Reference Configurations Versus Optimal Rotations: A Derivation of Linear Elasticity from Finite Elasticity for all Traction Forces

We rigorously derive linear elasticity as a low energy limit of pure traction nonlinear elasticity. Unlike previous results, we do not impose any restrictive assumptions on the forces, and obtain a full Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-convergence result. The analysis relies on identifying the correct reference configuration to linearize about, and studying its relation to the rotations preferred by the forces (optimal rotations). The Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-limit is the standard linear elasticity model, plus a term that penalizes for fluctuations of the reference configurations from the optimal rotations. However, on minimizers this additional term is zero and the limit energy reduces to standard linear elasticity.

one can only obtain, after moving to a subsequence, that e(u ε ) e(u), and √ ε∇u ε → W for some u ∈ W 1,2 and W ∈ M n×n skew (Maddalena et al. 2019a, Theorem 2.2). 2. The limiting u does not minimize the expected linear elastic functional (1.1), but rather the energỹ This energy is further investigated in a sequel paper, Maddalena et al. (2019b). 3. Unlike Dal Maso et al. (2002), there is no full -limit, but rather a statement about approximate minimizers.

Reference Configurations and Optimal Rotations
The above-mentioned works defined the displacement with respect to a reference configuration that is dictated by the problem; that is, by the boundary conditions or the forces. In this work, we show that by choosing, for a given deformation, the rigid motion closest to it as its reference configuration, one can obtain stronger and more general results. More precisely, we define the reference configuration of a deformation y ∈ W 1,2 ( ; R n ) as the map Rx + c, R ∈ SO(n), c ∈ R n that minimizes the displacement, that is 1 In this case, one should distinguish between the reference configuration induced by a deformation y, and the preferred rotations of the forces, which we call optimal rotations. Formally, for the energy (1.2), we define the set of optimal rotations as R := arg max R∈SO(n) {F(R)} , where F ∈ M n×n * is the linear functional defined by the forces, that is, (1.4) In this setting, the correct normalization of the energy to consider is that is, the deviation ofJ ε from its value on optimal rotations. By rotating the system, we can always assume that I ∈ R and thus, define J ε (y) :=J ε (y) −J ε (id). As shown in Corollary 4.2, the compatibility assumption of Maddalena et al. (2019a) is equivalent to saying that R = {I }.

