

Proof of Lemma 2.11

For the “if”-part we have

Conversely, in the distributional sense implies by (2.49):

$$\begin{aligned} {\text {D}}^2\zeta \equiv 0 \qquad \text {and}\qquad {\text {D}}^2A\equiv 0\text { in the distributional sense}; \end{aligned}$$
(2.51)

thus,

$$\begin{aligned} \zeta (x)=\big \langle b,x\big \rangle +\beta \qquad \text {and}\qquad A = \mathfrak {A}_n(B\,x+d) \text { a.e.} \end{aligned}$$
(2.52)

for some \(b\in \mathbb {R}^n\), \(\beta \in \mathbb {R}\), \(B\in \mathbb {R}^{\frac{n(n-1)}{2}\times n}\) and \(d\in \mathbb {R}^{\frac{n(n-1)}{2}}\), and we have

as well as

$$\begin{aligned} {\text {Curl}}_{n}A= {\text {Curl}}_{n}(\mathfrak {A}_n\left( B\,x\right) ) = B^T + C^T, \end{aligned}$$

where \(C\in \mathbb {R}^{\frac{n(n-1)}{2}\times n}\) has only possibly nonzero entries at those positions at which the matrix has zeros. Hence, the condition, gives:

(2.53)

implying that \(C\equiv 0\) and almost everywhere in \(\Omega \). \(\square \)

Remark 2.13

The expression of the kernel follows also from the consideration for the classical trace-free Korn inequalities. Indeed, on simply connected domains, \({\text {Curl}}_{n}P\equiv 0\) implies that \(P={\text {D}}u\) for a vector field \(u\in W^{1,\,p}(\Omega ,\mathbb {R}^n)\). Thus, the condition \({\text {dev}}_n{\text {sym}}P\equiv 0\) reads \({\text {dev}}_n{\text {sym}}{\text {D}}u\equiv 0\), whose solutions are well known as infinitesimal conformal mappings, cf.  [1, 3, 6, 12, 14,15,16].

2.3 Function spaces

Having above relations at hand, we can now catch up the arguments from [9]. Let \(\Omega \subseteq \mathbb {R}^n\); we start by defining the space

$$\begin{aligned} W^{1,\,p}({\text {Curl}}_{n}; \Omega ,\mathbb {R}^{n\times n})&:=\{P\in L^p(\Omega ,\mathbb {R}^{n\times n})\mid {\text {Curl}}_{n} P \in L^p(\Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}})\} \end{aligned}$$
(2.54a)

equipped with the norm

$$\begin{aligned} \Vert P\Vert _{ W^{1,\,p}({\text {Curl}}_{n}; \Omega ,\mathbb {R}^{n\times n})}&:=\left( \Vert P\Vert ^p_{L^p(\Omega ,\mathbb {R}^{n\times n})} + \Vert {\text {Curl}}_{n} P\Vert ^p_{L^p(\Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}})} \right) ^{\frac{1}{p}} \end{aligned}$$
(2.54b)

and its subspace \(W^{1,\,p}_0({\text {Curl}}_{n}; \Omega ,\mathbb {R}^{n\times n})\) as the completion of \(C^\infty _0(\Omega ,\mathbb {R}^{n\times n})\) in the \(W^{1,\,p}({\text {Curl}}_{n}; \Omega ,\mathbb {R}^{n\times n})\)-norm.

In our proofs, we shall use an important equivalence of norms due to Nečas [13, Théorème 1] valid on bounded Lipschitz domains, cf. also discussion in [9, 11] and the references cited therein.

