Optimal incompatible Korn–Maxwell–Sobolev inequalities in all dimensions

We characterise all linear maps A:Rn×n→Rn×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {A}}:\mathbb R^{n\times n}\rightarrow \mathbb R^{n\times n}$$\end{document} such that, for 1≤p<n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p<n$$\end{document}, PLp∗(Rn)≤c(A[P]Lp∗(Rn)+CurlPLp(Rn))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\Vert P\right\Vert _{{\text {L}}^{p^{*}}(\mathbb R^{n})}\le c\,\Big (\left\Vert {\mathscr {A}}[P]\right\Vert _{{\text {L}}^{p^{*}}(\mathbb R^{n})}+\left\Vert {\text {Curl}}P\right\Vert _{{\text {L}}^{p}(\mathbb R^{n})} \Big ) \end{aligned}$$\end{document}holds for all compactly supported P∈Cc∞(Rn;Rn×n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P\in {\text {C}}_{c}^{\infty }(\mathbb R^{n};\mathbb R^{n\times n})$$\end{document}, where CurlP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Curl}}P$$\end{document} displays the matrix curl. Being applicable to incompatible, that is, non-gradient matrix fields as well, such inequalities generalise the usual Korn-type inequalities used e.g. in linear elasticity. Different from previous contributions, the results gathered in this paper are applicable to all dimensions and optimal. This particularly necessitates the distinction of different combinations between the ellipticities of A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {A}}$$\end{document}, the integrability p and the underlying space dimensions n, especially requiring a finer analysis in the two-dimensional situation.


I
One of the most fundamental tools in (linear) elasticity or uid mechanics are Korn-type inequalities.Such inequalities are pivotal for coercive estimates, leading to well-posedness and regularity results in spaces of weakly di erentiable functions; see [12,14,16,25,27,29,33,34,39,40,48,61] for an incomplete list.In their most basic form they assert that for each n ≥ 2 and each 1 < q < ∞ there exists a constant c = c(q, n) > 0 such that holds for all u ∈ C ∞ c (R n ; R n ).Here, ε(u) = sym Du is the symmetric part of the gradient.Within linearised elasticity, where ε(u) takes the role of the in nitesimal strain tensor for some displacement u : Ω → R n , variants of inequality (1.1) imply the existence of minimisers for elastic energies u → ˆΩ W (sym Du) dx (1.2) in certain subsets of Sobolev spaces W 1,q (Ω; R n ) provided the elastic energy density W satis es suitable growth and semiconvexity assumptions, see e.g.F M¨ [24].Variants of (1.1) also prove instrumental in the study of (in)compressible uid ows [4,23,48] or in the momentum constraint equations from general relativity and trace-free in nitesimal strain measures [2,17,26,43,44,63,64].
Originally, (1.1) was derived by K [39] in the L 2 -se ing and later on generalised to all 1 < q < ∞.Inequality (1.1) is non-trivial because it strongly relies on gradients not being arbitrary matrix elds; note that there is no constant c > 0 such that holds.As such, (1.3) fails since general matrix elds need not be Curl-free.It is therefore natural to consider variants of (1.3) that quantify the lack of curl-freeness or incompatibility.Such inequalities prove crucial in view of in nitesimal (strain-)gradient plasticity, where functionals typically involve integrands Du → W (sym e) + | sym P | 2 + V (Curl P ) based on the additive decomposition of the displacement gradient Du into incompatible elastic and irreversible plastic parts: Du = e + P and sym e representing a measure of (in nitesimal) elastic strain while sym P quanti es the plastic strain; see G et al. [28], M¨ et al. [42,51] and N et al. [21,52,54,57,58,60] for related models.Another eld of application are generalised continuum models, e.g. the relaxed micromorphic model, cf.[55,56,65].A key tool in the treatment of such problems are the incompatible Korn-Maxwell-Sobolev inequalities which, in the context of (1.3), read as where the reader is referred to Section 3.1 for the de nition of the n-dimensional matrix curl operator.
For brevity, we simply speak of KMS-inequalities.Scaling directly determines q in terms of p (or vice versa), e.g.leading to the Sobolev conjugate q = n p n−p if 1 ≤ p < n.Variants of (1.4) on bounded domains have been studied in several contributions [2,15,28,32,42,43,44,45,46,59].Inequality (1.4) asserts that the symmetric part of P and the curl of P are strong enough to control the entire matrix eld P .However, it does not clarify the decisive feature of the symmetric part to make inequalities as (1.4) work and thus has le the question of the sharp underlying mechanisms for (1.4) open.
In this paper, we aim to close this gap.Di erent from previous contributions, where such inequalities were studied for speci c combinations (p, n) or particular choices of matrix parts A [P ] such as sym P or dev sym P , 1 the purpose of the present paper is to classify those parts A [P ] of matrix elds P such that (1.4) holds with sym P being replaced by A [P ] for all possible choices of 1 ≤ p ≤ ∞ depending on the underlying space dimension n.We now proceed to give the detailled results and their context each.