Main Results
In this paper we address the pure traction elastic problem (1.2), using the definitions of reference configurations, optimal rotations, and normalized energy as discussed above. That is, for a given deformation y ε ∈ W 1,2 ( ; R n ), whose reference configuration according to (1.3) is R ε x + c ε , we define its rescaled displacement by We obtain the following: 1. First, we prove that the set of optimal rotations R is a totally geodesic submanifold of SO(n) (Proposition 4.1). This geometric observation is important for the following analysis. We also give a complete classification of the possible optimal rotations in dimensions n = 2, 3 (Sect. 6). 2. Compactness (Theorem 5.1): If 1 ε 2 J ε (y ε ) is bounded, then, modulo a subsequence, we have is the projection of R ε onto R, and A 0 is an element of the normal bundle at R 0 of R in SO(n). We can write A 0 = R 0 W 0 for some W 0 ∈ M n×n skew . 3. -convergence (Theorem 5.2): Under the above notion of convergence y ε → (u 0 , R 0 , W 0 ), the functional J ε -converges to where F is defined in (1.4). 2 It turns out that this viewpoint, compared to the one of Maddalena et al. (2019a), provides better compactness properties, a full -convergence result, and it is valid for all equilibrated forces (in particular, the assumption R = {I } is not necessary for a rigorous validation of linear elasticity). On a more technical point, our proofs are simpler and work for any dimension n, whereas the proofs in Maddalena et al. (2019a) rely on the Rodrigues rotation formula (see (A.1)), which is only valid for n = 2, 3.
Our approach also gives a geometric interpretation to the difference between the Dirichlet and Neumann derivations of linear elasticity: Whereas in the Dirichlet case, the rotational part R ε of the reference configuration differs from the rotation prescribed by the boundary data by an order of ε (see Dal Maso et al. 2002, equation (3.14)), in the Neumann case the distance between R ε and the optimal rotations prescribed by the forces is only of order √ ε. 3 From a mechanical point of view, it means that a low energy pure traction elastic body can fluctuate more compared to a low energy elastic body which is clamped in part of its boundary. Finally, we note that the term − 1 2 F(R 0 W 2 0 ) that appears in the limiting energy, does not appear in the standard linear elastic energy, such as (1.1) (this can be viewed as a manifestation of the "gap", as it is called in Maddalena et al. (2019a), between standard linear elasticity and its rigorous derivation from finite elasticity for pure traction problems). This term represents the elastic cost of fluctuations of the reference configurations from the optimal rotations; in the Dirichlet case, these fluctuations are smaller, and their elastic cost does not appear in this energy scaling. However, note that the term − 1 2 F(R 0 W 2 0 ) is non-negative, since R 0 is an optimal rotation (see (4.1) below); therefore, from a minimization point of view, we can always choose W 0 = 0, thus eliminating it. More precisely, we show that minimizers of J ε converge to minimizers of I of the form (u 0 , R 0 , 0), which reduces I to the standard linear elasticity energy (see Theorem 5.3), with the slight difference that formal derivations of linear elasticity typically focus on linearization about a fixed optimal rotation and thus do not consider R 0 explicitly. In other words, the standard linear elasticity energy gives the correct asymptotic description of minimizers of finite elasticity for small forces not only in the Dirichlet case, but also for all pure traction problems.
After this work was essentially complete, we learned about the papers (Mainini and Percivale 2020; Jesenko and Schmidt 2020), where the authors study the derivation of pure traction linear elasticity from finite elasticity for incompressible materials. In Mainini and Percivale (2020), the external forces are assumed to satisfy the same compatibility condition as in Maddalena et al. (2019a), that is, in our language R = {I }. In Jesenko and Schmidt (2020), the assumptions on the forces imply the other extreme, namely that R = SO(n). We believe that our approach, adapted to the incompressible case, should be able to unify these two results and extend them to all forces. Structure of this Paper In Sect. 2, we describe in more detail the elastic energy J ε that we are considering, and define the set of optimal rotations R induced by it. In Sect. 3, we give some standard preliminary estimates, regarding (a) the distance between a deformation and its reference configuration (Lemma 3.1, in which the Friesecke-James-Müller rigidity theorem comes into play), and (b) the scaling of the infimum of elastic energy J ε (Proposition 3.2), which justifies the energy scaling considered. In Sect. 4, we treat the geometry of the set of optimal rotations R, and show that it is a totally geodesic submanifold of SO(n) (Proposition 4.1). In Sect. 5, we state and prove our main results-compactness (Theorem 5.1), -convergence (Theorem 5.2), and convergence of minimizers (Theorem 5.3). In Sect. 6, we give a full classification of the possible sets of optimal rotations that can arise in two and three dimensions, and provide examples for each.

The Model
Let ⊂ R n be a Lipschitz domain, and consider the energyJ ε : where W : ×M n×n → [0, ∞] is the elastic energy density, a Carathéodory function satisfying the following assumptions: A ∈ M n×n and a.e. x ∈ . (d) Regularity: There exists a neighborhood of SO(n) in which W(x, ·) is C 2 uniformly in x: A W(·, I ) is a bounded function in . We note that assumptions (b) and (c) imply that for all B ∈ M n×n and a.e. x ∈ . We assume that the forces f and g are equilibrated, that is, Without this assumption, by changing y → y + c we can makeJ ε arbitrarily small, i.e., infJ ε = −∞. Let and define the set of optimal rotations R by R := arg max R∈SO(n) {F(R)} .
FixR ∈ R. By changing f →R T f , g →R T g and y →R T y, we can assume without loss of generality thatR = I . In particular, we have with equality holding if and only if R ∈ R. Let I ε be the elastic part ofJ ε , i.e.,

