$L^p$-trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions

For $1<p<\infty$ we prove an $L^p$-version of the generalized trace-free Korn inequality for incompatible tensor fields $P$ in $ W^{1,\,p}_0(\operatorname{Curl}; \Omega,\mathbb{R}^{3\times3})$. More precisely, let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that \[ \|{ P }\|_{L^p(\Omega,\mathbb{R}^{3\times3})}\leq c\,\left(\|{\operatorname{dev} \operatorname{sym} P }\|_{L^p(\Omega,\mathbb{R}^{3\times3})} + \|{ \operatorname{dev} \operatorname{Curl} P }\|_{L^p(\Omega,\mathbb{R}^{3\times3})}\right) \] holds for all tensor fields $P\in W^{1,\,p}_0(\operatorname{Curl}; \Omega,\mathbb{R}^{3\times3})$, i.e., for all $P\in W^{1,\,p}(\operatorname{Curl}; \Omega,\mathbb{R}^{3\times3})$ with vanishing tangential trace $ P\times \nu=0 $ on $ \partial\Omega$ where $\nu$ denotes the outward unit normal vector field to $\partial\Omega$ and $\operatorname{dev} P := P -\frac13 \operatorname{tr}(P)\,\mathbb{1}_3$ denotes the deviatoric (trace-free) part of $P$. We also show the norm equivalence \[ \|{ P }\|_{L^p(\Omega,\mathbb{R}^{3\times3})}+\|{\operatorname{Curl} P }\|_{L^p(\Omega,\mathbb{R}^{3\times3})}\leq c\,\left(\|{\operatorname{dev} \operatorname{sym} P }\|_{L^p(\Omega,\mathbb{R}^{3\times3})} + \|{ \operatorname{dev}\operatorname{Curl} P }\|_{L^p(\Omega,\mathbb{R}^{3\times3})}\right) \] for tensor fields $P\in W^{1,\,p}_0(\operatorname{Curl}; \Omega,\mathbb{R}^{3\times3})$. These estimates 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.


Introduction
Korn-type inequalities are crucial for a priori estimates in linear elasticity and fluid mechanics. They allow to bound the L p -norm of the gradient Du in terms of the symmetric gradient, i.e. Korn's first inequality states ∃ c > 0 ∀ u ∈ W 1, p 0 (Ω, R n ) : Du L p (Ω,R n×n ) ≤ c sym Du L p (Ω,R n×n ) . (1.1) Generalizations to many different settings have been obtained in the literature, including the geometrically nonlinear counterpart [24,39,23], mixed growth conditions [15], incompatible fields (also with dislocations) holds for each u ∈ W 1, p 0 (Ω, R 2 ), 1 but, again, this result ceases to be valid if the Dirichlet conditions are prescribed only on a part of the boundary, cf. the counterexample in [6, sec. 6.6].
Korn-type inequalities fail for the limiting cases p = 1 and p = ∞. Indeed, from the counterexamples traced back in [16,38,61,47] it follows that Ω |sym Du|dx does not dominate each quantity Ω |∂ i u j |dx for any vector field u ∈ W 1, 1 0 (Ω, R n ). Hence, also trace-free versions fail for p = 1 and p = ∞. On the other hand, Poincaré-type inequalities estimating certain integral norms of the deformation u in terms of the total variation of the symmetric strain tensor sym Du are still valid. In particular, for Poincaré-type inequalities for functions of bounded deformation involving the deviatoric part of the symmetric gradient we refer to [26].
The classical Korn's inequalities need compatibility, i.e. a gradient Du; giving up the compatibility necessitates controlling the distance of P to a gradient by adding the incompatibility measure (the dislocation density tensor) Curl P . We showed in [43] the following quantitative version of Korn's inequality for incompatible tensor fields P ∈ W 1, p (Curl; Ω, R 3×3 ): inf A∈so(3) P − A L p (Ω,R 3×3 ) ≤ c sym P L p (Ω,R 3×3 ) + Curl P L p (Ω,R 3×3 ) .