M
2.1.All dimensions estimates.To approach (1.4) in light of the failure of (1.3), one may employ a Helmholtz decomposition to represent P as the sum of a gradient (hence curl-free part) and a curl (hence divergence-free part).Under suitable assumptions on A , the gradient part can be treated by a general version of Korn-type inequalities implied by the usual C ´ Z theory [9].On the other hand, the div-free part is dealt with by the fractional integration theorem (cf.Lemma 3.4 below) in the superlinear growth regime p > 1.If p = 1, then the particular structure of the div-free term in n ≥ 3 dimensions allows to reduce to the B B theory [3].As a starting point, we thus rst record the solution of the classi cation problem for combinations (p, n) = (1, 2): eorem 2.1 (KMS-inequalities for (p, n) = (1, 2)).Let n = 2 and 1 < p < 2 or n ≥ 3 and 1 ≤ p < n.
Given a linear map A : R m×n → R N , with m, N ∈ N, the following are equivalent: (a) ere exists a constant c = c(p, n, A ) > 0 such that the inequality holds for all P ∈ C ∞ c (R n ; R m×n ).(b) A induces an elliptic di erential operator, meaning that Au := A [Du] is an elliptic di erential operator; see Section 3.2 for this terminology.For the particular choice A [P ] = sym P , this gives us a global variant of [15, m. 2] by C G .Speci cally, eorem 2.1 is established by generalising the approach of S and the rst author [32] from n = 3 to n ≥ 3 dimensions, and the quick proof is displayed in Section 5.As a key point, though, we emphasize that ellipticity of A su ces for (2.1) to hold.
As mentioned above, if n ≥ 3, the borderline estimate for p = 1 is a consequence of a Helmholtz decomposition and the B B estimate Even if n = 3, interchanging div and curl in (2.2) is not allowed here, as can be seen by considering regularisations of the gradients of the fundamental solution we then have sup i∈N ∆g i L 1 (R 3 ) < ∞, and validity of the corresponding modi ed inequality would imply the contradictory estimate However, as can be seen from Example 2.2 below, inequality (2.2) fails to extend to n = 2 dimensions.Still, G et al. [28, §5] proved validity of a variant of inequality (1.4) in n = 2 dimensions for the particular choice of the symmetric part of a matrix, and so one might wonder whether eorem 2.1 remains valid for the remaining case (p, n) = (1, 2) as well.is, however, is not the case, as can be seen from where dev sym induces the usual trace-free symmetric gradient ε D (u) = dev sym Du which is elliptic (cf.Section 3.2).Let us now consider for f ∈ C ∞ c (R 2 ) the matrix eld en we have and this inequality is again easily seen to be false by taking regularisations and smooth cut-o s of the fundamental solution of the Laplacian, Φ(x) = 1 2π log(|x|).In conclusion, eorem 2.1 does not extend to the case (p, n) = (1, 2).As the strength of Curl decreases when passing from n ≥ 3 to n = 2, one expects that validity of inequalities in general requires stronger conditions on A .In this regard, our second main result asserts that the critical case (p, n) = (1, 2) necessitates very strong algebraic conditions on the matrix parts A indeed: Here, we understand by inducing a di erential operator A that the associated di erential operator , satis es the corresponding properties.For the precise meaning of C-ellipticity and ellipticity and cancellation, the reader is referred to Section 3.2.
Properties (b) and (c) express in which sense mere ellipticity has to be strengthened in order to yield the criticial Korn-Maxwell-Sobolev inequality (2.4).Working from Example 2.2, condition (c) is natural from the perspective of limiting L 1 -estimates on the entire space (cf.V S [71]).It is then due to the speci c dimensional choice n = 2 that C-ellipticity, usually playing a crucial role in boundary estimates, coincides with ellipticity and cancellation; see R ¸˘ and the rst author [30] and Lemma 3.1 below for more detail.In establishing necessity and su ciency of these conditions to yield (a) we will however employ direct consequences of the corresponding features on the induced di erential operators, which is why eorem 2.3 appears in the above form.Lastly, the dimensional description of C-ellipticity in the sense of (d) is a by-product of the proof, which might be of independent interest.
Since A = sym induces a C-elliptic di erential operator in n = 2 dimensions whereas A = dev sym does not, this result clari es validity of (2.4) for A = sym and its failure for A = dev sym.
, by the non-availability of (2.2), inequalities (1.3) cannot be approached by invoking Helmholtz decompositions and using critical estimates on the solenoidal parts.As known from [28], in the particular case of A (and their complementary parts, the skew-symmetric matrices) to deduce estimate (2.4).e use of this particularly simple structure also enters proofs of various variants of Korn-Maxwell inequalities.A typical instance is the use of N 's formula, allowing to express DP in terms of Curl P in three dimensions provided P takes values in the set of skew-symmetric matrices so(3); also see the discussion in [42,Sec. 1.4] by M¨ and the second named authors.However, in view of classifying the parts A [P ] for which (2.4) holds, we cannot utilise the simple structure of symmetric matrices.To resolve this issue, we will introduce so-called almost complementary parts for the sharp class of parts A [P ] for which (2.4) can hold at all, and establish that these almost complementary parts have a su ciently simple structure to get suitable access to strong Bourgain-Brezis estimates.