Preliminary Estimates
We begin with some preliminary calculations: In Lemma 3.1, we show that if J ε (y ε ) ≤ Cε 2 , then the W 1,2 -distance between y ε and its reference configuration is of order ε. In Proposition 3.2 we show that for some C > 0 depending on the forces f , g and the energy density W . These motivate the study of the -limit of 1 ε 2 J ε . In this section, we use the notation A ε B ε if A ε ≤ C B ε for some constant C > 0 that is independent of ε, but can depend on , the constant c in the coercivity assumption (c), and other fixed quantities.
Lemma 3.1 If J ε (y ε ) ≤ Cε 2 , then I ε (y ε ) = O(ε 2 ) and there exist a sequence R ε ∈ SO(n) and constants c ε ∈ R n such that If R ε ∈ SO(n) is another sequence with respect to which this holds, then |R ε − R ε | ε.
Remark As we will show later, the fact that |R ε − R ε | ε implies that we can regard any sequence R ε x + c ε for which this lemma holds as reference configurations of the sequence y ε , without changing the results of this paper.
Proof By the Friesecke-James-Müller rigidity theorem (Friesecke et al. 2002, Theorem 3.1), the coercivity assumption (c) on W implies that there exist R ε ∈ SO(n) such that ∇ y ε − R ε L 2 (I ε (y ε )) 1/2 . This also implies that, for an appropriate constant c ε , From the trace theorem, a similar bound also holds for L 2 -norm of the trace of Y ε . Therefore, we only need to prove that I ε (y ε ) = O(ε 2 ). Using the inequalities above, (2.3) and (2.4), we have which completes the proof by choosing δ small enough. Finally, the last statement follows since Proof The upper bound follows since J ε (id) = 0. For the lower bound, consider a sequence of approximate minimizers y ε , that is for some C > 0. In particular, J ε (y ε ) ≤ C ε 2 , hence the results of Lemma 3.1 hold. We therefore have for some constant C > 0.

Geometry of the Set of Optimal Rotations R
We recall that the tangent space to SO(n) at the identity is the space of skew-symmetric matrices, and at R ∈ SO(n) it is {RW : W ∈ M n×n skew }. Moreover, for a fixed R, we have that SO(n) = {Re W : W ∈ M n×n skew }, and for every R ∈ SO(n), there exists W ∈ M n×n skew such that R = Re W and the map t ∈ [0, 1] → Re t W is a minimizing geodesic in SO(n) connecting R and R .
Let now R ∈ R and W ∈ M n×n skew . From the definition of R the function φ(t) := F(Re t W ) satisfies φ (0) = 0 and φ (0) ≤ 0. Thus, we deduce that for every W ∈ M n×n skew and R ∈ R. We note that the first equation in (4.1) for R = I , together with (2.3), provides the usual balance condition in linearized elasticity: for every W ∈ M n×n skew and c ∈ R n . Our main result of this section is the following characterization of the set of optimal rotations: Proposition 4.1 R is a closed, connected, boundaryless, totally geodesic submanifold of SO(n), and the tangent space of R at R 0 is In particular, T R R 0 is a linear space.
Recall that a totally geodesic submanifold M of a manifold N is a submanifold, such that a length-minimizing curve in M between any two elements in M is also a length-minimizing curve in N (e.g., a hyperplane in Euclidean space).