( 1.4) Note that the constant skew-symmetric matrix fields (restricted to Ω) represent the elements from the kernel of the right-hand side of (1.4). For compatible P = Du recover from (1.4) the quantitative version of the classical Korn's inequality, namely for u ∈ W 1, p (Ω, R 3 ): inf A∈so(3) Du − A L p (Ω,R 3×3 ) ≤ c sym Du L p (Ω,R 3×3 ) (1.5) and for skew-symmetric matrix fields P = A ∈ so(3) the corresponding Poincaré inequality for squared skew-symmetric matrix fields A ∈ W 1, p (Ω, so(3)) (and thus for vectors in R 3 ): inf A∈so(3) A − A L p (Ω,R 3×3 ) ≤ c Curl A L p (Ω,R 3×3 ) ≤c DA L p (Ω,R 3×3 2 ) , (1.6) where in the last step we have used that Curl consists of linear combinations from D. Interestingly, for skew-symmetric A also the converse is true, more precisely, the entries of DA are linear combinations of the entries from Curl A, cf. e.g. [ where L(.) denotes a corresponding linear operator with constant coefficients, not necessarily the same in any two places in the present paper. In fact, the mentioned results also hold in higher dimensions n > 3, see [42] and the discussion contained therein. In our proof of (1.4) we were highly inspired by a proof of (1.5) advocated by P. G. Ciarlet and his collaborators [11,12,13,14,29,19,10], which uses the Lions lemma resp. Nečas estimate, the compact embedding W 1, p ⊂⊂ L p and the representation of the second distributional derivatives of the displacement u by a linear combination of the first derivatives of the symmetrized gradient Du: D 2 u = L(D sym Du). (1.8) It is worth mentioning that the role of the latter ingredient (1.8) was taken over by (1.7) in our proof of (1.4) in [43] resp. [42]. In n = 3 dimensions the relation (1.7) is an easy consequence of the so called Nye's formula [60, eq. (7)]: where we identify the vectorspace of skew-symmetric matrices so(3) and R 3 via axl : so(3) → R 3 which is defined by the cross product: and associates with a skew-symmetric matrix A ∈ so(3) the vector axl A := (−A 23 , A 13 , −A 12 ) T . The relation (1.9a) admits moreover a counterpart on the group of orthogonal matrices O(3) and even in higher spatial dimensions, see [54]. In fact, Nye's formula is (formally) a consequence of the following algebraic identity: where the vector product of a matrix and a vector is to be seen row-wise and Anti : R 3 → so(3) is the inverse of axl. Despite the absence of the simple algebraic relations in the higher dimensional case a corresponding relation to (1.7) also holds true in n > 3, see e.g. [42]. Moreover, the kernel in quantitative versions of Korn's inequalities is killed by corresponding boundary conditions, namely by a vanishing trace condition u | ∂Ω = 0 in the case of (1.5) and (1.6) and by a vanishing tangential trace condition P × ν | ∂Ω = 0 in the general case (1.4), cf. [43,42].
The objective of the present paper is to improve on inequality (1.4) by showing that it already suffices to consider the deviatoric (trace-free) parts on the right-hand side, hence, further contributing to the problems proposed in [58]. More precisely, the main results are where dev X := X − 1 3 tr(X)·1 denotes the deviatoric part of a square tensor X ∈ R 3×3 and K dS,dC represent the kernel of the right-hand side and is given by (1.13) By killing the kernel with tangential trace conditions (note that dev(P × ν) = 0 iff P × ν = 0) we arrive at the following Korn's first type inequality Theorem 1.2. Let Ω ⊂ R 3 be a bounded Lipschitz domain and 1 < p < ∞. There exists a constant c = c(p, Ω) > 0 such that we have for all The appearance of the term dev Curl P on the right hand side of (1.14) would suggest to consider pintegrable tensor fields P with 'only' p−integrable dev Curl P . However, this would not lead to a new Banach space, since we show that for all m ∈ Z it holds that (1.15) The estimate (1.14) generalizes the corresponding result in [6] from the L 2 -setting to the L p -setting, whereas the trace-free second type inequality (1.12) is completely new. Generalizations to different right hand sides and higher dimensions have been obtained in the recent papers [40,41]. Note however that the estimates (1.12) and (1.14) are restricted to the case of three dimensions since the deviatoric operator acts on square matrices and only in the three-dimensional setting the matrix Curl returns a square matrix.