Thus, in what follows \(\Omega \subset \mathbb {R}^n\) will be a bounded domain with Lipschitz boundary and we are allowed to define boundary conditions in the distributional sense, so that

$$\begin{aligned} \begin{aligned} W^{1,\,p}_0({\text {Curl}}_{n}; \Omega ,\mathbb {R}^{n\times n})&= \{P\in W^{1,\,p}({\text {Curl}}_{n}; \Omega ,\mathbb {R}^{n\times n}) \mid P \times _{n} \nu = 0 \text { on } \partial \Omega \}\\&= \{P\in W^{1,\,p}({\text {Curl}}_{n}; \Omega ,\mathbb {R}^{n\times n}) \mid P\,\tau _l = 0 \text { on } \partial \Omega \text { for all } l=1,\ldots ,n-1\}, \end{aligned} \end{aligned}$$
(2.55)

where \(\nu \) stands for the outward unit normal vector field and \(\{\tau _l\}_{l=1,\ldots , n-1}\) denotes a moving tangent frame on \(\partial \Omega \), cf. [10]. Here, the generalized tangential trace \(P\times _{n} \nu \) is understood in the sense of \(W^{-\frac{1}{p},\, p}(\partial \Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}})\) which is justified by partial integration, so that its trace is defined by

(2.56)

having denoted by \(\widetilde{Q}\in W^{1,\,p'}(\Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}})\) any extension of Q in \(\Omega \), where \(\big \langle .,.\big \rangle _{\partial \Omega }\) indicates the duality pairing between \(W^{-\frac{1}{p},\,p}(\partial \Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}})\) and \(W^{1-\frac{1}{p'},\,p'}(\partial \Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}})\). Indeed, for smooth P and Q on \(\overline{\Omega }\) we have

(2.57)

where in \((*)\) we have used the fact that we only deal with linear combinations of partial derivatives and from the classical divergence theorem it holds

$$\begin{aligned} \int \limits _{\partial \Omega } \nu _i\, \zeta \,\mathrm {d}S = \int \limits _{\Omega }\partial _i\, \zeta \,\mathrm {d}x \end{aligned}$$
(2.58)

for a smooth scalar function \(\zeta \) on \(\overline{\Omega }\).

Further, following [11] we introduce also the space \(W^{1,\,p}_{\Gamma ,0}({\text {Curl}}_{n};\Omega ,\mathbb {R}^{n\times n})\) of functions with vanishing tangential trace only on a relatively open (non-empty) subset \(\Gamma \subseteq \partial \Omega \) of the boundary by completion of \(C^\infty _{\Gamma ,0}(\Omega ,\mathbb {R}^{n\times n})\) with respect to the \(W^{1,\,p}({\text {Curl}}_{n};\Omega ,\mathbb {R}^{n\times n})\)-norm.

3 Trace-free Korn inequalities for incompatible tensors in higher dimensions

With the auxiliary results in hand, we can now catch up with the higher-dimensional versions of the results presented in [9].

Lemma 3.1

Let \(n\ge 3\), \(\Omega \subset \mathbb {R}^n\) be a bounded Lipschitz domain and \(1<p<\infty \). Then, \(P\in \mathscr {D}'(\Omega ,\mathbb {R}^{n\times n})\), \({\text {dev}}_n{\text {sym}}P\in L^p(\Omega ,\mathbb {R}^{n\times n})\) and \({\text {Curl}}_{n} P \in W^{-1,\,p}(\Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}})\) imply \(P\in L^p(\Omega ,\mathbb {R}^{n\times n})\). Moreover, we have the estimate

(3.1)

with a constant \(c=c(n,p,\Omega )>0\).

Proof

We have to show that follows from the assumptions of the lemma. By the linearity of differential operator \({\text {D}}{\text {Curl}}_{n}\) and the orthogonal decomposition holding in \(\mathscr {D}'(\Omega ,\mathbb {R}^{n\times n})\), we obtain

(3.2)

Thus, by the assumed regularity of the right-hand side, it follows that the left-hand side belongs to \( W^{-2,\,p}(\Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}\times n})\). Furthermore, we have

(3.3)

By Lemma 2.9 we obtain and an application of the Lions lemma resp. Nečas estimate [9, Thm. 2.7 and Cor. 2.8] to yield the conclusions. \(\square \)

Eliminating the first term on the right-hand side of (3.1) gives:

Theorem 3.2

Let \(n\ge 3\), \(\Omega \subset \mathbb {R}^n\) be a bounded Lipschitz domain and \(1<p<\infty \). There exists a constant \(c=c(n,p,\Omega )>0\), such that for all \(P\in L^p(\Omega ,\mathbb {R}^{n\times n})\) we have

$$\begin{aligned} \inf _{T\in K_{dS,C_n}}\Vert P-T\Vert _{L^p(\Omega ,\mathbb {R}^{n\times n})}\le c\,\Big (\Vert {\text {dev}}_n {\text {sym}}P \Vert _{L^p(\Omega ,\mathbb {R}^{n\times n})}+ \Vert {\text {Curl}}_{n} P \Vert _{W^{-1,\,p}(\Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}})}\Big ), \end{aligned}$$
(3.4)

where the kernel is given by

(3.5)

Remark 3.3

This result does not directly extend to \(n=2\), since in that case the condition \({\text {dev}}_2{\text {sym}}{\text {D}}u\equiv 0\) becomes the system of Cauchy–Riemann equations \(\{u_{1,x}=u_{2,y} \ \wedge \ u_{1,y}=-u_{2,x} \}\) so that the corresponding nullspace is infinite-dimensional.

Proof of Theorem 3.2

The characterization of the kernel of the right-hand side gives

so that \(P\in K_{dS,C_n}\) if and only if and . Hence, (3.5) follows by virtue of Lemma 2.11 and the conclusion follows in a similar way to [9, Thm. 3.8]. \(\square \)

Remark 3.4

For compatible displacement gradients \(P={\text {D}}u\), we get back from (3.4) the quantitative version of the classical trace-free Korn’s inequality, cf. [3, 15, 16].

Finally, we examine the effect of tangential boundary conditions \(P\times _{n}\nu \equiv 0\).

Theorem 3.5

Let \(n\ge 3\), \(\Omega \subset \mathbb {R}^n\) be a bounded Lipschitz domain and \(1<p<\infty \). There exists a constant \(c=c(n,p,\Omega )>0\), such that for all \(P\in W^{1,\,p}_0({\text {Curl}}_{n}; \Omega ,\mathbb {R}^{n\times n})\) we have

$$\begin{aligned} \Vert P \Vert _{L^p(\Omega ,\mathbb {R}^{n\times n})}\le c\,\Big (\Vert {\text {dev}}_n{\text {sym}}P \Vert _{L^p(\Omega ,\mathbb {R}^{n \times n})}+ \Vert {\text {Curl}}_{n} P \Vert _{L^p(\Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}})}\Big ). \end{aligned}$$
(3.6)

Proof

We follow the same argumentation scheme as in the proof of [11, Theorem 3.5] and consider a sequence \(\{P_k\}_{k\in \mathbb {N}}\subset W^{1,\,p}_0({\text {Curl}}_{n};\Omega ,\mathbb {R}^{n\times n})\) converging weakly in \(L^p(\Omega ,\mathbb {R}^{n\times n})\) to some \(P^*\) so that \({\text {dev}}_n{\text {sym}}P^* = 0\) almost everywhere and \({\text {Curl}}_{n} P^* = 0\) in the distributional sense, i.e., \(P^*\in K_{dS,C_n}\). By (2.56) we obtain that \(\big \langle P^*\times _{n} (-\nu ),Q\big \rangle _{\partial \Omega }=0\) for all \(Q\in W^{1,\,p'}(\Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}})\). However, the boundary condition \(P^*\times _{n}\nu =0\) is also valid in the classical sense, since \(P^*\in K_{dS,C_n}\) has an explicit representation. Using the explicit representation of , we conclude using Observation 2.7 that, in fact, \(P^*\equiv 0\):

\(\square \)

Remark 3.6

Estimate (3.6) should persist also in \(n=2\) dimensions. So, the case \(p=2\) is already contained in [1]. However, for the general case \(p\in (1,\infty )\) we need a different approach and it will be the subject of a forthcoming note.

Remark 3.7

For compatible \(P={\text {D}}u\) we recover from (3.6) a tangential trace-free Korn inequality.

Remark 3.8

For \(n\ge 3\), the previous results also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset \(\Gamma \subseteq \partial \Omega \) of the boundary, cf. discussion in [11]. But, this is not the case in \(n=2\) dimensions. Indeed, already the trace-free version of Korn’s first inequality (1.1) with only partial boundary condition is false in the \(n=2\) case, cf., e.g., the counterexample contained in [1, section 6.6].