Corollary 4.2
An immediate corollary is that strict inequality in (4.1) is equivalent to saying that R is a singleton, i.e., R = {I }. This strict inequality is the compatibility assumption on the forces in Maddalena et al. (2019a) Proposition 4.1 is what we need for the compactness and -convergence results. Later on, in Sect. 6, we give more details on the structure of R; in particular, we show that the second fundamental form of SO(n) in M n×n in the direction F is negative semi-definite, and that the number of its zero principal curvatures corresponds to the dimension of R. This yields a complete classification of the possible optimal rotations in two and three dimensions.
We will prove Proposition 4.1 at the end of the section, after a few preliminaries. For later use, we denote . Also, we define the projection operator Since R is a closed submanifold, P is well defined in a neighborhood of R. Here, dist SO(n) is the intrinsic distance in the manifold SO(n); that is, Note that this distance is equivalent to the regular (Frobenius) distance in M n×n (since SO(n) is a compact submanifold), and moreover, Towards the proof of Proposition 4.1, we start by recalling a few linear algebra facts: any W ∈ M n×n skew can be written as R T R, where R ∈ SO(n) and (4.6) From this, we have the following: Lemma 4.3 Given a rotation R ∈ SO(n), any rotation R ∈ SO(n) can be written as R = Re W , where W ∈ M n×n skew and the values λ 1 , . . . , λ k in the representation (4.6) of W belong to the interval (−π, π]. Proof We prove for the case R = I , that is, that for each W ∈ M n×n skew there exists W ∈ M n×n skew such that e W = e W , and whose nonzero eigenvalues {±λ i i} k i=1 satisfy λ i ∈ (−π, π]. For a general R, the result follows by multiplying everything from the left by R. First, note that (4.6) implies that where λ i and R ∈ SO(n) are as in (4.6), and We note that e W = cosh(W ) + sinh(W ), and this is exactly the decomposition of e W into a symmetric (cosh(W )) and a skew-symmetric (sinh(W )) matrices. Formulae (4.7) imply, in particular, that if Thus, it is possible to choose the λ i s in any interval of length 2π . This completes the proof.
Next, we note that for every R 0 ∈ R, R ∈ SO(n) and i, Assume otherwise; without loss of generality, assume that We have hence, for every t ∈ (0, 2π), using that sinh(t W ) ∈ M n×n skew and thus, F(R 0 sinh(t W )) = 0 by (4.1), Now we can easily prove the following two lemmas, which are the main building blocks towards Proposition 4.1. Lemma 4.4 states that for any W ∈ T R R 0 (see (4.2)), the whole SO(n)-geodesic emanating from R 0 in direction W belongs to R; Lemma 4.5 states that for any two elements R 0 , R 1 ∈ R, there exists a geodesic between them that belongs to R.
Remark In dimensions n = 2, 3, we can actually obtain that any geodesic between R 0 and R 1 lies in R; for n > 3, this is no longer the case due to conjugate points. See Appendix A for details.
Proof Let R 0 , R 1 ∈ R, and pick W ∈ M n×n skew such that R 1 = R 0 e W , with W of the form of Lemma 4.3. We therefore have, for some R ∈ SO(n), that where we used (4.8). Since λ i ∈ (−π, π]\{0}, it follows that a i = 0 for all i. But then, for every t ∈ R, Finally, we prove Proposition 4.1.

Proof of Proposition 4.1:
We first prove that the set is a vector space. It is obvious that T is closed under scalar multiplication; the idea is to "zoom in" near the origin, where we can effectively treat the geodesics that connect two matrices in SO(n) as straight lines in the linear space of matrices: Assume that W 1 , W 2 ∈ T ; Lemma 4.4 implies that e taW 1 , e tbW 2 ∈ R for every a, b ∈ R and t > 0. We will show that for small t, the midpoint of the geodesic between e taW 1 and e tbW 2 belongs to R, and that this midpoint is exp t 2 (aW 1 + bW 2 + O(t)) . The previous lemmata will then imply that t 2 (aW 1 + bW 2 + O(t)) ∈ T ; we will then "zoom out" and obtain that aW 1 + bW 2 ∈ T . Indeed, consider, for small t, the geodesic between e taW 1 and e tbW 2 . We can write it as where e Z = e −taW 1 e tbW 2 , hence Since |Z | = O(t), we obtain that for small enough t, all the eigenvalues of Z are close to zero; hence, Lemma 4.5 implies that this geodesic belongs to R. In particular, we have that the midpoint of this geodesic, e taW 1 e Z /2 , belongs to R; we can write it as Using Lemma 4.5 again, we have that e τ Z ∈ R for every τ , from which we obtain that Z ∈ T . Since T is closed to scalar multiplication, we have that 2Z /t ∈ T ; thus, for every t > 0, and since T is a closed set, we have that aW 1 + bW 2 ∈ T . We now claim that at the vicinity of I , R is the image of the exponential map restricted to T . Indeed, Lemma 4.4 implies that the image of the exponential map, restricted to T , is in R. On the other hand, Lemma 4.5 implies that if R ∈ R, then R = e W for some W ∈ T . This tells us that at the vicinity of I , R is a manifold whose tangent space is T .
However, we can do this analysis around any R 0 ∈ R, and thus, R is indeed a manifold whose tangent space is T R R 0 . Lemma 4.5 implies that it is connected. Since for each R 0 , R is locally homeomorphic to an open neighborhood of the zero element of the vector space T R R 0 , we have that R has no boundary; since, by definition, R is a set of maximizers of a continuous function, it is closed. We therefore deduce that R is a closed manifold.
Finally, Lemma 4.4 implies that for any W ∈ T R 0 R, the SO(n)-geodesic R 0 e t W stays on the submanifold R; hence, R is totally geodesic a small square.

