Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}-trace-free version of the generalized Korn inequality for incompatible tensor fields in arbitrary dimensions

For n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 3$$\end{document} and 1<p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<\infty $$\end{document}, we prove an Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}-version of the generalized trace-free Korn-type inequality for incompatible, p-integrable tensor fields P:Ω→Rn×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P:\Omega \rightarrow \mathbb {R}^{n\times n}$$\end{document} having p-integrable generalized Curln\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Curl}}_{n}$$\end{document} and generalized vanishing tangential trace Pτl=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P\,\tau _l=0$$\end{document} on ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document}, denoting by {τl}l=1,…,n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\tau _l\}_{l=1,\ldots , n-1}$$\end{document} a moving tangent frame on ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document}. More precisely, there exists a constant c=c(n,p,Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=c(n,p,\Omega )$$\end{document} such that ‖P‖Lp(Ω,Rn×n)≤c‖devnsymP‖Lp(Ω,Rn×n)+‖CurlnP‖LpΩ,Rn×n(n-1)2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert P \Vert _{L^p(\Omega ,\mathbb {R}^{n\times n})}\le c\,\left( \Vert {\text {dev}}_n {\text {sym}}P \Vert _{L^p(\Omega ,\mathbb {R}^{n \times n})}+ \Vert {\text {Curl}}_{n} P \Vert _{L^p\left( \Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}}\right) }\right) , \end{aligned}$$\end{document}where the generalized Curln\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Curl}}_{n}$$\end{document} is given by (CurlnP)ijk:=∂iPkj-∂jPki\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\text {Curl}}_{n} P)_{ijk} :=\partial _i P_{kj}-\partial _j P_{ki}$$\end{document} and denotes the deviatoric (trace-free) part of the square matrix X. The improvement towards the three-dimensional case comes from a novel matrix representation of the generalized cross product.


Introduction
The estimate ∃ c > 0 ∀ u ∈ W 1, p 0 (Ω, R n ) : Du L p (Ω,R n×n ) ≤ c dev n sym Du L p (Ω,R n×n ) , (1.1) for n ≥ 2 and p ∈ (1, ∞) where dev n X := X − 1 n tr(X) · 1 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 ∃ c > 0 ∀ P ∈ W 1, p 0 (Curl; Ω, R 3×3 ) : has been given. The main objective of the present paper is to extend (1.2) to the trace-free case for n ≥ 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 D 2 (A+ζ·1 3 ) = L(DCurl(A+ζ·1 3 )) denoting by L a constant coefficients linear operator, for a skew-symmetric matrix field A and scalar field ζ. Here, we catch up with a corresponding linear expression in all dimensions n ≥ 3.
127 Page 2 of 14 P. Lewintan and P. Neff ZAMP 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 ∈ L 1 (Ω, R n×n ) with Curl n P ∈ L 1 (Ω, R n× n(n−1) 2 ) the sharp geometric rigidity estimate inf R∈SO(n) with a constant c = c(n, Ω), the Sobolev-conjugate exponent 1 * := n n−1 , and where the generalized Curl n is seen without a matrix representation as (Curl n P ) ijk := ∂ i P kj − ∂ j P ki . Replacing the geometric rigidity by Korn's inequality, they deduced from (1.3) furthermore inf A∈so(n) (1.4) These estimates remain true for Curl n P being a Radon measure. In that case, the L 1 -norm of Curl n P has to be substituted by the total variation of the measure Curl n P , cf. [2]. Lauteri and Luckhaus [7] obtained the rigidity estimate (1.3) in the Lorentz space L 1 * ,∞ . 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.

Preliminaries and auxiliary results
By . ⊗ . we denote the dyadic product and by ., . the scalar product, so(n) := {A ∈ R n×n | A T = −A} is the Lie-algebra of skew-symmetric matrices and Sym(n) := {X ∈ R n×n | X T = X}.
Recall that usually the higher-dimensional generalization of the curl is an operation curl n : Thus, in order to express this operation using the Hamiltonian formalism as a generalized cross product with ∇, we focus on the following bijection a n : so(n) → R n(n−1) 2 given by a n (A) := (A 12 , A 13 , A 23 , . . . , A 1n , . . . , A (n−1)n ) T for A ∈ so(n) (2.2a) as well as its inverse A n : R n(n−1) 2 → so(n), so that A n (a n (A)) = A for all A ∈ so(n) and a n (A n (a)) = a for all a ∈ R n(n−1) 2 (2.2b) ZAMP L p -trace-free version of the generalized Korn inequality Page 3 of 14 127 and in coordinates it looks like Moreover, we will make use of the following notations and (2.4b)

