Sharp conditions for the linearization of finite elasticity

Calc. Var. (2021) 60:164 https://doi.org/10.1007/s00526-021-02037-y Calculus of Variations Sharp conditions for the linearization of finite elasticity Edoardo Mainini1 · Danilo Percivale1 Received: 21 January 2021 / Accepted: 13 June 2021 / Published online: 24 July 2021 © The Author(s) 2021 Abstract We consider the topic of linearization of finite elasticity for pure traction problems. We characterize the variational limit for the approximating sequence of rescaled nonlinear elastic energies. We show that the limiting minimal value can be strictly lower than the minimal value of the standard linear elastic energy if a strict compatibility condition for external loads does not hold. The results are provided for both the compressible and the incompressible case. Mathematics Subject Classification 49J45 · 74K30 · 74K35 · 74R10 1 Introduction Let  ⊂ R3 be the reference configuration of a hyperelastic body. If y :  → R3 is the deformation field and h > 0 is an adimensional parameter, we introduce the scaled global energy of the body, including the stored elastic energy and the work of external forces, by  −2 Gh (y) := h W (x, ∇y) dx − h −1 L(y − i). (1.1)  R3×3 Here, W :  × → [0, +∞] is the strain energy density and i denotes the identity map. For every x ∈ , the function W (x, ·) is assumed to be frame indifferent and uniquely minimized at rotations with value 0. It is also assumed to be C 2 -smooth around rotations and to satisfy a suitable coercivity condition to be introduced later on. Moreover, the load functional L is defined by   L(v) := f · v dx + g · v d H2 (x),  ∂ Communicated by A. Malchiodi. B Edoardo Mainini [email protected] Danilo Percivale [email protected] 1 Dipartimento di Ingegneria meccanica, energetica, gestionale e dei trasporti, Università degli studi di Genova, Via all’Opera Pia, 15, 16145 Genova, Italy 123 164 Page 2 of 31 E. Mainini, D. Percivale where f :  → R3 is a volume force field, g : ∂ → R3 is a surface force field, and H2 denotes the surface measure. In a pure traction problem, in order to have uniform bounds from below for functionals Gh we have to assume that L(y − i) ≤ 0 for every deformation y such that   W (x, ∇y) dx = 0. (1.2) Under our assumptions on W , (1.2) holds true if and only if ∇y is a constant rotation matrix, i.e., y(x) = Rx + c for some R ∈ S O(3) and some c ∈ R3 , where S O(3) denotes the special orthogonal group. Thus, we need to assume that L((R − I)x + c) ≤ 0 (1.3) for every R ∈ S O(3) and every c ∈ R3 , and by taking R = I we get L(c) ≤ 0 for every c ∈ R3 , that is, L(c) = 0 for every c ∈ R3 . Hence, (1.3) is equivalent to the following two conditions L(c) = 0 L((R − I)x) ≤ 0 ∀ c ∈ R3 , ∀ R ∈ S O(3). (1.4) (1.5) We observe that indeed if R ∈ S O(3) exists such that L((R − I)x) > 0, then Gh is not uniformly bounded from below with respect to h, that is, inf Gh → −∞ as h → 0 (see also Remark 2.6 below). It is worth noting that (1.4) says that external loads have null resultant while it will be shown in Remark 2.1 that (1.5) implies they have null momentum (without being equivalent to the null momentum condition). The choice of the scaling powers in (1.1) depends on the behavior of the elastic strain energy density and of the work expended by external loads for deformations which are close to a suitable rotation of the reference configuration, say y = R(i + hu), where u :  → R3 and where R belongs to the following rotation kernel associated to L (that satisfies (1.4)–(1.5)) 0 SL := {R ∈ S O(3) : L((R − I)x) = 0}. Indeed, by frame indifference we obtain  Gh (y) = h −2 W (x, I + h∇u(x)) dx − L(Ru) − h −1 L((R − I)x), (1.6) (1.7)  0 , by a Taylor expansion of W (x, ·) around the identity matrix we formally get and if R ∈ SL for every fixed u  lim Gh (y) = h→0  Q(x, E(u)) dx − L(Ru), (1.8) where E(u) := 21 (∇uT + ∇u) and where we have introduced the quadratic form 1 T 2 F D W (x, I) F, F ∈ R3×3 , x ∈ . 2 Therefore, it is natural to guess that the variational limit G of Gh as h → 0 can be obtained 0 , namely from (1.8) through a minimization among all R ∈ SL  G (u) := Q(x, E(u)) dx − max L(Ru). (1.9) Q(x, F) :=  123 0 R∈SL Sharp conditions for the linearization of finite elasticity Page 3 of 31 164 We stress that, if R = I, this corresponds to the usual formal derivation of linearized elasticity. 0 the work done by external loads for going from  to R is null, so But for any other R ∈ SL that it might be energetically convenient to consider deformations near Rx rather than near 0 this heuristic argument fails. Indeed, the identity. On the other hand it is clear that if R ∈ / SL 0 is not energetically convenient due to the behavior of the last term choosing R ∈ S O(3) \ SL in the right hand side of (1.7) as h → 0. 