Abstract
For \(n\ge 3\) and \(1<p<\infty \), we prove an \(L^p\)-version of the generalized trace-free Korn-type inequality for incompatible, p-integrable tensor fields \(P:\Omega \rightarrow \mathbb {R}^{n\times n}\) having p-integrable generalized \({\text {Curl}}_{n}\) and generalized vanishing tangential trace \(P\,\tau _l=0\) on \(\partial \Omega \), denoting by \(\{\tau _l\}_{l=1,\ldots , n-1}\) a moving tangent frame on \(\partial \Omega \). More precisely, there exists a constant \(c=c(n,p,\Omega )\) such that
where the generalized \({\text {Curl}}_{n}\) is given by \(({\text {Curl}}_{n} P)_{ijk} :=\partial _i P_{kj}-\partial _j P_{ki}\) 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.
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1 Introduction
The estimate
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
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
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
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
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
as well as its inverse \(\mathfrak {A}_n:\mathbb {R}^{\frac{n(n-1)}{2}}\rightarrow \mathfrak {so}(n)\), so that
and in coordinates it looks like
Moreover, we will make use of the following notations
and
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
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
The matrices can be characterized inductively, and for \(a=(\overline{a},a_n)^T\) the matrix has the form
so,
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
which differs from the expression and from \(\mathfrak {A}_3(a)\) which reads
Moreover, we have
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
and
In such a way, we obtain for all \(b\in \mathbb {R}^n\):
Furthermore, for \(a,b\in \mathbb {R}^n\) it holds
end especially for \(a=b\):
Hence, for all \(a,b\in \mathbb {R}^n\):
by induction over n, and, especially for \(a=b\):
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:
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
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
and
Thus, for all \(b\in \mathbb {R}^3\)
and consequently from the expression (2.19) we conclude
Now, assume for the inductive step that for all \(\overline{b}\in \mathbb {R}^{n-1}\) with \(n\ge 4\) we have
For \(b\in \mathbb {R}^n\) we have
Surely, by the expression (2.14). The induction hypothesis gives
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
-
(i)
\(A\times _{n}b = L(\mathfrak {a}_n (A) \otimes b)\)
-
(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
and on the other hand
Thus, the conclusions of the lemma follow by induction over the dimension n. Indeed, for the base case \(n=2\) we have
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:
-
(i)
\(\overline{A}\times _{n-1}\overline{b} = L(\mathfrak {a}_{n-1}(\overline{A}) \otimes \overline{b})\) and
-
(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,
-
(i)
\(\overline{A}\times _{n-1}\overline{b} = L(\mathfrak {a}_{n} (A) \otimes b)\) and
-
(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,
On the other hand, we have
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:
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
and for a general matrix \(P\in \mathbb {R}^{n\times n}\)
Thus, especially for a symmetric matrix \(S\in {\text {Sym}}(n)\) and a skew-symmetric matrix \(A\in \mathfrak {so}(n)\) we obtain
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
so that taking the trace of the symmetric part on both sides we obtain
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
In a similar way, the generalized curl is related to the generalized cross product
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
This differential operator kills the Jacobian matrix of a vector field (a compatible field), indeed
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
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
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
so that
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.
,
- 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 \(P\in \mathscr {D}'(\Omega ,\mathbb {R}^{n\times n})\):
in the style of the incompatibility operator known from the three-dimensional case. In regard of (2.32), we see
so that substituting b by \(-\nabla \) in Lemma 2.3 we obtain
Moreover, with regard to (2.34)\(_2\) we have for a skew-symmetric matrix field \(A\in \mathscr {D}'(\Omega ,\mathfrak {so}(n))\):
concluding for the 1. part that
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
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):
thus,
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
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:
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
equipped with the norm
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
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
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
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 \(\zeta \) 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
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
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
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
where the kernel is given by
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
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].
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Acknowledgements
The authors thank the referees for their valuable comments. This work was initiated in the framework of the Priority Programme SPP 2256 “Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials” funded by the Deutsche Forschungsgemeinschaft (DFG, German research foundation), Project ID 422730790. The second author was supported within the project “A variational scale-dependent transition scheme–from Cauchy elasticity to the relaxed micromorphic continuum” (Project ID 440935806). Moreover, both authors were supported in the Project ID 415894848 by the Deutsche Forschungsgemeinschaft.
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Lewintan, P., Neff, P. \(L^p\)-trace-free version of the generalized Korn inequality for incompatible tensor fields in arbitrary dimensions. Z. Angew. Math. Phys. 72, 127 (2021). https://doi.org/10.1007/s00033-021-01550-6
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DOI: https://doi.org/10.1007/s00033-021-01550-6