Subcritical KMS-inequalities and other variants.
e inequalities considered so far scale and thus the exponent p * cannot be improved for a given p.On balls B r (0), one might wonder which conditions on A need to be imposed for inequalities for all P ∈ C ∞ c (B r (0); R m×n ) to hold with c = c(p, q, n, A ) > 0 provided q is strictly less than the optimal exponent p * .Since the exponent q here is strictly less than the non-improvable choice p * , one might anticipate that even in the case n = 2 ellipticity of the operator Au := A [Du] alone su ces.Indeed we have the following eorem 2.4 (Subcritical KMS-inequalities).Let n ≥ 2, m, N ∈ N, 1 ≤ p < ∞ and 1 ≤ q < p * .en the following hold: (a) Let q = 1.
) holds if and only if there exists a linear map L : R N ×m → R N ×m such that

5) holds if and only if A induces an elliptic di erential operator by virtue of Au
Let us remark that eorems 2.1-2.4 all deal with exponents p < n.Inequalities that address the case p ≥ n and herea er involve di erent canonical function spaces, are discussed in Sections 5 and 6.We conclude this introductory section by discussing several results and generalisations which appear as special cases of the results displayed in the present paper: Remark 2.5.
2. Sharp conditions on the non-di erential part P → A [P ] to yield subcritical KMS-inequalities in n-dimensions.

2.3.
Organisation of the paper.Besides the introduction, the paper is organised as follows: In Section 3 we x notation and gather preliminary results on di erential operators and other auxiliary estimates.Based on our above discussion, we rst address the most intricate case (p, n) = (1, 2) and thus establish eorem 2.3 in Section 4. eorems 2.1 and 2.4, which do not require the use of almost complementary parts, are then established in Sections 5 and 6.Finally, the appendix gathers speci c Helmholtz decompositions used in the main part of the paper, contextualises the approaches and results displayed in the main part with previous contributions and provides auxiliary material on weighted Lebesgue and Orlicz functions.

P
In this preliminary section we x notation, gather auxiliary notions and results and provide several examples that we shall refer to throughout the main part of the paper.
3.1.General notation.We denote ω n := L n (B 1 (0)) the n-dimensional Lebesgue measure of the unit ball.For m ∈ N 0 and a nite dimensional (real) vector space X, we denote the space of Xvalued polynomials on R n of degree at most m by P m (R n ; X); moreover, for 1 < p < ∞, we denote Ẇ1,p (R n ; X) the homogeneous Sobolev space (i.e., the closure of Finally, to de ne the matrix curl Curl P for P : R n → R n×n , we recall from [41,44] the generalised cross product e generalised cross product a × n • is linear in the second component and thus can be identi ed with a multiplication with a matrix denoted by a ×n ∈ R n(n−1) 2 ×n so that For a vector eld a : R n → R n and matrix eld P : R n → R m×n , with m ∈ N, we nally declare curl a and Curl P via Notions for di erential operators.Let A be a rst order, linear, constant coe cient di erential operator on R n between two nite-dimensional real inner product spaces V and W .In consequence, there exist linear maps For Ω ⊆ R n open and 1 ≤ q ≤ ∞, we set With an operator of the form (3.4), we associate the symbol map Following [7], we denote In the speci c case where V = R m , W = R m×n and A = D is the derivative, this gives us back the usual tensor product Moreover, if {a 1 , ..., a N } is a basis of R N and {e 1 , ..., e n } is a basis of R n , then by linearity the set and hence contains a basis of R(A).For our future objectives, it is important to note that this result holds irrespectively of the particular choice of bases.
We now recall some notions on di erential operators gathered from [7,35,66,67,71].An operator of the form (3.4) is called elliptic provided the Fourier symbol 0 is strictly positive de nite.A strengthening of this notion is that of C-ellipticity.Here we require that for any ξ ∈ C n \{0} the associated symbol map 30,31,71]).Let A be a rst order di erential operator of the form (3.4) with n = 2. en the following are equivalent: (a) A is C-elliptic.
(b) A is elliptic and cancelling, meaning that A is elliptic in the above sense and [31,Sec. 3] for a discussion; for general dimensions n ≥ 2, one has (b) ⇔ (c), see [71], but only (a) ⇒ (b), see [30,31]).e following lemma is essentially due to S [66]; also see [19,37] for the explicit forms of the underlying Poincaré-type inequalities: Lemma 3.2.An operator A of the form (3.4) is C-elliptic if and only if there exists m ∈ N 0 such that ker(A) ⊂ P m (R n ; V ).Moreover, for any open, bounded and connected Lipschitz domain Ω ⊂ R n there exists c = c(q, n, A, Ω) > 0 such that for any u ∈ W A,q (Ω) there is We conclude with the following example to be referred to frequently, pu ing the classical operators ∇, ε and ε D into the framework of (3.4).For this, it is convenient to make the identi cation Example 3.3 (Gradient, deviatoric gradient, symmetric gradient and deviatoric symmetric gradient).In the following, let n = 2, V = R 2 and W = R 2×2 .e derivative ts into the framework (3.4) by taking whereas the symmetric gradient is recovered by taking Similar representations can be found in higher dimensions.However, let us remark that whereas ∇, ∇ D and ε are C-elliptic in all dimensions n ≥ 2, the trace-free symmetric gradient is C-elliptic precisely in n ≥ 3 dimensions (cf.[7, Ex. 2.2]).Another class of operators that arises in the context of in nitesimal elastoplasticity but is handled most conveniently by the results displayed below, is discussed in Example 5.1.