0 ≡ {I}, then (1.9) reduces to In the case that (1.4)–(1.5) hold and SL  E (u) := Q(x, E(u)) dx − L(u),  i.e., to the form of the total potential energy that follows from the standard derivation of the linear theory for hyperelastic bodies, which is based on a linearization around the identity, see [7, Chapters IX-X]. It has been shown in [10] (see also [9]) that in this case lim (inf Gh ) = min E h→0 (1.10) and if Gh (yh ) − inf Gh → 0 as h → 0 (i.e., if (yh ) is a sequence of quasi-minimizers of Gh ) then uh := h −1 (yh − i) → u0 ∈ argmin E (1.11) in a suitable sense. In particular, u0 satisfies the equilibrium conditions ⎧ ⎨ − div Q′ (x, E(u0 )) = f in  ⎩ Q′ (x, E(u0 )) n = g (1.12) on ∂, where Q′ (x, F) := D 2 W (x, I) F and n is the outer unit normal to ∂. In [12] we have extended the results of [10] to incompressible elasticity. Indeed, it is shown in [12] that (1.10) and (1.11) hold true by substituting Gh with the scaled incompressible global energy GhI , defined by replacing W with W I in the right hand side of (1.1), where  W (x, F) if det F = 1 W I (x, F) := +∞ otherwise, and by substituting E with E I (v) := where   Q I (x, E(v)) − L(v), ⎧ ⎨ 1 FT D 2 W (x, I) F if Tr F = 0 Q (x, F) := 2 ⎩ +∞ otherwise. I (1.13) Roughly speaking, these results can be interpreted by saying that, if (1.4) holds along with the strict compatibility condition L((R − I)x) < 0 ∀ R ∈ S O(3) \ {I}, then linear elasticity can be viewed as the variational limit of finite elasticity both in the compressible and in the incompressible case. 0 need not be reduced to the identity matrix, By assuming only (1.4) and (1.5), since SL a minimizer of functional (1.9) is not expected to satisfy (1.12) in general. On the other 123 164 Page 4 of 31 E. Mainini, D. Percivale hand, we may ask if its energy level equals the minimal value of E : this fact is still an open question and it represents our main focus, along with the analogous comparison between optimal energy levels in the incompressible case. A consequence of a recent result shown by Maor and Mora in [13, Theorem 5.3] is that if (1.4) and (1.5) hold along with a quadratic growth condition from below for W (x, ·), then indeed lim (inf Gh ) = min G . h→0 In this paper we extend this result to the incompressible case and to more general coercivity assumptions on the strain energy density W , but more than anything else we exhibit examples in which min G < min E and min G I < min E I , (1.14) where G I is defined by replacing Q with Q I in (1.9). Surprisingly enough, this shows that, at least under the sole assumptions (1.4)–(1.5), the energy level of the minimizer of E does not necessarily provide the minimal value of the variational limit of the scaled finite elasticity functional Gh . A gap between lim h→0 (inf Gh ) (resp. lim h→0 (inf GhI )) and min E (resp. min E I ) may appear. In detail, by assuming the coercivity condition ∀ F ∈ R3×3 W (x, F) ≥ C g p (d(F, S O(3))) for some p ∈ (1, 2], where g p (t) := ⎧2 ⎪ ⎨t (1.15) if 0 ≤ t ≤ 1 (1.16) p ⎪2t − 2 + 1 if t ≥ 1 ⎩ p p and d(·, S O(3)) denotes the distance function from rotations, for the incompressible case we will prove the following result (see Theorem 2.5 below). If (yh ) ⊂ W 1, p (, R3 ) is a sequence of quasi-minimizers of GhI , then by defining the generalized rescaled displacements   g p (|∇yh − R|) dx : R ∈ S O(3) , uh (x) := h −1 (RhT y(x)−x), where Rh ∈ argmin  there is a (not relabeled) subsequence such that ∇uh ⇀∇u∗ weakly in L p (, R3×3 ) as h → 0, where u∗ ∈ H 1 (, R3 ) and u∗ is a minimizer of G I over W 1, p (, R3 ). Moreover, GhI (yh ) → G I (u∗ ) and inf W 1, p (,R3 ) GhI → min W 1, p (,R3 ) GI as h → 0. Here, the precise characterization of G I is ⎧ I 1 3 ⎪ ⎪ ⎨ Q (x, E(u)) dx − max0 L(Ru) if u ∈ Hdiv (, R ) R∈SL  I G (u) = ⎪ ⎪ ⎩ +∞ otherwise in W 1, p (, R3 ), 1 (, R3 ) denotes the space of divergence-free H 1 (, R3 ) vector fields. Such a where Hdiv result improves the one in [12], as it allows to obtain the characterization of the limit energy 0 ≡ {I}. even without the assumption SL 123 Sharp conditions for the linearization of finite elasticity Page 5 of 31 164 It also generalizes a recent result of Jesenko and Schmidt [8] and reduces to it when 0 ≡ S O(3). We will provide the same statement for the compressible case in Theorem 2.4, SL thus obtaining an analogous of [13, Theorem 5.3] for the case of the p-growth assumption (1.15) (see also Remark 2.12 below). On top of that, we will show in Theorem 2.7 that there are configurations and external loads such that the strict inequalities (1.14) hold, the minimization problems being cast on W 1, p (, R3 ). We will end our analysis by remarking 0 is not reduced to the identity matrix: that (1.10) might be true even if (1.4)–(1.5) hold and SL indeed, it is always possible to rotate the external forces in such a way that (1.10) holds for the problem with rotated forces, see Theorem 2.10. Let us mention that our results about convergence