A generalized cross product
Regarding our goal to express curl n by the Hamiltonian formalism, we apply the following generalization of the cross product for n ≥ 2 acting as × n : Since for a fixed a ∈ R n the operation a × n . is linear, in the second component there exists a unique matrix denoted by a ×n ∈ R n(n−1) 2 ×n such that The matrices . ×n can be characterized inductively, and for a = (a, a n ) T the matrix a ×n has the form (2.6) so, (2.7) 127 Page 4 of 14 P. Lewintan and P. Neff ZAMP Remark 2.1. There are many possible identifications of skew-symmetric matrices with vectors. However, it is this matrix representation . ×n of the generalized cross product × 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 × 3 b, with a, b ∈ R 3 , are permutations (with a sign) of the entries of the classical cross product a × b. Recall that also the operation a × . can be identified with a multiplication with the skew-symmetric matrix which differs from the expression a ×3 and from A 3 (a) which reads Moreover, we have (2.10) this allows us to define a generalized cross product of a vector b ∈ R n and a matrix P ∈ R n×m from the left and with a matrix B ∈ R m×n from the right via b × n P := b ×n P ∈ R n(n−1) 2 ×m to be seen as column-wise cross product, to be seen as row-wise cross product.
In such a way, we obtain for all b ∈ R n : Furthermore, for a, b ∈ R n it holds (2.13) end especially for a = b: (2.14) ZAMP L p -trace-free version of the generalized Korn inequality Page 5 of 14 127 Hence, for all a, b ∈ R n : by induction over n, and, especially for a = b: The entries of b ×n × n b are, by definition, linear combinations of b i b j , the entries of b⊗b. Interestingly, for n ≥ 3 also the converse holds true, i.e., the entries of b ⊗ b are linear combinations of the entries of b ×n × n b 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 D 2 (A + ζ · 1 n ) = L(DCurl n (A + ζ · 1 n )) for n ≥ 3, so that we can follow the argumentation scheme presented in n = 3 dimensions, cf. [9], also in the higher-dimensional case.
Proof of Lemma 2.3 by induction over n ≥ 3 For the base case n = 3, we have and consequently from the expression (2.19) we conclude by the expression (2.14). The induction hypothesis gives By definition (2.11b), the entries of B × 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.
denoting by L a corresponding constant coefficients linear operator which can differ in both cases.
Proof. For A ∈ so(n) and b ∈ 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 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 A ∈ so(n − 1) and all b ∈ R n−1 , i.e., it holds: Thus, returning to A ∈ so(n) and b ∈ R n we have by the expressions (2.26) and (2.27), respectively, and the conclusion of part (i) of the lemma follows then from the definition of the generalized cross product, indeed, On the other hand, we have . This finishes the proof of (ii) since we have shown that all the entries of a n (A) ⊗ b can be written as linear combinations of the entries of A × n b.
Remark 2.6. The identity (2.30) is not a new result, and usually it is expressed using coordinates: 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 coordinatefree proof and provides a deeper insight into the algebra needed in the present paper.
For a square matrix P ∈ R n×n , we can take the generalized cross product with a vector b ∈ R n both left and right, and simultaneously we obtain for the identity matrix and for a general matrix P ∈ R n×n Thus, especially for a symmetric matrix S ∈ Sym(n) and a skew-symmetric matrix A ∈ so(n) we obtain (2.34) Observation 2.7. Let A ∈ so(n) and α ∈ R. Then, (A + α · 1 n ) × n b = 0 for b ∈ R n \{0} implies A = 0 and α = 0.
Proof. Taking the generalized cross product from the left on both sides of 0 = A × n b + α · 1 n × n b gives so that taking the trace of the symmetric part on both sides we obtain which implies α = 0. Consequently, we moreover have A × n b = 0 which by Lemma 2.5 (ii) yields a n (A) ⊗ b = 0, and thus a n (A) = 0 and A = 0.