Miscellaneous bounds.
In this section, we record some auxiliary material on integral operators.Given 0 < s < n, the s-th order Riesz potential I s (f ) of a locally integrable function f is de ned by where c s,n > 0 is a suitable nite constant, the precise value of which shall not be required in the sequel.We now have (a) For any 1 < p < ∞ with s p < n the Riesz potential I s is a bounded linear operator For any 1 ≤ q < np n−sp if s p < n or all 1 ≤ q < ∞ if sp ≥ n and any r > 0 there exists c = c(p, q, n, s, r) > 0 such that we have In the regime s = 0 we require two other ingredients as follows.e rst one is Whereas the preceding lemma proves particularly useful in the context of bounded domains (also see eorem 4.8 below), the situation on the entire R n can be accessed by Calderón-Zygmund estimates [9] (see [71,Prop. 4.1] or [16,Prop. 4.1] for related arguments) and implies Lemma 3.6 (Calderón-Zygmund-Korn).Let A be a di erential operator of the form (3.4) and let 1 < p < ∞. en A is elliptic if and only if there exists a constant c = c(p, n, A) > 0 such that we have roughout this section, let n = 2. e complementary part of P with respect to A : R 2×2 → R 2×2 is given by P − A [P ].In the present section we establish that, if A induces a C-elliptic di erential operator, then for a suitable linear map L : R 2×2 → R 2×2 the image ) is one-dimensional, in fact a line spanned by some invertible matrix.We then shall refer to P − LA [P ] as the almost complementary part.
To this end, note that if holds.Within this framework, we now have Proposition 4.1.Suppose that A is C-elliptic.en there exists a linear map L : R 2×2 → R 2×2 and some G ∈ GL(2) such that , it su ces to show that there exist a linear map L : R 2×2 → R 2×2 , G ∈ GL(2) and numbers γ ij ∈ R not all equal to zero such that If we can establish (4.3), we express X ∈ R 2×2 as x ij e i ⊗ e j and use (4.3) in conjunction with (4.1) to nd Since not all γ ij 's equal zero, this gives (4.2).Let us therefore establish (4.3).We approach (4.3) by distinguishing several dimensional cases.First note that A [R 2×2 ] is at least two dimensional.In fact, and since A is elliptic, ) and hence A fails to be cancelling.Since ellipticity together with cancellation is equivalent to C-ellipticity for rst order operators in n = 2 dimensions by Lemma 3.1, we infer that For any given G ∈ GL(2) we may simply declare L by its action on these basis vectors via L(e i ⊗ A e j ) := (e i ⊗ e j ) − G.
(ii) If a 22 = 0, a 12 = 0, then necessarily a 11 = 0 and so G is invertible.Indeed, if a 11 = 0, then (4.1) and (4.In conclusion, G de ned by (4.8) satis es G ∈ GL(2) and moreover where we recall that (i 0 , j 0 ) = (2, 1).By assumption and because of (4.7), e 1 ⊗ A e 1 , e 1 ⊗ A e 2 and e 2 ⊗ A e 2 form a basis of A [R 2×2 ], and we may extend these three basis vectors by a vector f ∈ R 2×2 to a basis of R 2×2 .We then declare the linear map L by its action on these basis vectors via L(e i ⊗ A e j ) := e i ⊗ e j if (i, j) = (i 0 , j 0 ), Combining our above ndings, we obtain = (i,j) =(i0,j0) = e i0 ⊗ e j0 − G, and so, in light of (4.12), we may choose to see that (4.3) is ful lled; in particular, not all γ ij 's vanish, and G ∈ GL(2).e proof is complete.
e next lemma shows that in two dimensions the operator dev sym is indeed a typical example of an elliptic but not C-elliptic operator: Lemma 4.2 (Description of rst order C-elliptic operators in two dimensions).Suppose that A is elliptic.en the operator A given by Au : Proof.
e su ciency part is already contained in the proof of the previous Proposition 4.1.For the necessity part consider for a, b ∈ C 2 with a = (a 1 , a , then all e i ⊗ A e j are linearly independent over R, and the condition a And for b = 0 we deduce a = 0, meaning that the operator A is C-elliptic in this case. In the case dim(A [R 2×2 ]) = 3 we assume without loss of generality, that there exist α, β, γ ∈ R such that If b 2 = 0, then by b = 0 we must have b 1 = 0 and we obtain a 1 = a 2 = 0, meaning that the operator A is C-elliptic.Otherwise, with b 2 = 0 we have a 1 + β a 2 = 0.If a 2 = 0 then we are done.We show, that the case a 2 = 0 cannot occur.Indeed, if a 2 = 0 then (4.15) yields: which cannot be ful lled since b = 0 and α + βγ = 0.
Let us now be more precise and explain in detail where the proof of Proposition 4.1 fails if A is elliptic but not C-elliptic.Ellipticity implies that for any ξ ∈ R 2 , dim(A[ξ](R 2 )) = 2 and by the foregoing lemma we must have dim(A [R 2×2 ]) = 2. Hence, if A is not C-elliptic, then precisely two pure Atensors e i ⊗ A e j , (i, j) ∈ I := {(i 0 , j 0 ), (i 1 , j 1 )}, are linearly independent.In order to introduce the linear map L, we de ne it via its action on these basis vectors.At a rst glance, this seems to only yield two conditions, but this is not so because of (4.3) as we here pose compatibility conditions for all indices that ought to be ful lled.In particular, the de nition of L on the xed basis vectors must be compatible with (4.3) also for (i, j) / ∈ I, and since (4.3) includes all rank-one-matrices, this is non-trivial.As can be seen explicitly from Example 4.4 below, non-C-elliptic operators do not satisfy this property.
Example 4.3 (Gradient and symmetric gradient).For the (C-elliptic) gradient, the set {e i ⊗ A e j } is just 1 0 0 0 , 0 1 0 0 , 0 0 1 0 , 0 0 0 1 and so the gradient falls into the case dim(A [R 2×2 ]) = 4 in the case distinction of the above proof.For the (C-elliptic) deviatoric or symmetric gradients, the set {e i ⊗ A e j } is just and so both the deviatoric and the symmetric gradient falls into the case dim(A [R 2×2 ]) = 3 in the case distinction of the above proof.
Example 4.4 (Trace-free symmetric gradient).For the (non-C-elliptic) trace-free symmetric gradient, the set {e i ⊗ A e j } is given by and so the rst and the third element are linearly dependent.We now verify explicitly that the relation (4.3) cannot be achieved in this case for some G ∈ GL(2).Suppose towards a contradiction that (4.3) can be achieved.en From (4.16) we infer that G needs to be a multiple of the identity matrix, whereas we infer from (4.17) that G is contained in the skew-symmetric matrices, and this is impossible.