of minimizers (Theorem 2.4 and Theorem 2.5) can also be adapted to higher space dimension, up to slight modifications to the proofs. Let us finally mention that several other results about variational linearization of finite elasticity, including Dirichlet problems, incompressibility constraints or even theories for multiwell potentials are found in [1–4,8,11,15]. Plan of the paper In Sect. 2 we introduce the assumptions of the theory and state the main results. Section 3 collects some preliminary results. In Sect. 4 we provide the proof of the variational convergence results. Eventually, Sect. 5 delivers the main example with a limiting energy that is below the minimal value of the standard linearized elasticity functional. 2 Main results We introduce the setting for compressible and incompressible elasticity, then we state the main results. In the following, the reference configuration  is always assumed to be a bounded open connected Lipschitz set in R3 . 3×3 is the set of 3 × 3 real matrices, endowed with the Euclidean norm As basic √ notation, R 3×3 |F| = FT F. Rsym (resp. R3×3 skew ) denotes the subset of symmetric (resp. skew-symmetric) 3×3 we define sym F := 21 (F + FT ) and skew F := 21 (F − FT ). By matrices. For every F ∈ R S O(3) we denote the special orthogonal group and for every R ∈ S O(3) there exist ϑ ∈ R 2 2 2 and W ∈ R3×3 skew , such that |W| = |W | = 2 and such that the following Euler-Rodrigues representation formula holds R = I + sin ϑ W + (1 − cos ϑ) W2 . (2.1) Assumptions on the elastic energy density We let W :  × R3×3 → [0, +∞] be L3 ×B9 - measurable satisfying the following assumptions, see also [2,11]: W (x, RF) = W (x, F) ∀R ∈ S O(3) ∀ F ∈ R3×3 , for a.e. x ∈ , min W = W (x, I) = 0 for a.e. x ∈ . (W 1) (W 2) 123 164 Page 6 of 31 E. Mainini, D. Percivale Concerning the regularity of W , we assume that there exist an open neighborhood U of S O(3) in R3×3 , an increasing function ω : R+ → R satisfying limt→0+ ω(t) = 0 and a constant K > 0 such that for a.e. x ∈  W (x, ·) ∈ C 2 (U ), |D 2 W (x, I)| ≤ K and |D 2 W (x, F) − D 2 W (x, G)| ≤ ω(|F − G|) ∀ F, G ∈ U . (W 3) We assume in addition the following growth property from below: there exist C > 0 and p ∈ (1, 2] such that for a.e. x ∈  W (x, F) ≥ C g p (d(F, S O(3))) ∀ F ∈ R3×3 , (W 4) where g p : [0, +∞) → R is the strictly convex function defined by (1.16). We notice that a standard application of the Hölder inequality shows that for every η ∈ L p () and every h ∈ (0, 1)    h −2 g p (h|η|) dt ≥ |η|2 dt + h p−2 |η| p dt  |η|≤h −1 |η|≥h −1   2 2− p p p−2 ≥ |η| dt + h || |η| p dt − (2.2) −1 p |η|≤h −1 p |η|≥h  2− p ≥ |η| p dt − ||. p  In order to consider incompressible elasticity models, starting from a function W as above we also introduce the incompressible strain energy density by letting, for a.e. x ∈ ,  W (x, F) if det F = 1 W I (x, F) := +∞ otherwise. Assumptions on the external forces 3p We introduce a body force field f ∈ L 4 p−3 (, R3 ) and a surface force field g ∈ 2p L 3 p−3 (∂, R3 ), where p is such that (W 4) holds. From here on, f and g will always be understood to satisfy such summability assumptions. The load functional is the following linear functional   L(v) := f · v dx + g · v d H2 (x), v ∈ W 1, p (, R3 ). (2.3)  ∂ We note that since  is a bounded Lipschitz domain, the Sobolev embedding 3p W 1, p (, R3 ) ֒→ L 3− p (, R3 ) and the Sobolev trace embedding W 1, p (, R3 ) ֒→ 2p L 3− p (∂, R3 ) imply that L is a bounded functional over W 1, p (, R3 ). We assume that external loads have null resultant L(c) = 0 ∀ c ∈ R3 (L1) and that they satisfy the following weak compatibility condition L((R − I)x) ≤ 0 ∀ R ∈ S O(3). (L2) 0 associated to a functional L satisfying A crucial object in our results is the rotation kernel SL the above assumptions, which is the set defined by (1.6). Such a kernel includes at least the identity matrix and represents the set of rotations that realize equality in (L2). 123 Sharp conditions for the linearization of finite elasticity Page 7 of 31 164 Remark 2.1 Thanks to the the Euler-Rodrigues representation formula for rotations (2.1), it is readily seen that (L2) may be rewritten as h W (θ ) := L(Wx) sin θ + (1 − cos θ )L(W2 x) ≤ 0 for every θ ∈ [0, 2π] and for every W ∈ R3×3 skew with |W| = 2. Since h W (0) = h W (2π) = 0, then 0 ≤ h ′W (2π) = L(Wx) = h ′W (0) ≤ 0, that is, by linearity of L, 3×3 ∀ W ∈ Rskew , L(Wx) = 0 and so if (L2) holds then external loads have null momentum. Therefore, (L2) is equivalent to L(Wx) = 0, L(W2 x) ≤ 0 ∀ W ∈ R3×3 (2.4) skew , and we mention that formulation (2.4) of the compatibility condition (with strict inequality) is the one appearing in [9,10,12]. On the other hand it is worth noting that the null momentum condition does not imply the second relation in (2.4). Indeed, let f(x) = −x, g ≡ 0 and let  be the open unit ball in R3 . Then  L(W2 x) = |Wx|2 dx > 0  for every W ∈ momentum. R3×3 skew , W  ≡ 0, despite that external loads have null resultant and null Remark 2.2 The characterization (2.4) of (L2) and Euler-Rodrigues formula entail 0 SL = {R ∈ S O(3) : L((R − I)x) = 0} = eW : W ∈ XL0 where 2 XL0 := W ∈ R3×3 skew : L(Wx) = L(W x) = 0 . 