Again, for compatible P = Du we get back a tangential trace-free classical Korn inequality for the displacement gradient, namely as well as inf where Π denotes an arbitrary projection operator from W 1, p (Ω, R 3 ) onto the space of conformal Killing vectors, here the finite dimensional kernel of dev sym D, which is given by quadratic polynomials of the form namely the infinitesimal conformal mappings, cf. [63,49,33,17,64,65], see Figure 1 for an illustration in 2D. A first proof of (1.18), even in all dimensions n ≥ 3, was given by Reshetnyak [63] over domains which are star-like with respect to a ball. Over bounded Lipschitz domains the trace-free Korn's second inequality in all dimensions n ≥ 3, namely was justified by Dain [17] in the case p = 2 and by Schirra [65] for all p > 1. Their proofs use again the Lions lemma and the "higher order" analogues of the differential relation (1. However, the differential operators sym D and dev n sym D are particular cases of the so-called coercive elliptic operators whose study began with Aronszajn [5]. co nfo rmal map Figure 1: In the planar case, the condition dev 2 sym Du = 0 coincides with the Cauchy-Riemann equations for the function u (see Appendix). Therefore, infinitesimal conformal mappings in 2D are holomorphic functions which preserve angles exactly. This ceases to be the case for 3D infinitesimal conformal mappings defined by dev 3 sym Du = 0.
Let us go back to whose first proof for P ∈ W 1, 2 0 (Curl; Ω, R 3×3 ) was given in [6] via the trace-free classical Korn's inequality, a Maxwell estimate and a Helmholtz decomposition and is not directly amenable to the L p -case. Here, we catch up with the latter.
In the following section we start by summarizing the notations and collect some preliminary results from algebraic calculations which are needed in the subsequent vector calculus to establish relations of the type: for skew-symmetric tensor fields A and scalar functions ζ, where L denotes a corresponding constant coefficients linear operator. Based on this "higher order" analogue of the differential relation (1.7) we prove our main results in the last section using a similar argumentation as in [17,65] which argue by the Lions lemma resp. Nečas estimate and the compact embedding W 1,p (Ω) ⊂⊂ L p (Ω) .

Notations and preliminaries
Let n ≥ 2. We consider for vectors a, b ∈ R n the scalar product a, b := n i=1 a i b i ∈ R, the (squared) norm a 2 := a, a and the dyadic product a ⊗ b := (a i b j ) i,j=1,...,n ∈ R n×n . Similarly, we define the scalar product for matrices P, Q ∈ R n×n by P, Q := n i,j=1 P ij Q ij ∈ R and the (squared) Frobenius-norm by P 2 := P, P . We highlight by . · . the scalar multiplication of a scalar with a matrix, whereas matrix multiplication is denoted only by juxtaposition.
Moreover, P T := (P ji ) i,j=1,...,n denotes the transposition of the matrix P = (P ij ) i,j=1,...,n . The latter decomposes orthogonally into the symmetric part sym P := 1 2 P + P T and the skew-symmetric part skew P := 1 2 P − P T . We will denote by so(n) := {A ∈ R n×n | A T = −A} the Lie-Algebra of skewsymmetric matrices.
For the identity matrix we will write 1, so that the trace of a squared matrix P is given by tr P := P, 1 . The deviatoric (trace-free) part of P is given by dev n P := P − 1 n tr(P )·1 and in three dimensions its index will be suppressed, i.e. we write dev instead of dev 3 .
We will denote by D (Ω) the space of distributions on a bounded Lipschitz domain Ω ⊂ R n and by W −k, p (Ω) the dual space of W k, p 0 (Ω), where p = p p−1 is the Hölder dual exponent to p. Throughout the paper we use c as a generic positive constant, which is not necessarily the same in any two places, and we use L(.) as a generic linear operator with constant coefficients, which also may differ in any two places within the paper.