4.2.
e implication '(b) ⇒ (a)' of eorem 2.3.Proposition 4.1 crucially allows to adapt the reduction to B B -type estimates as pursued in [28] in the symmetric gradient case.We state a version of the result that will prove useful in later sections too: Lemma 4.5 ([3, 8, 70]).Let n ≥ 2. en the following hold: (a) ere exists a constant c n > 0 such that For completeness, let us note that by embedding for any 1 < p < ∞ and A ∈ R n×n there exists c = c(p, n, A) > 0 such that we have the elementary inequality Based on the preceding lemma and the results from the previous section we may now pass on to Proof of the implication '(b) ⇒ (a)' of eorem 2.3.Let P ∈ C ∞ c (R 2 ; R 2×2 ).By Proposition 4.1, there exists a linear map L : R 2×2 → R 2×2 , G ∈ GL(2) and a linear function γ : R 2×2 → R such that Realising that the pointwise relation (4.21) gives In consequence, for some scalar, linear, homogeneous second order di erential operator B. Now, based on inequality (4.18) and recalling n = 2, we infer By de nition of f, cf.(4.23), the previous inequality implies and (4.20) then yields by virtue of G ∈ GL(2) ) and consider the solution u = (u 1 , u 2 ) of the equation −∆u = f obtained as u = Φ 2 * f , where Φ 2 is the two-dimensional Green's function for the negative Laplacian.By classical elliptic regularity estimates we have on the one hand (4.28) On the other hand, se ing we nd by (4.28) and (4.27) and We may thus write T − P = Dv for some v : R2 → R 2 , for which Lemma 3.6 yields 2 e proof is then concluded by spli ing and using (4.30), (4.29) for the rst and (4.29) for the second term.e proof is complete.