0 ⇒ R T ∈ S 0 , because Therefore, we have R ∈ SL L L((R T − I)x) = L((R T − R)x) + L((R − I)x) = 0 holds true since R T − R is skew-symmetric. Moreover, if Wi ∈ XL0 , i = 1, 2, then by (L2) 0 ≥ L((W1 ± W2 )2 x) = L(W12 x) + L(W22 x) ± L((W1 W2 + W2 W1 )x) = ±L((W1 W2 + W2 W1 )x), 0 . By recalling that I ∈ S 0 , we conclude that that is, W1 ± W2 ∈ XL0 , hence eW1 ±W2 ∈ SL L 0 0 can be obtained SL is a subgroup of S O(3). A more detailed characterization of the set SL from the results in [13]. Energy functionals The rescaled finite elasticity functionals Gh : W 1, p (, R3 ) → R ∪ {+∞} are defined by (1.1) 123 164 Page 8 of 31 E. Mainini, D. Percivale and the limit energy functional G : W 1, p (, R3 ) → R ∪ {+∞} is defined as ⎧ ⎪ ⎪ Q(x, E(u)) dx − L(u) − max L((R − I)u) if u ∈ H 1 (, R3 ) ⎨ 0 R∈SL  G (u) := ⎪ ⎪ ⎩ +∞ otherwise in W 1, p (, R3 ), where Q(x, F) := 21 FT D 2 W (x, I) F. By introducing the standard functional of linearized elasticity E : W 1, p (, R3 ) → R ∪ {+∞}, namely ⎧ ⎪ ⎪ ⎨ Q(x, E(u)) dx − L(u) if u ∈ H 1 (, R3 )  E (u) := ⎪ ⎪ ⎩ +∞ otherwise in W 1, p (, R3 ), 0 and L(0) = 0. It is well-known that E we immediately see that G ≤ E , since I ∈ SL admits a unique minimizer up to infinitesimal rigid displacements (i.e., up to the addition of a displacements field v such that E(v) = 0). Since the optimization problem in the definition 0 , it is not difficult to check that G is invariant under the addition of G is among rotations in SL of infinitesimal rigid displacements, i.e., G (u + v) = G (u) whenever E(v) ≡ 0. On the other hand, in general minimizers of G are not unique up to infinitesimal rigid displacements (see Proposition 5.2 later on). When considering incompressible elasticity, the functional GhI : W 1, p (, R3 ) → R ∪ {+∞}, representing the scaled total energy, is defined by  GhI (y) := h −2 W I (x, ∇y) dx − h −1 L(y − i),  while the limit functional G I : W 1, p (, R3 ) → R ∪ {+∞} is defined by ⎧ I 1 3 ⎪ ⎪ ⎨ Q (x, E(u)) dx − L(u) − max0 L((R − I)u) if u ∈ H (, R ) R∈ S  L G I (u) := ⎪ ⎪ ⎩ +∞ otherwise in W 1, p (, R3 ), where Q I is defined by (1.13). We also introduce the functional of incompressible linearized elasticity E I : W 1, p (, R3 ) → R ∪ {+∞}, namely ⎧ ⎪ I ⎪ ⎨ Q (x, E(u)) dx − L(u) if u ∈ H 1 (, R3 ) I  E (u) := ⎪ ⎪ ⎩ +∞ otherwise in W 1, p (, R3 ), and again G I ≤ E I . Functional E I admits a unique minimizer up to infinitesimal rigid displacements. Before moving to the statement of the main results, we introduce a couple of definitions. For every y ∈ W 1, p (, R3 ) we define   A p (y) := argmin g p (|∇y − R|) dx : R ∈ S O(3) .  Moreover, we may combine this definition with the rigidity inequality by Friesecke, James and Müller [5], in the general form appearing in [2,6], to get the following estimate. There exists 123 Sharp conditions for the linearization of finite elasticity Page 9 of 31 164 a constant C p = C p () > 0 such that for every y ∈ W 1, p (, R3 ) and every R ∈ A p (y)   g p (|∇y − R|) dx ≤ C p g p (d(∇y, S O(3))) dx, (2.5)   where d(F, S O(3)) := inf{|F − R| : R ∈ S O(3)}. Moreover, we introduce the following Definition 2.3 Given a vanishing sequence (h j ) j∈N ⊂ (0, 1), we say that (y j ) j∈N ⊂ W 1, p (, R3 ) is a sequence of quasi-minimizers of Gh j if Gh j (y j ) − lim j→+∞ inf W 1, p (,R3 ) Gh j = 0. Sequences of quasi-minimizers of GhI j are defined in the same way. Convergence results We are ready for the statement of the convergence result in the compressible case. Theorem 2.4 Assume (L1), (L2), (W 1), (W 2), (W 3), (W 4). Let (h j ) j∈N ⊂ (0, 1) be a vanishing sequence. Then we have inf W 1, p (,R3 ) Gh j ∈ R for any j ∈ N. Moreover, if (y j ) j∈N ⊂ W 1, p (, R3 ) is a sequence of quasi-minimizers of Gh j , and if R j ∈ A p (y j ) for any j ∈ N, then by defining T u j (x) := h −1 j (R j y(x) − x) there is a (not relabeled) subsequence such that ∇u j ⇀∇u∗ weakly in L p (, R3×3 ) where u∗ ∈ H 1 (, R3 ) is a minimizer of G over Gh j (y j ) → G (u∗ ), inf W 1, p (,R3 ) as j → +∞, W 1, p (, R3 ), Gh j → min W 1, p (,R3 ) and G as j → +∞. The same statement holds in the incompressible case. Indeed, under an additional assumption on ∂, which is needed in the proof when invoking a technical approximation lemma from [12], we have the following result. Theorem 2.5 Assume that ∂ has a finite number of connected components. Assume (L1), (L2), (W 1), (W 2), (W 3), (W 4). Let (h j ) j∈N ⊂ (0, 1) be a vanishing sequence. Then we have inf GhI j ∈ R for any j ∈ N. (2.6) W 1, p (,R3 ) Moreover, if (y j ) j∈N ⊂ W 1, p (, R3 ) is a sequence of quasi-minimizers of GhI