Neˇcas–Lions lemma revisited: An L p -version of the generalized Korn inequality for incompatible tensor fields

For 1 < p < ∞ , we prove an L p -version of the generalized Korn inequality for incompatible tensor fields P in W 1 , p 0 ( Curl ; Ω , R 3 × 3 ) . More precisely, let Ω ⊂ R 3 be a bounded Lipschitz domain. Then there exists a constant c = c ( p , Ω ) > 0 such that holds for all tensor fields P ∈ W 1 , p 0 ( Curl ; Ω , R 3 × 3 ) , that is, for all P ∈ W 1 , p ( Curl ; Ω , R 3 × 3 ) with vanishing tangential trace P × 𝜈 = 0 on 𝜕 Ω where 𝜈 denotes the outward unit normal vector field to 𝜕 Ω . For compatible P = D u , this recovers an L p -version of the classical Korn's first inequality and for skew-symmetric P = A an L p -version of the Poincaré inequality.

For 1 < p < ∞, we prove an L p -version of the generalized Korn inequality for incompatible tensor fields P in W holds for all tensor fields P ∈ W 1, p 0 (Curl; Ω, R 3×3 ), that is, for all P ∈ W 1, p (Curl; Ω, R 3×3 ) with vanishing tangential trace P × = 0 on Ω where denotes the outward unit normal vector field to Ω. For compatible P =