4.3.
e implication '(a) ⇒ (b)' of eorem 2.3.Again, let n = 2 and p = 1.As in [32], one directly obtains the necessity of A being induced by an elliptic di erential operator for (2.4) to hold; in fact, testing (2.4) with gradient elds P = Du gives us the usual Korn-type inequality . en one uses Lemma 3.6 to conclude the ellipticity; hence, in all of the following we may tacitly assume A to be induced by an elliptic di erential operator.
e necessity of C-ellipticity requires a re ned argument, which we address now: Lemma 4.7.In the situation of eorem 2.3, validity of (2.4) implies that A induces a C-elliptic di erential operator.
Proof.Suppose that A is not C-elliptic, meaning that there exist Writing this last equation by separately considering real and imaginary parts, we obtain For f ∈ C ∞ c (R 2 ), we then de ne x ξ := ( x, Re(ξ) , x, Im(ξ) ) and We rst claim that there exists a constant c > 0 such that To this end, we rst note that we may assume Re(ξ) and Im(ξ) to be linearly independent over R. is argument is certainly clear to experts, but since it is crucial for our argument, we give the quick proof.Suppose that Re(ξ) and Im(ξ) are not linearly independent.en there are three options: (i) If Im(ξ) = 0, then Re(ξ) = 0 (as otherwise ξ = 0).en (4.32) 1 implies Re(v) = 0 by ellipticity of A, and (4.32) we may imitate the argument from (i) to arrive at a contradiction.(iii) Based on (i) and (ii), we may assume that Im(ξ), Re(ξ) = 0 and that there exists λ = 0 such that Re(ξ) = λIm(ξ).Inserting this relation into (4.32),we arrive at By our assumption, λ = 0, and ellipticity of A implies that λRe(v) = Im(v) and Re(v) + λIm(v) = 0, so (1 + λ 2 )Re(v) = 0, so Re(v) = Im(v) = 0, and this is at variance with our assumption v = 0. Secondly, we similarly note that Re(v) and Im(v) can be assumed to be linearly independent over R. Suppose that this is not the case.en there are three options: (i') If Re(v) = 0, then necessarily Im(v) = 0. en (4.32) 1 implies that Im(ξ) = 0. Inserting this into (4.32) 2 yields Re(ξ) = 0 and so ξ = 0, which is at variance with our assumption of ξ ∈ C 2 \{0}.(ii') If Im(v) = 0, we may imitate the argument from (i') to arrive at a contradiction.(iii') Based on (i') and (ii'), we may assume that Im(v), Re(v) = 0 and that there exists λ = 0 such that Re(v) = λIm(v).Inserting this relation into (4.32)yields But then Im(ξ) = 0 and, by λ = 0, Re(ξ) = 0, so ξ = 0, in turn being at variance with ξ = 0. Based on (i)-(iii) and (i')-(iii'), we conclude that the set

and that
. is norm is equivalent to any other norm on R 2×2 , and hence (4.34) follows by the de nition of P f and a change of variables, again recalling that Re(ξ), Im(ξ) and Re(v), Im(v) are linearly independent.
In a next step, we record that for any To conclude the proof, we will now establish that the Korn-Maxwell-Sobolev inequality in this situation yields the contradictory estimate To this end, note that for any ϕ ∈ C ∞ c (R 2 ) and any X ∈ R 2×2 we have as in (4.22): Moreover, writing Re(ξ) = (ξ 11 , ξ 12 ) and Im(ξ) = (ξ 21 , ξ 22 ) , we nd As a consequence of (4.38), we have for j ∈ {1, 2} Based on (4.37), the previous two identities imply by de nition of P f (cf.(4.33)) Now de ne α := det ξ 11 ξ 12 ξ 21 ξ 22 so that Hence, in view of (4.34), (4.35) and (4.40), starting with the Korn-Maxwell-Sobolev inequality and a change of variables (recall that Re(ξ), Im(ξ) are linearly independent), we end up at the contradictory estimate (4.36).us, A has to be C-elliptic and the proof is complete.
Based on (4.45), we may conclude the proof as follows.If P ∈ N ⊥ , we use Lemma 4.5 (a) for Lipschitz domains and imitate (4.23) to nd with f := G −1 Curl P (4.47) In the general case, we apply inequality (4.47) to P − m0 j=1 ´Ω g j , P dx g j ∈ N ⊥ .Hence (b) follows.Ad '(b)⇒(a)'.Inserting gradients P = Du into (4.41)yields the Korn-type inequality By routine smooth approximation, this inequality extends to all u ∈ W A,2 (Ω).In particular, if Au ≡ 0 in Ω, then u must coincide with a polynomial of xed degree.Since this is possible only if A is C-elliptic by Lemma 3.2, (a) follows and the proof is complete.