j , and if R j ∈ A p (y j ) for any j ∈ N, then by defining T u j (x) := h −1 j (R j y(x) − x) there is a (not relabeled) subsequence such that ∇u j ⇀∇u∗ weakly in L p (, R3×3 ) as j → +∞, 123 164 Page 10 of 31 E. Mainini, D. Percivale where u∗ ∈ H 1 (, R3 ) is a minimizer of G I over W 1, p (, R3 ), and GhI j (y j ) → G I (u∗ ), inf W 1, p (,R3 ) GhI j → GI min W 1, p (,R3 ) as j → +∞. Remark 2.6 Inequality in (L2) can never be reversed. Indeed let us assume (L1), (W 1), (W 2), (W 3), (W 4) and that there exists R∗ ∈ S O(3) such that L((R∗ − I)x) > 0. (2.7) Then it is readily seen that by setting y∗j (x) = R∗ x we get ∗ Gh j (y∗j ) = −h −1 j L((R − I)x) → −∞ as j → +∞. For instance, we notice that if the body is subject to a uniform boundary compressive force field then (2.7) occurs for every R ∈ S O(3), R  = I. Indeed, if n denotes the outer unit normal vector to ∂, and we choose g = λn with λ < 0 and f ≡ 0, then  g · (R − I)x d H2 (x) = λ (Tr(R − I)) || > 0 ∀ R ∈ S O(3), R  = I. ∂ The gap with linear elasticity 0 ≡ {I} then By summarizing, if (L1), (L2), (W 1), (W 2), (W 3), (W 4) are satisfied and SL I functionals G and G are the classical functionals of linear incompressible and compressible elasticity, respectively. On the other hand if (2.7) occurs then by Remark 2.6 no convergence result is possible. In all other cases, although a full description of the limit functionals is given by means of G and G I , it is not a priori clear if their minimal values coincide with the minimal values of linear (compressible or incompressible) elasticity. In order to complete the picture, we will show that there exist configurations and external forces satisfying the assumptions of Theorem 2.4 0 is not reduced to the identity matrix) for which and Theorem 2.5 (and such that SL min W 1, p (,R3 ) G< min E (2.8) min EI, (2.9) W 1, p (,R3 ) and min W 1, p (,R3 ) GI < W 1, p (,R3 ) Therefore, in such case, given any vanishing sequence (h j ) j∈N ⊂ (0, 1) and any sequence (y j ) j∈N ⊂ W 1, p (, R3 ) of quasi-minimizers of Gh j (resp. GhI j ) there is no subsequence such that Gh j (y j ) → min E (resp. GhI j (y j ) → min E I ). This shows that the minimal value of the usual functional of linearized elasticity E (resp. E I ) need not be the correct approximation of inf Gh (resp. inf GhI ) in the regime of small h. We are going to analyze in detail an example of validity of (2.8) and (2.9). The setting for such an example is the following. We assume that  is a cylinder:  := {(x, y, z) ∈ R3 : x 2 + y 2 < 1, 0 < z < 1}. (2.10) We consider a particular choice for functional L from (2.3), letting g ≡ 0 and letting f ∈ L 2 (, R3 ) have a specific form. Namely, we let  L(u) = f ·u with f(x, y, z) := (ϕx (x, y), ϕ y (x, y), ψ(z)), (2.11)  123 Sharp conditions for the linearization of finite elasticity Page 11 of 31 164 where ϕ and ψ satisfy the following restrictions (where B denotes the unit ball centered at the origin in the x y plane and  denotes the Laplacian in the x, y variables): ϕ ∈ C 2 (B) is radial, there exists (x, y) ∈ B such that ϕ(x, y)  = 0,  1 ′ φ(1) = φ (1) = r 2 φ ′ (r ) dr = 0, where φ denotes the radial profile of ϕ, (f1) 0 ψ ∈ C 0 ([0, 1]),  1 0 ψ(z) dz = 0,  1 0 zψ(z) dz ≥ 0. (f2) Condition (f1) can be satisfied by choosing for instance ϕ to be a suitable polynomial in the radial variable, like φ(r ) = 4r 6 − 9r 4 + 6r 2 − 1. As we will discuss in Sect. 5, with such 0 . We choices of  and L the conditions (L1) and (L2) are satisfied and moreover {I}  SL 1 1 0 0 also have SL  S O(3) if 0 zψ(z) dz > 0 and SL ≡ S O(3) if 0 zψ(z) dz = 0. Concerning the strain energy functionals, we choose any function W (x, F) = W (F) satisfying assumptions (W 1), (W 2), (W 3), (W 4) and being such that 1 T 2 F D W (I) F = 4 |F|2 2 ∀ F ∈ R3×3 . (2.12) Accordingly we let W I (x, F) = W I (F) be equal to W (F) if det F = 1 and equal to +∞ otherwise. An example is the homogeneous Kirchoff - Saint-Venant energy, obtained by setting W (x, F) = W (F) = |FT F − I|2 . We have the following Theorem 2.7 Assume (2.10), (2.11), (f1), (f2), (W 1), (W 2), (W 3), (W 4) and (2.12). Then, the assumptions of Theorem 2.4 and of Theorem 2.5 are satisfied, (2.8) holds true, and if ψ L 2 (0,1) is small enough (2.9) holds true as well. Remark 2.8 In the assumptions of Theorem 2.7, let us consider the rescaled displacement fields v j and the generalized rescaled displacement fields u j associated to a sequence of quasi-minimizers (y j ) ⊂ W 1, p (, R3 ) of Gh j (or GhI j ). Since v j (x) = h −1 j (y j (x) − x) and T u j (x) = h −1 j (R j y j (x) − x), where R j ∈ A p (y j ), we have E(v j ) = R Tj + R j − 2I 2h j − sym(R j ∇u j ). (2.13) Along a suitable subsequence we have ∇u j → ∇u weakly in L p (, R3×3 ) and R j → R∗ ∈ 0 as we will show in Sect. 4. However, along the same sequence, E(v ) is unbounded in SL j L p (, R3×3 ), otherwise the results