INTRODUCTION
In this paper, we generalize the main result from Neff et al 1 for n = 3 to the L p -setting. This is, we prove where, in the classical sense, the vanishing tangential trace reads P × = 0 on Ω and denotes the outward unit normal vector field on the boundary of a bounded Lipschitz domain Ω. The original proof of (1) in L 2 , compare Neff et al, 1 is rather technical and uses the classical Korn's inequality, the Maxwell-compactness property, and suitable Helmholtz decompositions of tensor fields together with a restricting assumption on the domain to be "slicable" and is, moreover, not directly amenable to the L p -case. Our new argument essentially uses only the Lions lemma resp. Nečas estimate (Theorem 1), the compactness of W 1, p 0 (Ω) ⊂⊂ L p (Ω), and the algebraic identity where Anti ∶ R 3 → (3) is the canonical identification of vectors with skew-symmetric matrices; see Section 2. For a Lipschitz domain (i.e., open connected with Lipschitz boundary) Ω ⊂ R n , the Lions lemma states that f ∈ L p (Ω) if and only if f ∈ W −1, p (Ω) and ∇ ∈ W −1, p (Ω, R n ), which is equivalently expressed by the Nečas estimate with a positive constant c = c(p, n, Ω).
Let Ω ⊂ R n be a bounded Lipschitz domain. In this text, we refer to Korn's first inequality (in L p ) with vanishing boundary values * as the statement It can be obtained from Korn's second inequality (in L p ), which does not require boundary conditions and which reads A further consequence of the latter inequality is the quantitative version These inequalities are crucial for a priori estimates in linear elasticity and fluid mechanics and hence cornerstones for well-posedness results in linear elasticity (L 2 -setting) and the Stokes problem (L p -setting). They can be generalized to many different settings, including the geometrically nonlinear counterpart, 2,3 mixed growth conditions, 4 incompatible fields (also with dislocations), 1,5-7 and trace-free infinitesimal strain measures. [8][9][10][11] Other generalizations are applicable to Orlicz spaces [12][13][14][15] and SBD functions with small jump sets, [16][17][18][19] thin domains, [20][21][22] and John domains [23][24][25] as well as the case of nonconstant coefficients. [26][27][28][29] Piecewise Korn-type inequalities subordinate to a FEM mesh and involving jumps across element boundaries have also been investigated; see, for example, literature. 30,31 Korn's inequalities fail for p = 1 and p = ∞; see literature. [32][33][34] There exist many different proofs of the classical Korn's inequalities; see the discussions in previous works 1, [35][36][37][38][39][40] as well as Ciarlet 41, Sect. 6.15 and the references contained therein. A rather concise and elegant argument uses the well-known representation of the second distributional derivatives of the displacement u by a linear combination of the first derivatives of the symmetrized gradient: that is, D (Du) = L(D sym Du), with a linear operator L.
Then symDu = 0 implies that u is a first-order polynomial. Furthermore, for 1 < p < ∞, the Lions lemma resp. Nečas estimate (Theorem 1) applied to (7) yields a variant of Korn's second inequality in W 1, p (Ω) from which, in turn, the first Korn's inequality (with boundary conditions) can be deduced [35][36][37][38] using an indirect argument together with the compactness of the dual spaces Furthermore, such an argumentation scheme also applies to obtain Korn-type inequalities on surfaces, [42][43][44] in Sobolev spaces with negative exponents, 45 and in weighted homogeneous Sobolev spaces. 24 Korn's inequalities can be generalized to incompatible square tensor fields P if one adds a term in Curl P on the right-hand side, compare Neff et al, 1 thus extending Korn's first inequality to incompatible tensor fields having vanishing * In fact, the estimate is true for functions with vanishing boundary values on a relatively open (non-empty) subset of the boundary. restricted tangential trace on (a relatively open subset of) the boundary. For recent refined estimates which involve only the deviatoric (i.e., trace free) part of sym P and Curl P, see Bauer et al. 5 In the two-dimensional case, an even stronger estimate holds true for fields P ∈ L 1 (Ω, R 2×2 ) with Curl P ∈ L 1 (Ω, R 2 ); then P ∈ L 2 (Ω, R 2×2 ) and under the normalization condition ∫ Ω skewP dx = 0; compare Garroni et al. 46 However, for applications, it is preferable to work in the three-dimensional case and under more natural tangential boundary conditions. Our new inequality (1) is originally motivated from infinitesimal gradient plasticity with plastic spin. There, one introduces the additive decomposition Du = e + P of the displacement gradient Du ∈ R 3×3 into incompatible nonsymmetric elastic distortion e and incompatible plastic distortion P. Then the thermodynamic potential generically has the form where ∈ L 2 (Ω, R 3 ) describes the body force; see, for example, previous studies. [47][48][49][50] Here, ||sym e|| 2 represents the elastic energy, ||sym P|| 2 induces linear hardening, and Curl P is known as the dislocation density tensor. The L 2 -generalized Korn's inequality establishes coercivity of (10) with respect to displacements and plastic distortions, for example, if Dirichlet boundary condition u | Γ D = 0 and consistent tangential boundary conditions P × | Γ D = 0 are prescribed. Crucial for plasticity theories with spin is that the plastic contribution cannot be reduced to a dependence on the symmetric plastic strain p ∶= sym P alone, as can be done in classical plasticity. The system of equations connected to (10) reads Div(sym (Du − P)) = , (balance of forces) together with appropriate initial and boundary conditions, where I K is the indicator function of a convex domain K. The L p -version presented in this article may then serve to show well-posedness results for nonlinear dislocation mediated hardening. Notably, analytic examples suggest to use ||Curl P|| q with 1 < q < 2 for the nonlocal dislocation backstress; compare previous studies. 46,51 Another field of application of the generalized Korn's inequality for incompatible tensor fields is the so-called relaxed micromorphic model; see literature. [52][53][54] In this generalized continuum model, the task is to find the macroscopic displacement u ∶ Ω ⊂ R 3 → R 3 and the (still macroscopic) microdistortion tensor P ∶ Ω ⊂ R 3 → R 3×3 minimizing the elastic energy The equilibrium equations are the Euler-Lagrange equations to (12), which read Div sym(Du − P) = (balance of forces), Curl Curl P + sym P = sym (Du − P) (generalized balance of angular momentum).
Here, (13) 2 represents a tensorial Maxwell problem in which, due to the appearance of sym P (instead of P), the equations are strongly coupled. Note that the appearance of sym P in (10) and (12) is dictated by invariance of the model under infinitesimal rigid body motions. The well-posedness of the weak formulation depends on Korn-type inequalities for incompatible tensor fields; see previous works. 1,5 Dynamic versions of this model allow for the description of frequency band-gaps as observed in metamaterials; compare Madeo et al. 53 The band-gap property crucially depends on using Curl P in the model. Let us mention as third application the p-CurlCurl problem 55-57 appearing in modeling the magnetic field in a high-temperature superconductor. By the definition of the Banach space W 1, p (Curl; Ω, R 3×3 ), it is clear that with G ∈ L p′ (Ω, R 3×3 ) admits a unique solution for 1 < p ≤ 2, since (14) is the Euler-Lagrange equation to the strictly convex minimization problem Our new a priori estimate (1) then allows us to show existence and uniqueness for 1 < p ≤ 2 of weak solutions P ∈ resp.