Considerations from vector calculus
Subsequently, we make use of the algebraic behavior of the vector differential operator ∇ as a vector for formal calculations. So, the derivative and the divergence of a vector field a ∈ D (Ω, R n ) can be seen as Da = a ⊗ ∇ and div a = a, ∇ = Da, 1 n = tr(Da). This differential operator kills the Jacobian matrix of a vector field (a compatible field), indeed Furthermore, for a scalar field ζ ∈ D (Ω, R) we obtain Curl n (ζ · 1 n ) = 1 n ∇ζ ×n T = ∇ζ ×n T . (2.41) For P ∈ D (Ω, R n×m ), we consider also the column-wise differential operator of first order coming from a cross product from the left, namely ∇ × n P = ∇ ×n P, It is a well-known result and follows from the linear expression (ii) in Lemma 2.5 replacing b by −∇ 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 ≥ 3, A ∈ D (Ω, so(n)) and ζ ∈ D (Ω, R). Then, the entries of D 2 (A + ζ · 1 n ) are linear combinations of the entries of DCurl n (A + ζ · 1 n ).

Remark 2.10.
The statement is false in n = 2 dimensions. Indeed, with α, ζ ∈ D (Ω, R) we have and we cannot extract ∂ 1 ∂ 1 α from the components of (2.44).
Proof of Lemma 2.9. The proof is divided into the two observations 1. D 2 ζ = L(DCurl n (A + ζ · 1 n )), 2. D 2 A = L(DCurl n (A + ζ · 1 n )) denoting by L a corresponding constant coefficients linear operator which can differ in both cases.
To show that the entries of the Hessian D 2 ζ can be written as linear combinations of the entries of DCurl n (A + ζ · 1 n ), we introduce the following second-order derivative operator for square matrix fields P ∈ D (Ω, R n×n ): inc n P := ∇ × n P × n ∇ = −∇ × n (Curl n P ) (2.45) in the style of the incompatibility operator known from the three-dimensional case. In regard of (2.32), we see Moreover, with regard to (2.34) 2 we have for a skew-symmetric matrix field A ∈ D (Ω, so(n)): concluding for the 1. part that D 2 ζ = L(inc n (ζ · 1 n )) = L(sym(inc n (A + ζ · 1 n ))) = L(inc n (A + ζ · 1 n )) = L(DCurl n (A + ζ · 1 n )) (2.49) where in the last step we have used that the entries of inc n P = −∇ × n (Curl n P ) are, of course, linear combinations of the entries from DCurl n P . To establish part 2, recall that the entries of DA for a skew-symmetric matrix field are linear combinations of the entries of Curl n A, giving The conclusion follows by taking the ∂ j -derivative of (2.50) together with the observation from the 1. part: In the last result of this section, we focus on the kernel of dev n sym and Curl n :

Remark 2.12.
The "only if"-part is false in n = 2 dimensions. To see this, take in (2.43) α and ζ to be the real and imaginary part of a holomorphic function.
Proof of Lemma 2.11. For the "if"-part we have Conversely, Curl n (A + ζ · 1 n ) ≡ 0 in the distributional sense implies by (2.49): , and we have as well as where C ∈ R n(n−1) 2 ×n has only possibly nonzero entries at those positions at which the matrix b ×n has zeros. Hence, the condition, Curl n (A + ζ · 1 n ) ≡ 0 gives: implying that C ≡ 0 and B = b ×n almost everywhere in Ω.
127 Page 10 of 14 P. Lewintan and P. Neff ZAMP 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, Curl n P ≡ 0 implies that P = Du for a vector field u ∈ W 1, p (Ω, R n ). Thus, the condition dev n sym P ≡ 0 reads dev n sym Du ≡ 0, whose solutions are well known as infinitesimal conformal mappings, cf. [1,3,6,12,14-16].