of [10] and [12] would entail convergence to the minimal value of the standard linearized elasticity functional, in contrast with Theorem 2.7. In fact, Theorem 2.7 shows that R∗  = I: the optimal rotation at a minimizer u of G (or G I ) is the limit of the rotations R j , it is not the identity matrix and then (2.13) confirms that E(v j ) is unbounded in L p (, R3×3 ). Rotated external forces with no gap Going back to the general setting, it is clear that a necessary condition for the validity of 0 is not reduced to the identity matrix. However, such a condition (2.8) and (2.9) is that SL 123 164 Page 12 of 31 E. Mainini, D. Percivale is not sufficient for the presence of a gap with linear elasticity. This is immediately seen by 0 ≡ S O(3) but of course all the four minima choosing f ≡ 0 and g ≡ 0, in which case SL appearing in (2.8)–(2.9) are equal to zero. A much more general result holds, showing that the appearance of the gap is strongly influenced by the initial choice of the external loads. Before providing such result, we introduce some further notation. For every R ∈ S O(3), let  Gh,R (y) := h −2 W (x, ∇y) dx − h −1 LR (y(x) − x), y ∈ W 1, p (, R3 ),  ⎧ ⎪ ⎪ Q(x, E(v)) dx − max LR (R̂v) if v ∈ H 1 (, R3 ) ⎨ GR (v) := 0 R̂∈SL  ⎪ ⎪ ⎩ R otherwise in W 1, p (, R3 ), +∞ where LR : W 1, p (, R3 ) → R is defined as   LR (v) := L(Rv) = R T f · v dx −  ∂ R T g · v d H2 (x), (2.14) so that LR is just the usual load functional associated to the external forces R T f, R T g. Similarly, ER is defined by replacing L with LR in the definition of E . The corresponding I , G I , E I of incompressible elasticity are also defined by replacing L with functionals Gh,R R R LR in the definition of GhI , G I and E I , respectively. 0 there holds Remark 2.9 Since L satisfies (L1) and (L2), it is clear that given R ∈ SL 3 LR (c) = 0 for every c ∈ R and LR ((S − I)x) = L((RS − I)x) − L((R − I)x) = L((RS − I)x) ≤ 0 (2.15) for every S ∈ S O(3), thus showing that LR satisfies (L1) and (L2) as well. Therefore, by Theorem 2.4 and Theorem 2.5, functionals GR and GRI can be viewed as the limit of I functionals Gh,R and Gh,R respectively. 0 , the rotated load Theorem 2.10 Assume (L1),(L2), (W 1), (W 2), (W 3), (W 4). If R ∈ SL 0 0 functional LR still satisfies (L1) and (L2), and SL ≡ SLR . Moreover, if u minimizes G (resp. 0 realizes the maximum in the definition of G (u) (resp. G I ) over W 1, p (, R3 ) and R ∈ SL I G (u)), then u minimizes GR (resp. GRI ) over W 1, p (, R3 ) and min W 1, p (,R3 ) GR = min W 1, p (,R3 ) ER (resp. min W 1, p (,R3 ) GRI = min W 1, p (,R3 ) ERI ). Remark 2.11 Given any external forces f, g satisfying (L1)–(L2), the above theorem yields the existence of new external forces satisfying (L1)–(L2), having the same rotation kernel as f, g, for which there is no gap with linear elasticity. We include in this section the straightforward proof of Theorem 2.10. Proof of Theorem 2.10. We give the proof for the compressible case, the arguments for the incompressible case being the very same. 0 we notice that, since S 0 and By Remark 2.11, LR satisfies (L1) and (L2). If R ∈ SL L 0 SLR are subgroups of S O(3) as shown in Remark 2.2, inequality (2.15) is an equality as 0 , so that S 0 ⊇ S 0 . Still assuming R ∈ S 0 , we may also prove the opposite soon as S ∈ SL LR L L 123 Sharp conditions for the linearization of finite elasticity Page 13 of 31 164 0 0 we deduce that R ∈ S 0 , so that again inclusion: indeed, by the inclusion SL ⊇ SL LR R 0 0 0 , thus T T Remark 2.2 implies R ∈ SLR and R S ∈ SLR for every S ∈ SL R L((S − I)x) = LR (R T (S − I)x) = LR ((R T S − I)x) − LR ((R T − I)x) = 0 0 , proving that S 0 ⊆ S 0 . for every S ∈ SL LR L R 0 realize the maximum in the definition of G (u), where u minimizes G over Let now R ∈ SL W 1, p (, R3 ). We conclude by checking that u is also a minimizer of GR over W 1, p (, R3 ) and that the identity matrix realizes the maximum in the definition of GR (u). We proceed by 0 such that contradiction, supposing that there are ũ ∈ W 1, p (, R3 ) and R̃ ∈ SL R   Q(x, E(ũ)) dx − LR (R̃ũ) < Q(x, E(u)) dx − LR (u). min GR = W 1, p (,R3 )   0, SL Then, having shown that RR̃ ∈ we deduce   G (ũ) = Q(x, E(ũ)) dx − max L(Sũ) ≤ Q(x, E(ũ)) dx − L(RR̃ũ) 0 S∈SL  = =    Q(x, E(ũ)) dx − LR (R̃ũ) <    Q(x, E(u)) dx − LR (u) Q(x, E(u)) dx − L(Ru) = G (u), ⊔ ⊓ which is a contradiction with the minimality of u for G . Remark 2.12 We close this section by mentioning a difference between our approach to Theorem 2.4 and the one in [13], where the authors introduce the set R := argmaxR∈S O(3) L(Rx). Under the usual assumptions on W (with p = 2) and assuming only (L1), it follows from [13, Theorem 5.3] that for every U ∈ R the infimum of  Jh,U (y) := h −2 W (x, ∇y) dx − h −1 L(y − Ux) (2.16)  among all u ∈ H 1 (, R3 ) converges as h → 0 to the minimum over u ∈ H 1 (, R3 ) of  JU (u) = Q(x, E(u)) dx − max L(URu).  