In 3-dimensions we make use of the vector product × : R 3 × R 3 → R 3 . Since the vector product a × . with a fixed vector a ∈ R 3 is linear in the second component, there exists a unique matrix Anti(a) such that and direct calculations show that for a = (a 1 , a 2 , a 3 ) T the matrix Anti(a) has the form The inverse of Anti : The matrix representation of the cross product allows for a generalization towards a cross product of a matrix P ∈ R 3×3 and a vector b ∈ R 3 via so, especially, for P = 1 it holds We repeat the following crucial algebraic identity: Proof. We decompose P into its symmetric and skew-symmetric part, i.e., P = S + A = S + Anti(a), for some S ∈ Sym(3), A ∈ so(3) and with a = axl(A).
For a symmetric matrix S it holds tr(S × b) = 0 for any b ∈ R 3 , since 2 Thus, using the decomposition P = S + Anti(a), we have: Moreover, for any matrix P ∈ R 3×3 we note that and the conclusion follows from the identity An application of the Cauchy-Bunyakovsky-Schwarz inequality on the right hand side of (2.11) shows that Proof. By (2.5) and (2.4) we have: Taking the trace on both sides we obtain Thus, reinserting b, a = 0 in (2.13) and applying sym on both sides, this implies sym(b ⊗ a) = 0. Since and b = 0 we must have a = 0. Hence, by (2.13) also α = 0.
Formally the gradient and the curl of a vector field a : Ω → R 3 can be seen as Da = a ⊗ ∇ and curl a = a × (−∇).
The latter also generalizes to (3 × 3)-matrix fields P : Ω → R 3×3 row-wise: 3 Remark 2.3. Formal calculations (e.g. replacing b by ∇) have to be performed very carefully. Indeed, they are allowed in algebraic identities but fail, in general, for implications, e.g. for A ∈ so(3) and b ∈ R 3 we have A × b = 0 if and only if dev(A × b) = 0, since the following expression holds true, cf. Observation 2.1 and However, dev(Curl A) = dev(A × (−∇)) = 0 does not imply already that Curl A = A × (−∇) = 0, due to the counterexample A = Anti(x), since by Nye's formula (2.16) we have Curl(Anti(x)) = 2 · 1. Of course, we can interpret (2.17) also in the sense of vector calculus, which gives then an expression for ∆ Curl A in terms of the second distributional derivatives of dev(Curl A), but, the latter would have no meaning for the relation of Curl A and dev Curl A. (3)) and ζ ∈ D (Ω, R). Then Proof. Observe that applying (2.4) to the vector field ∇ζ we obtain: so that taking the Curl of the transpositions on both sides gives In other words, we have that Curl([dev Curl A] T ) ∈ so (3), and applying axl on both sides of (2.20) we obtain Taking the ∂ j -derivative of (2.19) for j = 1, 2, 3 we conclude The proof of part (a) is divided into the following two key observations: To show that each entry of the Hessian matrix D 2 ζ is a linear combination of the entries of DCurl(A + ζ · 1) we make use of the second-order differential operator inc given for is symmetric. On the other hand, for a skew-symmetric matrix field A ∈ D (Ω, so (3)) we have that is skew-symmetric. Hence, sym(inc (A + ζ · 1)) = ∆ζ · 1 − D 2 ζ and tr(inc (A + ζ · 1)) = 2 ∆ζ. (2.26) In other words, the entries of the Hessian matrix of ζ are linear combinations of entries from inc (A + ζ · 1): where we have used that the entries of inc B are, of course, linear combinations of entries of DCurl B.
To establish (a.ii) from (a.i) , recall that for a skew-symmetric matrix field A the entries of DA are linear combinations of the entries from Curl A: We conclude by taking the ∂ j -derivative of (2.28) for j = 1, 2, 3, namely Finally, we establish part (c) arguing in a similar way by showing the following linear combinations: Regarding (2.18) Transposing and taking the Curl on both sides yields and we obtain, similar to the decomposition in (2.27): On the other hand, taking inc of the transpositions on both sides of (2.29) gives yielding the relation · 1)). (2.33) Considering the second distributional derivatives in (2.29) we conclude · 1)).