NOTATION AND PRELIMINARIES
We denote by ⟨., .⟩ the scalar product and by .⊗. the dyadic product. In R 3 , we moreover make use of the cross product .×. . Since for a fixed vector a ∈ R 3 , the vector product a×. is linear in the second component, there exists a unique matrix Anti(a) with the property For a = (a 1 , a 2 , a 3 ) T , the matrix Anti(a) has the form so that Anti ∶ R 3 → (3) identifies R 3 with the space of skew-symmetric matrices (3) canonically and allows for a generalization towards a vector product of a matrix P ∈ R 3×3 and a vector b ∈ R 3 via which is seen as taking the row-wise cross product with b. Of crucial importance in our considerations is the relation An easy consequence is: Proof. Taking the squared norm on both sides of (2), we obtain ||(Anti(a)) × b|| 2 = ||b ⊗ a|| 2 + 3⟨b, a⟩ 2 − 2⟨b, a⟩ 2 = ||a|| 2 ||b|| 2 + ⟨a, b⟩ 2 , and the right-hand side is bounded from below by ||a|| 2 ||b|| 2 and from above by 2 ||a|| 2 ||b|| 2 . These bounds are sharp if a is perpendicular to b and if a is parallel to b, respectively.
These operations generalize to (3 × 3)-matrix fields row-wise. (In fact, we understand by DP the full gradient of P∈ ′ (Ω, R 3×3 ).) So, especially, the matrix Curl∶ ′ (Ω, R 3×3 )→ ′ (Ω, R 3×3 ) is introduced as Comparable with the classical representation of the second derivatives of the displacement by linear combinations of first derivatives of its symmetrized gradient, see (7), the fundamental relation (2) implies that the full gradient of a skew-symmetric matrix is already determined by its Curl: thus, Therefore, the entries of DA = DAnti(a) are linear combinations of the entries from Curl A: for any skew-symmetric matrix field A. (27) is also known as Nye's formula, compare Nye, 58, eq. (7) but is, in fact, a special case of the algebraic relation (2). Interestingly, relation (27) admits also a counterpart on SO(3) and even in higher spatial dimensions; see Neff and Münch. 59