R∈R 0 and It is readily seen that if L satisfies (L1) and I ∈ R then L satisfies (L2), R ≡ SL 1 3 JI (u) = G (u) for every u ∈ H (, R ). Moreover Jh,U (y) = Gh (y) + h −1 L(Ux − x) ≥ Gh (y), and equality holds if and only if I ∈ R so that in this case inf Gh → min G = min JI . /R Therefore the results of [13] imply Theorem (2.4) when p = 2. On the other hand, if I ∈ (so that (L2) does not hold) then inf Gh → −∞ as h → 0 (see Remark 2.6) and therefore hypothesis (L2) cannot be dropped in Theorem (2.4). In addition, we claim that whenever the sole condition (L1) holds along with the usual assumptions on W with p ∈ (1, 2], Theorem (2.4) implies that for every U ∈ R there holds inf Jh,U → min J as h → 0, where  J (u) = Q(x, E(u)) dx − max L(Ru).  R∈R 123 164 Page 14 of 31 E. Mainini, D. Percivale Indeed, we first notice that if L satisfies (L1), then U ∈ R implies that LU satisfies both (L1) and (L2) and that 0 R = {UR : R ∈ SL }. (2.17) U 0 it is immediately seen that US ∈ R, and given any S ∈ R In fact, given U ∈ R, if S ∈ SL ∗ U 0 , thus proving we may write S∗ = U UT S∗ and it is immediately seen that UT S∗ ∈ SL U (2.17). Moreover we notice that  Jh,U (y) = Gh,U (UT y) = h −2 W (x, ∇(UT y)) dx − h −1 LU (UT y − x),  thus inf Jh,U = inf Gh,U , and by recalling that LU satisfies both (L1) and (L2), it follows from Theorem 2.4 that inf Jh,U converges, as h → 0, to the minimum of  GU (u) := Q(x, E(u)) dx − max LU (Ru) 0 R∈SL  U among all u ∈ H 1 (, R3 ). By recalling (2.14) and by exploiting (2.17) we also get max LU (Ru) = max L(URu) = max L(Ru), 0 R∈SL 0 R∈SL U R∈R U that is, GU (u) ≡ J (u) and inf Jh,U → min J thus proving the claim (i.e., min J = min JU for every U ∈ R). In any case, we observe that if  is the reference configuration of the elastic body, the second term on the right hand side of (2.16) represents the work expended by the given external forces f, g if and only I ∈ R or equivalently if and only if (L2) is satisfied by f, g. 3 Preliminary results Some properties of W The frame indifference assumption (W 1) implies that there exists a function V such that for a.e. x ∈  (3.1) W (x, F) = V (x, 21 (FT F − I)) ∀ F ∈ R3×3 . By (W 3), for a.e. x ∈ , we have W (x, R) = D W (x, R) = 0 for any R ∈ S O(3). By (3.1), for a.e. x ∈ , given B ∈ R3×3 and h > 0 we have W (x, I+hB) = V (x, h symB+ 21 h 2 BT B) and (W 3) again implies lim h −2 W (x, I + hB) = h→0 1 1 symB D 2 V (x, 0) symB = BT D 2 W (x, I) B, 2 2 ∀ B ∈ R3×3 . By the latter and by (W 4), for a.e. x ∈ , the following holds for every B ∈ R3×3 with det B > 0: 1 T 2 B D W (x, I) B = lim h −2 W (x, I + hB) ≥ lim sup Ch −2 d 2 (I + hB, S O(3)) h→0 2 h→0  2  −2  T = lim sup Ch  (I + hB) (I + hB) − I = C|symB|2 . h→0 123 Sharp conditions for the linearization of finite elasticity Page 15 of 31 164 Moreover, as noticed also in [12], by expressing the remainder of Taylor’s expansion in terms of the x-independent modulus of continuity ω of D 2 W (x, ·) on the set U from (W 3), we have   2   W (x, I + hB) − h symB D 2 W (x, I) symB ≤ h 2 ω(h|B|)|B|2 (3.2)   2 for any small enough h (such that hB ∈ U ). Similarly, V (x, ·) is C 2 in a neighborhood of the origin in R3×3 , with an x-independent modulus of continuity η : R+ → R, which is increasing and such that limt→0+ η(t) = 0, and we have   2   V (x, hB) − h symB D 2 V (x, 0) symB ≤ h 2 η(h|B|)|B|2   2 (3.3) for any small enough h. Some functional inequalities Let p ∈ (1, 2]. Since  is a bounded open connected Lipschitz set, by Sobolev embedding, Sobolev trace embedding and by the Poincaré inequality for any v ∈ W 1, p (, R3 ) there exists c̄, d̄ ∈ R3 such that v − c̄ 3p L 3− p (,R3 ) + v − d̄ 2p L 3− p (∂,R3 ) ≤ K ∇v L p (,R3×3 ) , (3.4) where K is a constant only depending on , p. Moreover, the second Korn inequality (see for instance [14]), combined with Sobolev and trace inequalities, provides the existence of a further constant C K = C K (, p) such that for all v ∈ W 1, p (, R3 ) v − Pv 3p L 3− p (,R3 ) + v − Pv 2p L 3− p (∂,R3 ) ≤ C K E(v) L p (,R3×3 ) , (3.5) where P denotes the projection operator on infinitesimal rigid displacements, i.e., on the set of displacement fields v such that E(v) = 0. A useful consequence of (3.4), if (L1) holds true, is the following estimate. Since for any v ∈ W 1, p (, R3 ) and for every c, d ∈ R3 |L(v)| ≤ f 3p L 4 p−3 (,R3 ) v − c 3p L 3− p (,R3 ) + g 2p L 3 p−3 (∂,R3 ) v − d 2p L 3− p (∂,R3 ) then (3.4) implies |L(v)| ≤ CL ∇v L p (,R3×3 ) where CL := K f 3p L 4 p−3 (,R3 ) + g 2p L 3 p−3 (∂,R3 ) and K is the constant in (3.4). By Young inequality we then obtain |L(v)| ≤ p p−1 p (CL ε −1 ) p−1 + p −1 ε p ∇v L p (,R3×3 ) p (3.6) for every ε > 0. 