Main Results
Theorem 5.1 (Compactness) Let y ε ∈ W 1,2 ( ; R n ) be such that J ε (y ε ) ≤ Cε 2 , and let R ε x + c ε be a reference configuration of y ε , satisfying the results of Lemma 3.1. Denote the rescaled displacement of y ε by We then have the following, up to moving to a subsequence: where N R R 0 and P were defined in (4.3) and (4.4). Moreover, we have that R 0 , W 0 are independent of the choice of R ε , and u 0 is independent up to a change by an infinitesimal isometry Ax + b, where A ∈ M n×n skew and b ∈ R n . Theorem 5.2 ( -convergence) Under the convergence y ε → (u 0 , R 0 , W 0 ) as defined in Theorem 5.1, we have where Q is defined in (2.1). In particular, this means

Theorem 5.3 (Convergence of minimizers) Let y
Then, there exist a sequence R ε ∈ SO(n) and constants c ε ∈ R n such that, up to subsequences, the rescaled displacements converge to u 0 strongly in W 1,q ( ; R n ) for every 1 ≤ q < 2, R ε converge to R 0 ∈ R,

Remark
The results of Maddalena et al. (2019a) are an immediate consequence of Theorem 5.1. Indeed, let y ε ∈ W 1,2 ( ; R n ) be such that J ε (y ε ) ≤ Cε 2 and let v ε = 1 ε (y ε − id) be the displacement as defined in Maddalena et al. (2019a). By (5.1) we have that From this relation it is clear that in general one cannot expect v ε to be bounded in W 1,2 , since the limit R 0 of R ε may be different from I and, even if R 0 = I , the distance of R ε from R is only of order √ ε. Assume now that R = {I }. By Theorem 5.1 and Eq. (5.3), we deduce that √ ε∇v ε converge, up to subsequences, to W 0 strongly in L 2 . Moreover, writing R ε = e √ εW ε , with W ε a bounded sequence (Theorem 5.1), we hence, e(v ε ) converges, up to subsequences, to e(u 0 ) weakly in L 2 , and e(v 0 ) = e(u 0 ) + 1 2 W 2 0 . Thus, we recover the result of Maddalena et al. (2019a).
Proof of Theorem 5.1: Convergence of u ε and R ε . By Lemma 3.1, we have that u ε is bounded in W 1,2 , from which the first assertion follows. SO(n) is compact; hence, by moving to a subsequence, we have R ε → R 0 ∈ SO(n). Note that the boundedness of u ε implies that for some C > 0 we have If R 0 / ∈ R, then dist(R 0 , R) ≥ c for some constant c > 0, and since, from the definition of R, min{F(I − R) : R ∈ SO(n), dist(R, R) ≥ c} > 0, we obtain from (5.4) that ε −2 J ε (y ε ) → ∞, in contradiction. This proves the second assertion. Convergence of ε −1/2 (R ε −P(R ε )). First, note that R ε → R 0 ∈ R implies that P(R ε ) is well defined for small enough ε. We first show that dist SO(n) To simplify the notation, denote Q ε = P(R ε ) and we also have Q ε → R 0 , and therefore, by moving to a subsequence, we have that W ε → W , where |W | = 1 and W ∈ N R R 0 . From (5.4) and (4.1), we have that for some constant C > 0, Putting this all together we have which completes the proof as W 0 = αW ∈ N R R 0 . Uniqueness of R 0 and e(u 0 ). We now show that R 0 is independent of the choice of R ε , and that u 0 is also independent up to a change by a linear function Ax + b, with A ∈ M n×n skew . Indeed, assume that R ε is an alternative choice of rotations, u ε are the associated displacements, and let u 0 be their limit. From Lemma 3.1, we know that |R ε −R ε | < Cε for some C > 0; thus, lim R ε = lim R ε = R 0 .
Moreover, writing R ε = R ε e ε A ε for some uniformly bounded matrices A ε ∈ M n×n skew , we have Here, O(ε) is with respect to the L 2 norm. By passing to the limit, using the fact that A ε is antisymmetric and R T ε ∇ y ε → I strongly in L 2 (Lemma 3.1), we obtain that skew . Uniqueness of W 0 . It remains to show that W 0 is independent of the choice of R ε . Assume we have an alternative choice of rotations R ε . From Lemma 3.1, we have that We have already established the bounds From the definition of Q ε and Q ε , it therefore follows that Indeed, this follows from and similarly when reversing the roles of R ε and R ε . Our goal is to obtain |Q ε − Q ε | √ ε, which would imply the uniqueness of W 0 . If d ε √ ε, then we are done by (5.5), since the extrinsic and intrinsic distances on SO(n) are equivalent. We can therefore assume that d ε ≈ √ ε. Let us write where W ε ,W ε ∈ M n×n skew are of norm 1, and t ε = |Q ε − Q ε | + O(ε) (see (4.5)). In particular, t ε → 0.
Since both Q ε , Q ε ∈ R are optimal rotations, we obtain from Lemma 4.5 that for ε small enough, Q ε e tW ε ∈ R for any t ∈ R. We therefore have, for any t ∈ R, Let us restrict ourselves to |t| ≤ cd ε for some c > 0. Since d ε ≈ √ ε, we obtain that from which we obtain that On the other hand, we have Therefore, using again the fact that d ε ≈ √ ε, we have which completes the uniqueness proof a small square.