Remark 2.5. In the above proof we have used that the second-order differential operator inc does not change the symmetry property after application on square matrix fields, cf. the Appendix. Further properties are collected e.g. in [52,Appendix] The incompatibility operator inc arises in dislocation models, e.g., in the modeling of elastic materials with dislocations or in the modeling of dislocated crystals, since the strain cannot be a symmetric gradient of a vector field as soon as dislocations are present and the notion of incompatibility is at the basis of a new paradigm to describe the inelastic effects, cf. [20,4,3,46], cf. the Appendix for further comments. Moreover, the equation inc sym e ≡ 0 is equivalent to the Saint-Venant compatibility condition 5 defining the relation between the symmetric strain sym e and the displacement vector field u: inc sym e ≡ 0 ⇔ sym e = sym Du (2.34) over simply connected domains, cf. [1,46]. In the appendix we show that the operators inc and sym can be interchanged, so that inc sym e = sym inc e = sym Curl([Curl e] T ). (2.35) Investigations over multiply connected domains can be found e.g. in [66,30].
Returning to our proof, a crucial ingredient in our following argumentation is Theorem 2.6 (Lions lemma and Nečas estimate).
Let Ω ⊂ R n be a bounded Lipschitz domain. Let m ∈ Z and p ∈ (1, ∞). Then f ∈ D (Ω, R d ) and Df ∈ W m−1, p (Ω, R d×n ) imply f ∈ W m, p (Ω, R d ). Moreover, For the proof we refer to [2, Proposition 2.10 and Theorem 2.3], [7]. However, since we are dealing with higher order derivatives we also need a "higher order" version of the Lions lemma resp. Nečas estimate.

Corollary 2.7.
Let Ω ⊂ R n be a bounded Lipschitz domain, m ∈ Z and p ∈ (1, ∞). Denote by D k f the collection of all distributional derivatives of order k. Then f ∈ D (Ω, R d ) and with a constant c = c(m, p, n, d, Ω) > 0.
Proof. The assertion f ∈ W m, p (Ω, R d ) and the estimate (2.37) follow by inductive application of Theorem 2.6 to D l f with l = k − 1, k − 2, . . . , 0. Indeed, starting by applying Theorem 2.6 to D k−1 f gives D k−1 f ∈ W m−k+1, p (Ω, R d×n k−1 ) as well as Now, we can apply Theorem 2.6 to D k−2 f to deduce D k−2 f ∈ W m−k+2, p (Ω, R d×n k−2 ) and moreover Consequently, for all l = k − 1, k − 2, . . . , 0 we deduce D l f ∈ W m−l, p (Ω, R d×n l ) as well as Remark 2.8. The need to consider higher order derivatives is indicated by the appearance of linear terms in the kernel of Korn's quantitative versions, similar to the situation at the classical trace-free Korn inequalities [17,65]. In our case we have: Lemma 2.9. Let A ∈ L p (Ω, so (3)) and ζ ∈ L p (Ω, R). Then we have in the distributional sense Proof. Although the deductions have already been partially indicated in the literature, cf. e.g. [ Now, we focus on the "only if"-directions, starting with Taking the trace on both sides we obtain tr(Daxl A) = 0 and consequently hence sym(Daxl A) = 0. By the classical Korn's inequality (1.5) it follows that there exists a constant skew-symmetric matrix A ∈ so(3) so that Daxl A ≡ A, which implies A = Anti( Ax + b) with b ∈ R 3 . Furthermore, by (2.41) we obtain which establishes (a).
For part (b) we start with the relation dev Curl A ≡ 0 in (2.20) and have for some b ∈ R 3 and thus A = Anti(β x + b).
Finally, for part (c), let now dev Curl(A + ζ · 1) ≡ 0. Then considering the skew-symmetric parts of (2.30) we obtain Anti(∇ tr(Daxl A)) ≡ 0 ⇒ ∇ tr(Daxl A) ≡ 0. for some γ ∈ R, and we arrive at (c): We are now prepared to proceed as in the proof of the generalized Korn inequality for incompatible tensor fields .