Remark 2. Equation
It is remarkable and a deep result that a converse implication holds true as well.
For the proof, see Amrouche and Girault 60, Proposition 2.10 and Theorem 2.3 and Borchers and Sohr. 61 In our following discussions, the heart of the matter is the estimate (30); see Nečas. 62, Théorème 1 In fact, the case m = 0 is already contained in Cattabriga 63 ; for an alternative proof, see Mitrea and Wright 64, Lemma 11.4.1 and Boyer and Fabrie 65, Chapter IV as well as Bramble. 66 For further historical remarks, see the discussions in previous works 35,36 and the references contained therein.
Remark 3. Note that in the case d = n, it suffices to consider the symmetrized gradient operator symD instead of the full gradient D, since one can express the second distributional derivatives of a vector field by a linear combination of its first derivatives of the symmetrized gradient; see (7). Moreover, we deduce the estimate which is Korn's second inequality in L p or W 1, p with m = 0 resp. m = 1; compare literature 67,68 for the case p = 2.
For the subsequent considerations, we shall focus on the three-dimensional case n = 3. We will work in the Banach space equipped with the norm The density of (Ω, R 3×3 ) in W 1, p (Curl; Ω, R 3×3 ) follows by standard arguments. Furthermore, we consider the subspace W 1, p 0 (Curl; Ω, R 3×3 ) ∶= {P ∈ W 1, p (Curl; Ω, R 3×3 )|P × = 0 on Ω}, where denotes the outward unit normal vector field to Ω and the tangential trace P × is understood in the sense of W − 1 p , p ( Ω, R 3×3 ) which is justified by integration by parts, so that its trace is defined by
Remark 4. In the proof of our generalized Korn-type inequalities (Theorems 2 and 3), we make use of the compact embedding L p (Ω) ⊂ ⊂W −1, p (Ω), see (8), so that the W −1, p -norm of the first term on the right-hand side is of crucial importance and will not be estimated by the L p -norm.

(36)
By adding ||sym P|| L p (Ω,R 3×3 ) on both sides, the conclusion of the Lemma follows with regard to the orthogonal decomposition P = sym P+skew P.
By eliminating the first term on the right-hand side of (34), we will arrive at our generalized Korn-type inequalities; compare Theorems 2 and 3.
Remark 5. For compatible displacement gradients P = Du, we get back from (37) the quantitative version of the classical Korn's inequality (6) and for skew-symmetric matrix fields P = A the corresponding Poincaré inequality since DA = L(Curl A).
Remark 6. To deduce the kernel of the right-hand side of (37), we used Corollary 2 resp. 3. Interestingly, on simply connected domains, one can argue also in the following way: so one can apply the classical Korn's inequality (6) to infer from symD ≡ 0 that D ≡ const ∈ (3). Finally, we examine the effect of homogeneous boundary conditions.
Hence, the boundary condition P * × = Anti(a * ) × = 0 is also defined in the classical sense. By Corollary 1, we deduce a * = 0. We can now conclude as in the proof of Theorem 2.
Remark 7. The same argumentation scheme applies to show that (42) also holds true for functions with vanishing tangential trace only on a relatively open (non-empty) subset Γ ⊆ Ω of the boundary.
Remark 8. It is well known that Korn's inequality does not imply Poincaré's inequality; however, due to the presence of the Curl part, we get back both estimates from our generalization. Indeed, for compatible P = Du, we recover from (42) a tangential Korn inequality and for skew-symmetric P = A a Poincaré inequality.
Remark 9. The proof of Korn's inequality for the gradient of the displacement or for general incompatible tensor fields is mainly based on suitable representation formulas; compare (7) and Corollary 2, respectively. Cross-combining both conditions, we obtain only infinitesimal rigid body motions: See Smith 69 for comparable representation formulas and deduced coercive inequalities.
Remark 10. It is clear how to extend the present results to (n × n)-square tensor fields with n > 3. The corresponding generalized Curl and tangential trace operation have already been presented in literature. 7,59 This will be subject of a forthcoming note.

Open problems
An interesting question is whether our result holds on domains more general than Lipschitz. For example, the domain cannot have external cusps; indeed, both the classical Korn's inequality and the Lions lemma fail on such domains; compare previous works. 70,71 On the other hand, John domains support Korn-type inequalities; compare literature. [23][24][25]72 John domains generalize the concept of Lipschitz domains, allowing certain fractal boundary structures but excluding the formation of external cusps. However, there exist domains which are not John but allow for Korn and Poincaré inequalities; see the discussions in previous studies. 25,73,74 A similar result concerning the validity of the Lions lemma resp. Nečas estimate would be interesting.