Function spaces
Having above relations at hand, we can now catch up the arguments from [9]. Let Ω ⊆ R n ; we start by defining the space equipped with the norm and its subspace W 1, p 0 (Curl n ; Ω, R n×n ) as the completion of C ∞ 0 (Ω, R n×n ) in the W 1, p (Curl n ; Ω, R n×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 Ω ⊂ R n will be a bounded domain with Lipschitz boundary and we are allowed to define boundary conditions in the distributional sense, so that where ν stands for the outward unit normal vector field and {τ l } l=1,...,n−1 denotes a moving tangent frame on ∂Ω, cf. [10]. Here, the generalized tangential trace P × n ν is understood in the sense of W − 1 p , p (∂Ω, R n× n(n−1) 2 ) which is justified by partial integration, so that its trace is defined by ). Indeed, for smooth P and Q on Ω we have = Ω Curl n P, Q R n× n(n−1) 2 + P, Q ∇ ×n R n×n dx, (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 for a smooth scalar function ζ on Ω. Further, following [11] we introduce also the space W 1, p Γ,0 (Curl n ; Ω, R n×n ) of functions with vanishing tangential trace only on a relatively open (non-empty) subset Γ ⊆ ∂Ω of the boundary by completion of C ∞ Γ,0 (Ω, R n×n ) with respect to the W 1, p (Curl n ; Ω, R n×n )-norm.

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].
Proof. We have to show that skew P + 1 n tr P · 1 n ∈ L p (Ω, R n×n ) follows from the assumptions of the lemma. By the linearity of differential operator DCurl n and the orthogonal decomposition P = skew P + 1 n tr P · 1 n + dev n sym P holding in D (Ω, R n×n ), we obtain DCurl n (skew P + 1 n tr P · 1 n ) = DCurl n P − DCurl n dev n sym P in D (Ω, R n×n ). (3.2) Thus, by the assumed regularity of the right-hand side, it follows that the left-hand side belongs to W −2, p (Ω, R n× n(n−1) 2 ×n ). Furthermore, we have DCurl n (skew P + 1 n tr P · 1 n ) W −2, p ≤ DCurl n P W −2, p + DCurl n dev n sym P W −2, p ≤ c ( Curl n P W −1, p + dev n sym P L p ).
By Lemma 2.9 we obtain D 2 (skew P + 1 n tr P · 1 n ) ∈ W −2, p and an application of the Lions lemma resp. Nečas estimate [9, Thm. 2.7 and Cor. 2.8] to skew P + 1 n tr P · 1 n yield the conclusions. Eliminating the first term on the right-hand side of (3.1) gives: Theorem 3.2. Let n ≥ 3, Ω ⊂ R n be a bounded Lipschitz domain and 1 < p < ∞. There exists a constant c = c(n, p, Ω) > 0, such that for all P ∈ L p (Ω, R n×n ) we have 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 dev 2 sym Du ≡ 0 becomes the system of Cauchy-Riemann equations {u 1,x = u 2,y ∧ 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 K dS,Cn := {P ∈ L p (Ω, R n×n ) | dev n sym P = 0 a.e. and Curl n P = 0 in the dist. sense}, so that P ∈ K dS,Cn if and only if P = skew P + 1 n tr P · 1 n and Curl n (skew P + 1 n tr P · 1 n ) = 0. Hence, (3.5) follows by virtue of Lemma 2.11 and the conclusion follows in a similar way to [9, Thm. 3.8].
Finally, we examine the effect of tangential boundary conditions P × n ν ≡ 0. .

(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∈N ⊂ W 1, p 0 (Curl n ; Ω, R n×n ) converging weakly in L p (Ω, R n×n ) to some P * so that dev n sym P * = 0 almost everywhere and Curl n P * = 0 in the distributional sense, i.e., P * ∈ K dS,Cn . By (2.56) we obtain that P * × n (−ν), Q ∂Ω = 0 for all Q ∈ W 1, p (Ω, R n× n(n−1) 2 ). However, the boundary condition P * × n ν = 0 is also valid in the classical sense, since P * ∈ K dS,Cn has an explicit representation. Using the explicit representation of P * = A n − b ×n x + d + ( b, x + β) · 1 n , we conclude using Observation 2.7 that, in fact, P * ≡ 0: 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 ∈ (1, ∞) we need a different approach and it will be the subject of a forthcoming note.
Remark 3.7. For compatible P = Du we recover from (3.6) a tangential trace-free Korn inequality.
Remark 3.8. For n ≥ 3, the previous results also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset Γ ⊆ ∂Ω 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].