S KMS : T
(p, n) = (1, 2) For the conclusions of eorem 2.1 we only have to establish the su ciency parts; note that the necessity of ellipticity follows as in Section 4.3.
As in [32], the proof of the su ciency part is based on a suitable Helmholtz decomposition.In view of the explicit expressions of these parts (cf.(A.10) in the appendix), we have in all dimensions n ≥ 2 for all a ∈ C ∞ c (R n ; R n ) and all x ∈ R n : where we have used | b ×n | = √ n − 1 |b| for any b ∈ R n as a direct consequence of (3.2).Now, we decompose P ∈ C ∞ c (R n ; R m×n ) also in its divergence-free part P Div and curl-free part P Curl ; for the following, it is important to note that these matrix di erential operators act row-wise.us we may write P Curl = Du for some u : R n → R m .en by Lemma 3.6 we have (5.2) e remaining proofs then follow a er establishing To this end, we consider the rows of our incompatible eld P = (P 1 , . . ., P m ) .Using Lemma 3.4 with s = 1 yields (5.4) Combining (5.2) and (5.3) then yields eorem 2.1 for 1 < p < n.If n ≥ 3 and p = 1, estimate (5.3) is now a consequence of Lemma 4.5 (b).Indeed, using (4.19), we have where in the penultimate step we added P j curl and used the fact that curl P j curl = 0. Summarising, the proof of the eorem 2.1 is now complete.
Example 5.1 (Incompatible Maxwell/div-curl-inequalities).In the context of time-incremental in nitesimal elastoplasticity [57], the authors investigated the operator A : R n×n → R n×n given by: In n = 1 dimensions this operator clearly induces an elliptic di erential operator.We show, that the induced operator A is also elliptic in n ≥ 2 dimensions.To this end consider for ξ ∈ R n \{0}: Remark 5.2 (n = 1).We now brie y comment on the case n = 1 that has been le out so far.In this case, the only part maps A : R → R that induce elliptic operators are (non-zero) multiples of the identity.Since every ϕ ∈ C ∞ c (R) is a gradient, the corresponding KMS-inequalities read and hold subject to the ellipticity assumption on A .ese results only display a selection, and other scales such as smoothness spaces e.g.à la Triebel-Lizorkin [69] equally seem natural but are beyond the scope of this paper.However, returning to the above examples we single out the following Open estion 5.3 (On weighted Lebesgue and Orlicz spaces).(a) If one aims to generalise the above result for weighted Lebesgue spaces to the case p = 1 in n ≥ 3 dimensions, a suitable weighted version of Lemma 4.5 (b) is required.To the best of our knowledge, the only available weighted Bourgain-Brezis estimates are due to L [47] but work subject to di erent assumptions on the weights than belonging to A 1 .In this sense, it would be interesting to know whether (5.13)  w (R n ) boundedly.Similarly as this boundedness property can be improved on divergence-free maps to yield estimate (4.19), it would be interesting to know whether estimates of the form u L B (R n ) ≤ c curl u L A (R n ) hold for u ∈ C ∞ c,div (R n ; R n ) for if n ≥ 3 provided (5.14) 2 is in action (which is the case e.g. if A has almost linear growth) and A (n) dominates the ∆ 2 ∩ ∇ 2 -function B. is would imply (5.13) under slightly weaker conditions on A, B than displayed above.

S KMS
We conclude the paper by giving the quick proof of eorem 2.4.As to eorem 2.4 (a), let q = 1, ϕ ∈ C ∞ c (R n ; R m ) be arbitrary and choose r > 0 so large such that spt(ϕ) ⊂ B r (0).Since c > 0 as in (2.5) is independent of r > 0, we may apply this inequality to P = Dϕ.Sending r → ∞, we obtain By the sharp version of O 's Non-Inequality as given by K K [38, m. 1.3] (also see [13,22]), the existence of a linear map L ∈ L (R N ×m ; R N ×m ) with E i = L • A i for all 1 ≤ i ≤ n follows at once.If 1 < q < n, we may similarly reduce to the situation on the entire R n and use the argument given at the beginning of Section 4.3 to nd that ellipticity of Au := A [Du] is necessary for (2.5) to hold.
For the su ciency parts of eorem 2.4 (a) and (b), we note that by scaling it su ces to consider the case r = 1.Given P ∈ C ∞ c (B 1 (0); R m×n ), we argue as in Section 5 and Helmholtz decompose P = P Curl +P Div .Writing P Curl = Du, we have P Curl L q (B1(0)) ≤ c ( A [P ] L q (B1(0)) + P Div L q (B1(0)) ); for q = 1 this is a trivial consequence of E i = L • A i for all 1 ≤ i ≤ n, whereas for q > 1 this follows as in Section 5, cf.(5.2).Finally, the part P Div is treated by use of the subcritical fractional integration theorem, cf.Lemma 3.4 (b). is completes the proof of eorem 2.4.persists for all 1 < p < 2 and all 1 < q ≤ p * = 2p 2−p .One starts from (4.45) and estimates Curl P W −1,q (Ω) ≤ c sup ϕ∈W 1,q 0 (Ω;R 2 ) Dϕ L q (Ω) ≤1