Main results
We will make use of the Banach space equipped with the norm as well as its subspace where ν denotes the outward unit normal vector field to ∂Ω, and the tangential trace P × ν is understood in the sense of W − 1 p , p (∂Ω, R 3×3 ) which is justified by partial integration, so that its trace is defined by where Q ∈ W 1, p (Ω, R 3×3 ) denotes any extension of Q in Ω. Here, ., . ∂Ω indicates the duality pairing between W − 1 p , p (∂Ω, R 3×3 ) and W 1− 1 p , p (∂Ω, R 3×3 ). However, the appearance of the operator dev Curl on the right hand side of our designated results in this paper would suggest to work in but this is, surprisingly at first glance, not a new space: It is sufficient to show that the p-integrability of dev Curl P already implies the p-integrability of Curl P , and follows from the general case: Lemma 3.2. Let P ∈ D (Ω, R 3×3 ). Then we have for all m ∈ Z that Curl P ∈ W m, p (Ω, R 3×3 ) ⇔ dev Curl P ∈ W m, p (Ω, R 3×3 ). (3.4) Proof. We again consider the decomposition of P into its symmetric and skew-symmetric part, i.e. ∈ W m−1, p (Ω, R 3×3 ), (3.9) so that we obtain ∇ div a Since a = axl skew P ∈ D (Ω, R 3 ), we apply Theorem 2.6 to div a ∈ D (Ω, R) to conclude from (3.10) that div a ∈ W m, p (Ω, R). The statement of the Lemma then follows from the decompositions (3.5) and (3.6) which give the expression Remark 3.4 (Equivalence of norms). In view of (3.10) an application of the Lions lemma to div a, with a = axl skew P , gives us div a ∈ W m, p (Ω, R). Moreover, by the Nečas estimate we have ≤ c 3 ( P W m, p (Ω,R 3×3 ) + dev Curl P W m, p (Ω,R 3×3 ) ), (3.12) provided that P ∈ W m, p (Ω, R 3×3 ). Together with (3.11) we conclude: Curl P W m, p (Ω,R 3×3 ) ≤ c 4 ( P W m, p (Ω,R 3×3 ) + dev Curl P W m, p (Ω,R 3×3 ) ) (3.13) as well as P W m, p (Ω,R 3×3 ) + Curl P W m, p (Ω,R 3×3 ) ≤ c 5 ( P W m, p (Ω,R 3×3 ) + dev Curl P W m, p (Ω,R 3×3 ) ) (3.14) and especially for m = 0: for all P ∈ W 1, p (Curl; Ω, R 3×3 ). 6 Corollary 3.5. Let . and . be two norms on the same vector space X, with the following properties: both spaces (X, . ) and (X, . ) are complete, and there exists a constant C such that Then the two norms . and . are equivalent.
Remark 3.9. Of course, part (a) can also be proven independently of part (c). Indeed, using Lemma 2.4 (a) we obtain (3.24) and the conclusion follows from an application of Corollary 2.7 to skew P + 1 3 tr P · 1. The rigidity results now follow by elimination of the corresponding first term on the right-hand side.
Remark 3.18. In [28] the authors proved that in n = 2 dimensions, for p = 2 a Korn inequality for incompatibile fields also holds true when Curl P is only in L 1 (actually when it is a measure with bounded total variation) under the normalization condition Ω skew P dx = 0. In terms of scaling, it is interesting to involve in (3.34) the Sobolev exponent. So, we will show in a forthcoming paper that for 1 < p < 3 the following estimate holds true on an arbitrary open set Ω ⊆ R 3 : P L p * (Ω,R 3×3 ) ≤ c ( dev sym P L p * (Ω,R 3×3 ) + dev Curl P L p (Ω,R 3×3 ) ) (3.42) for all P ∈ C ∞ c (Ω, R 3×3 ), where p * = 3p 3−p . However, we do not know if such a result still holds in the borderline case p = 1.