Proof of Theorem 5.2: Lower Bound.
First consider the elastic part ε −2 I ε (y ε ). We have, using frame indifference, Taylor expanding W (I + A), we have from the regularity assumption (d) and (2.2) that where ω(t) is a non-negative function satisfying lim t→0 ω(t)/t 2 = 0. We therefore have Since u ε u 0 in W 1,2 , we have that χ ε → 1 in L 2 and therefore also χ 1/2 ε e(u ε ) e(u 0 ) in L 2 . Therefore, since Q(x, ·) is positive-semidefinite (and in particular, convex), we have that From this, and the fact that χ ε ω(ε|∇u ε |) ε 2 |∇u ε | 2 → 0 uniformly, we obtain that Putting all these together, we have which completes the proof of the lower bound. Upper Bound. For u 0 ∈ W 1,2 , choose a sequence u ε ∈ W 1,∞ such that u ε → u 0 in W 1,2 and ∇u ε ∞ < ε −1/2 . Define y ε := R 0 e √ εW 0 (x + εu ε ). In this case we have R ε = R 0 e √ εW 0 and u ε is indeed the displacement of y ε as in (5.1). Note that since R 0 ∈ R and W 0 ∈ N R R 0 , we have that R 0 = P(R ε ). It follows that y ε → (u 0 , R 0 , W 0 ) as needed. Now, similarly as in the lower bound, we have . Now, since u ε → u 0 strongly in W 1,2 and D 2 A W(·, I ) is in L ∞ , we have that Q(x, e(u ε )) dx → Q(x, e(u 0 )) dx. The forces part behaves exactly as in the lower bound, yielding Proof of Theorem 5.3: By Proposition 3.2 we have that J ε (y ε ) < Cε 2 ; hence, by Theorem 5.1 there exist u 0 ∈ W 1,2 ( ; R n ), R 0 ∈ R, and W 0 ∈ N R R 0 such that u ε u 0 in W 1,2 , R ε → R 0 , and lim inf 1 where we used the lower bound in Theorem 5.2. Let now v ∈ W 1,2 and R ∈ R. By the upper bound in Theorem 5.2 with W 0 = 0, there exists a sequence v ε ∈ W 1,2 such that Combining (5.2), (5.8), and (5.9), we deduce (5.10) Therefore, (u 0 , R 0 ) is a minimizer of the functional J on W 1,2 × R, and W 0 = 0 (this follows from (4.1) and the definition of N R R 0 ).
It remains to show that u ε converge to u 0 strongly in W 1,q for every 1 ≤ q < 2. Choosing v = u 0 and R = R 0 in (5.10), we obtain Equation (5.7) and the fact that I is an optimal rotation imply that 1 ε F(R ε − I ) → 0 and lim 1 ε 2 I ε (y ε ) = Q(x, e(u 0 )) dx. (5.11) Let now χ ε be defined as in (5.6). From the proof of the lower bound in Theorem 5.2, it follows that Therefore, by (5.11) we obtain lim Q(x, χ 1/2 ε e(u ε )) dx = Q(x, e(u 0 )) dx. (5.12) By the coercivity of Q, we have that We now use the weak convergence of χ 1/2 ε e(u ε ) to e(u 0 ) in L 2 , the boundedness of D 2 A W(x, I ), and equation (5.12), to deduce that χ 1/2 ε e(u ε ) → e(u 0 ) strongly in L 2 . Since χ ε → 1 in L p for every 1 ≤ p < ∞ and e(u ε ) is bounded in L 2 , we have that (1 − χ 1/2 ε )e(u ε ) → 0 strongly in L q for every 1 ≤ q < 2, hence e(u ε ) → e(u 0 ) strongly in L q for every 1 ≤ q < 2. By Korn's inequality, there exists, for every q ∈ (1, 2), a constant c q such that By the Rellich theorem u ε → u 0 strongly in L q ; hence, we conclude that u ε → u 0 strongly in W 1,q for every q ∈ (1, 2). The convergence for q = 1 follows immediately since is a bounded domain a small square.