ˆΩ
Curl P, ϕ dx ≤ c Curl P L p (Ω) by Hölder's inequality and ϕ L p (Ω) ≤ c Dϕ L q (Ω) for ϕ ∈ W 1,q 0 (Ω; R 2 ).If q < 2 and thus q > 2, this follows by Morrey's embedding theorem and by the estimate ϕ L s (Ω) ≤ c Dϕ L 2 (Ω) for all 1 < s < ∞ provided q = 2.If 2 < q ≤ 2p 2−p , we require p ≤ (q ) * , and this is equivalent to q ≤ 2p 2−p .We conclude with a link between the Orlicz scenario from Section 5 and eorem 2.4: Remark 6.2.As discussed by C et al. [5,10] for the (trace-free) symmetric gradient, if the Young function B is not of class ∆ 2 ∩∇ 2 , then Korn-type inequalities persist with a certain loss of integrability on the le -hand side.For instance, by [10,Ex. 3.8] one has for α ≥ 0 the inequality with the Zygmund classes L log α L. By the method of proof, the underlying inequalities equally apply to general elliptic operators.One then obtains e.g. the re ned subcritical KMS-type inequality P L log α L(B1(0)) ≤ c A [P ] L log 1+α L(B1(0)) + Curl P L p (B1(0)) (6.3) for all P ∈ C ∞ c (B 1 (0); R n×n ), where α ≥ 0, p ≥ 1 and c = c(p, α, n, A ) > 0 is a constant.e reader will notice that this inequality holds if and only if A induces an elliptic operator A. Note that for α = 0, the additional logarithm on the right-hand side of (6.3) is the key ingredient for (6.3) to hold for such A , while without the additional logarithm one is directly in the situation of eorem 2.4 (a) and this not only forces A = A [Du] to be elliptic but to trivialise.

n n = 2 n ≥ 3 F 1 .
[P ] = sym P one may use the speci c structure of symmetric matrices Dimension Exponent p p = 1 C-ellipticity/ ellipticity and cancellation ellipticity p > 1 ellipticity ellipticity Sharp conditions on the non-di erential part P → A [P ] to yield the optimal KMS-inequalities in n-dimensions.

Remark 4 . 6 .
Note that, despite of the special situation in three dimensions, in n = 2 dimensions we have in general div Curl P = −∂ 12 (P 11 − P 22 ) + ∂ 11 P 12 − ∂ 22 P 21 ≡ 0. (4.31)In fact, it is a crucial step in the previous proof to express div Curl P by a linear combination of second derivatives of A [P ]. is is clearly the case if A [P ] = dev P but is not ful lled for general C-elliptic operators A. For this reason we need Proposition 4.1 to ensure that there exists an invertible matrix G such that div G −1 Curl P = L(D 2 A [P ]).

4. 4 . 4 . 5 .
Proof of eorem 2.3.We now brie y pause to concisely gather the arguments required for the proof of eorem 2.3.e direction '(a)⇒(b)' is given by Lemma 4.7, whereas the equivalence of (b) and (c) is a direct consequence of Lemma 3.1.In turn, the equivalence '(b)⇔(d)' is established in Lemma 4.2.Finally, the remaining implication '(b)⇒ (a)' is proved at the end of Paragraph 4.2 above.In conclusion, the proof of eorem 2.3 is complete.KMS inequalities with non-zero boundary values.Even though the present paper concentrates on compactly supported maps, we like to comment on how Proposition 4.1 can be used to derive variants for maps with non-zero boundary values; its generalisation to higher space dimensions shall be pursued elsewhere.

( 4 . 43 )
Combining (4.42) and (4.43), we arrive at extends to X(R n ) = L n n−1 (R n , w) and Y = L 1 (R n ; w n−1 n ) for w ∈ A 1 .(b)e reader will notice that a slightly weaker bound can be obtained in the above Orlicz scenario provided one keeps the ∆ 2 -and ∇ 2 -assumptions on B but weakens the assumptions on A.By C[11, m. 2(i)] the Riesz potentialI 1 maps L A (R n ) → L B w (R n ) (with the weak Orlicz space L B w (R n ))precisely if (5.14) 2 holds and A (n) dominates B globally.For A(t) = t and herea erA(t) = 0 if 0 ≤ t ≤ 1, ∞ otherwise, together with the ∆ 2 ∩ ∇ 2 -function B(t) = t n n−1, this gives us the well-known mapping propertyI 1 : L 1 (R n ) → L n n−1