Classification and Examples of Optimal Rotations
In this section, we classify the possible sets R of optimal rotations, in dimensions n = 2, 3. The optimal rotations are derived from the functional F ∈ (M n×n ) * . Endowing M n×n with the Frobenius inner-product, we can identify F with an n × n matrix, which we will also denote by F; since F(W ) = 0 for any W ∈ M n×n skew , it follows that F is a symmetric matrix. Note that the assumption I ∈ R gives further restrictions on F, as seen in (4.8); in particular, it cannot be an arbitrary symmetric matrix. Proposition 6.1 (Classification of optimal rotations in 2D) When n = 2, the set of optimal rotations is either R = {I } or R = SO(2). The latter case happens if and only if tr F = 0.
Proof Since R is a complete, connected, closed, boundaryless submanifold of SO(2), and SO(2) is one dimensional, R is either a singleton or the whole SO(2). Since R ∈ SO(2) implies that −R ∈ SO(2), the case R = SO(2) happens if and only if F(R) = 0 for every R ∈ SO(2). Since F is symmetric and R = cos α − sin α sin α cos α for some angle α, this holds if and only if F is traceless.
Proposition 6.2 (Classification of optimal rotations in 3D) When n = 3, the set of optimal rotation is either R = {I } or one of the following: (3), if and only if F ≡ 0.
• R is isometric to the real projective plane P 2 (R) ∼ = S 2 ∼, where ∼ is the identification of antipodal points and S 2 is the round sphere. This happens if and only if the eigenvalues of F are a, a, −a for some a > 0. • R is a single closed geodesic (that is, it is isometric to SO(2) ∼ = S 1 ); this happens if and only if the eigenvalues of F are b, a, −a for some b > a ≥ 0.
In this case we can identify the second fundamental form as the quadratic correction of f , that is, In our case, the tangent and normal spaces of SO(n) at I are M n×n skew and M n×n sym , respectively. The map W → e W maps M n×n skew to SO(n); the decomposition of e W into skew and symmetric parts is given by Therefore, since W → sinh W is a diffeomorphism of M n×n sym at the vicinity of 0, we obtain that SO(n) is the graph of the function f : M n×n skew → M n×n sym , defined by for small enough W . Thus, the second form of SO(n) at the identity is II(W ) = 1 2 W 2 . The second fundamental form in a direction S ∈ M n×n sym is then the map W → 1 2 W 2 , S . If s 1 , . . . , s n are the eigenvalues of S, then a direct calculation shows that − 1 4 (s i + s j ), i < j are the principal curvatures of SO(n) at I in direction S. 6 Back to our case, we show that the second form of SO(n) at the identity in the direction F is negative semi-definite. That is, if f 1 , . . . , f n are the eigenvalues of F, then f i + f j ≥ 0 for all i = j. Assume otherwise, and without loss of generality assume that f 1 + f 2 < 0. This contradicts (4.8): Indeed, we can write F = R T diag( f 1 , . . . , f n )R for some R ∈ SO(n), and then, with the notation of (4.8), we obtain which is a contradiction to (4.8). (e i j − e ji ) of M n×n skew , where e i j is the standard matrix basis. If S is diagonal with entries s 1 , . . . , s n , then for W = i< j α i j W i j , we have that showing that the eigenvalues are − 1 4 (s i + s j ). For a general S, we have that S = R T D R for some rotation R and diagonal matrix D. The calculation is then similar, using the orthonormal basis R T W i j R. direction F at I , and so on. Since H is a hyperplane, it is totally geodesic in M 3×3 . It follows that II R,M 3×3 F vanishes:

Recall that II SO(3),M 3×3
F is a negative semi-definite quadratic form. Since it vanishes on a subspace of dimension dim R, it follows that at least dim R of the principal curvatures of SO(n) in the direction F vanish. As shown above, the principal curvatures are − 1 4 ( f 1 + f 2 ), − 1 4 ( f 2 + f 3 ) and − 1 4 ( f 1 + f 3 ), where f i are the eigenvalues of F. • If dim R = 3, it follows that f 1 = f 2 = f 3 = 0, and thus F = 0. Obviously, if F = 0 then R = SO(3) and thus dim R = 3. • If dim R = 2, we have that, without loss of generality f 1 = f 2 = − f 3 . Since

II SO(3),M 3×3
F is negative semi-definite, we have that f 1 + f 2 ≥ 0; if equality holds, then F = 0 and dim R = 3. We thus obtain that dim R = 2 implies that the eigenvalues of F are a, a, −a for some a > 0.
• If dim R = 1, we have that, without loss of generality, f 2 = − f 3 . Again, the negative semi-definiteness of II SO(3),M 3×3 F implies that f 1 ≥ | f 2 | = | f 3 |; thus, dim R = 1 implies that the eigenvalues of F are b, a, −a for some b > a ≥ 0.
In order to complete the proof, we need to show that if the eigenvalues of F are a, a, −a for some a > 0, then dim R = 2, and if they are b, a, −a for b > a ≥ 0, then dim R = 1. Assume that for some Q ∈ SO(3), F = Q T diag(a, a, −a) Q.
Thus, for a general matrix R ∈ SO(3), we have that F(Q T R Q) = a(R 11 + R 22 − R 33 ).
Writing R in a quaternion representation, that is R = p 1 + p 2 i + p 3 j + p 4 k for a unit vector p = ( p 1 , p 2 , p 3 , p 4 ), we obtain that F(Q T R Q) = a(1 − 4 p 2 4 ).
Thus, R is the two-dimensional submanifold Q{ p 4 = 0}Q T . Next, assume that for some Q ∈ SO(3) and b > a ≥ 0, we have In this case F(Q T R Q) is maximized for all rotations R around the x-axis. Thus, dim R ≥ 1, and since b > a, we have that dim R = 1.
Example 6.1 (Uniform tension) Let ⊂ R n be a Lipschitz domain, and denote by ν the outer normal of ∂ . Let the traction force f be f = ν, and set the body force g to be zero. We then have, using the divergence theorem, that It immediately follows that I is the unique maximizer of F on SO(n). That is, R = {I } in this case. 7 Example 6.2 (Uniform compression) Reversing the sign from the previous example, that is, taking f = −ν, we obtain F(A) = −| | tr(A).
In this case I is a minimizer of F among rotations; hence, in order to use the formalism of this paper, we first need to rotate the system by a maximizer of F. 8 If n = 2 (or more generally, if n is even), then −I is a maximizer, and rotating by it reduces this example to the previous one, with a unique maximizer.
If n = 3, we recall that for R = p 1 + p 2 i+ p 3 j+ p 4 k, tr(R) = 3−4( p 2 2 + p 2 3 + p 2 4 ). Thus, a maximizer of F in SO(3) is any rotation with p 1 = 0 (that is, a rotation by π around any axis). In particular, we obtain that R is two-dimensional in this case. Example 6.3 (Tangential forces) Consider now the two-dimensional case n = 2, and let the traction force be f = Z τ , where τ is the unit tangent to ∂ , and Z is a reflection matrix, say, a reflection by the x 2 axis. If there are no body forces, we have (by Green's theorem), whenever W ∈ M n×n skew , |W | = √ 2. 9 Let R 0 , R 1 ∈ R. If R 1 = R 0 e t 0 W for some t 0 = 0 and W ∈ M n×n skew , |W | = √ 2, then F(R 0 ) = F(R 1 ), together with (A.1) imply that F(R 0 W 2 ) = 0. Using (A.1) again (or Lemma 4.4), we have that R 0 e t W ∈ R for every t ∈ R. In other words, the assumption in Lemma 4.5, that W needs to be of the form of Lemma 4.3, can be dropped in dimensions n = 2, 3.