123 164 Page 16 of 31 E. Mainini, D. Percivale 4 Convergence of minimizers: Proof of Theorem 2.4 and Theorem 2.5 4.1 The incompressible case We give the proof of our convergence result regarding the incompressible case, which is the more difficult. We will briefly show how to adapt the arguments to the compressible case later on. The proof follows the standard line of a Ŵ-convergence argument: we prove compactness, a lower bound and an upper bound. Lemma 4.1 (Compactness). Assume (W 1),(W 2),(W 3),(W 4), (L1) and (L2). Let (h j ) j⊂N ⊂ (0, 1) be a vanishing sequence, let M > 0 and let (y j ) j∈N ⊂ W 1, p (, R3 ) be a sequence such that GhI j (y j ) ≤ M ∀ j ∈ N. (4.1) T Let R j ∈ A p (y j ) and u j (x) := h −1 j (R j y j (x)−x). Then, the sequence (∇u j ) j∈N is bounded p 3×3 in L (, R ) and any of its weak L p (, R3×3 ) limit points is of the form ∇u∗ for some u∗ ∈ H 1 (, R3 ). Moreover, any limit point of the sequence (R j ) j∈N ⊂ S O(3) belongs to 0. SL Proof By (4.1) we obtain det ∇y j = 1 for any j ∈ N. By (4.1), (L1), (L2) and (3.6) we get for every ε > 0   −2 −2 I W (x, ∇y j ) dx = h j W I (x, ∇y j ) dx ≤ M + h −1 hj j L(y j − x)   −1 = M + h −1 j L(y j − R j x) + h j L((R j − I)x) ≤ M + h −1 j L(y j − R j x) ≤ (4.2) p p−1 p (CL ε −1 ) p−1 + p −1 ε p R j ∇u j  L p (,R3×3 ) . p Moreover, by recalling (W 4), (2.5) and (2.2), there exists a constant C (only depending on p and ) such that for every ε > 0    −2 −2 −2 I I hj W (x, ∇y j ) dx = h j W (x, ∇y j ) dx ≥ Ch j g p (|∇y j − R j |) dx     (4.3) 2− p p g (h |R ∇y |) dx ≥ CR ∇u  − C ||, = Ch −2 p j j j j j j L p (,R3×3 ) p  which in combination with (4.2) entails, by taking small enough ε, p p R j ∇u j  L p (,R3×3 ) = ∇u j  L p (,R3×3 ) ≤ Q (4.4) for some suitable constant Q depending only on CL , p,  (and not on j). On the other hand  GhI (y j ) := h −2 W I (x, ∇y j ) dx − L(R j u j ) − h −1 j j L((R j − I)x) ≤ M  entails 0 ≤ −h −1 j L((R j − I)x) ≤ M + L(R j u j ) (4.5) and by (3.6), (4.4) we get L((R j − I)x) → 0 as j → +∞. Therefore, if R j → R∗ along a 0. suitable subsequence, we have L((R∗ − I)x) = 0 that is R∗ ∈ SL By (4.4), the sequence (∇u j ) j∈N is bounded in L p (, R3×3 ). As a consequence of the Poincaré inequality, any of its weak L p (, R3×3 ) limit points is of the form ∇u∗ for some 123 Sharp conditions for the linearization of finite elasticity Page 17 of 31 164 u∗ ∈ W 1, p (, R3 ). Assuming that ∇u∗ is the weak L p (, R3×3 ) limit point along a not relabeled subsequence, we are only left to prove that u∗ ∈ H 1 (; R3 ). To this aim we let √ B j := {x ∈  : h j |∇u j | ≤ 1} (4.6) so that  Bj |∇u j |2 dx ≤ h −2 j   g p (h j |∇u j |) dx = h −2 j   g p (|∇y j − R j |) dx, hence by (4.2), (4.3) and (4.4) we get uniform boundedness in L 2 (, R3×3 ) for the sequence (1 B j ∇u j ) j∈N thus up to subsequences 1 B j ∇u j ⇀w weakly in L 2 (, R3×3 ) as j → +∞. On the other hand for every q ∈ (1, p) we have  q B cj |∇u j | dx ≤  p B cj |∇u j | dx q/ p |B cj |( p−q)/ p where the right hand side vanishes as j → +∞ since |B cj | → 0 by Chebyshev inequality. By taking into account that ∇u j = 1 B cj ∇u j + 1 B j ∇u j and that 1 B j ∇u j ⇀w weakly in L 2 (, R3×3 ) we get ∇u j ⇀w weakly in L q (, R3×3 ) and recalling that ∇u j ⇀∇u∗ weakly in L p (, R3×3 ) we get w = ∇u∗ ∈ L 2 (, R3×3 ) thus proving that u∗ ∈ H 1 (, R3 ). ⊔ ⊓ Lemma 4.2 (Lower bound). Assume (L1), (L2), (W 1), (W 2), (W 3), (W 4). Let (y j ) j∈N ⊂ T W 1, p (, R3 ) be a sequence. For any j ∈ N, let R j ∈ A p (y j ) and u j (x) := h −1 j (R j y j (x) − 1, p 3 p x). Suppose that there exists u ∈ W (, R ) such that ∇u j ⇀∇u weakly in L (, R3×3 ). Then lim inf GhI j (y j ) ≥ G I (u). j→+∞ Proof We may assume wlog that GhI j (y j ) ≤ M for any j ∈ N hence u ∈ H 1 (; R3 ) by Lemma (4.1) and 1 = det ∇y j = det(R j (I + h j ∇u j )) = det(I + h j ∇u j ) = 1 = 1 + h j div u j − h 2j (Tr(∇u j )2 − (Tr ∇u j )2 ) + h 3j det ∇u j 2 a.e. in , that is, div u j = 1 h j (Tr(∇u j )2 − (Tr ∇u j )2 ) − h 2j det ∇u j . 2 By taking into account that ∇u j are uniformly bounded in L p we get h αj |∇u j | → 0 a.e. in  for every α > 0 hence div u j = 21 h j (Tr(∇u j )2 − (Tr ∇u j )2 ) − h 2j det ∇u j → 0 a.e. in . Since the weak convergence of ∇u j implies div u j ⇀ div u weakly in L p () we get div u = 0 a.e. in . By setting D j := E(u j ) + 21 h j ∇uTj ∇u j , 123 164 Page 18 of 31 E. Mainini, D. Percivale by (3.3), (3.1) (L1) and (L2), and by recalling that B j is defined in (4.6), we get for large enough j  1 V (x, h j D j ) dx − L(R j u j ) − h −1 GhI j (v j ) ≥ 2 j L((R j − I)x) h j Bj   1 T 2 D j D V (x, 0) D j dx − η(h j D j )|D j |2 dx − L(R j u j ) ≥ Bj 2 Bj   √ 1 (1 B j D j )T D 2 W (x, I) (1 B j D j ) dx − η( h j ) |1 B j D j |2 dx − L(R j u j ), ≥ 2   (4.7)     1 3/2 T since on B j we have h j |D j | ≤ hj h j |∇v j | + 2 h j |∇v j ||∇v j | ≤ 2 h j for large enough j (so that indeed (3.3) can be applied) and since η is increasing. Since h j ∇uTj ∇u j → 0 a.e. in  and |B cj | → 0 as j → +∞, and since |1 B j h j ∇uTj ∇u j | ≤ 1, we get 1 B j h j ∇uTj ∇u j ⇀0 weakly in L 2 (, R3×3 ). By taking into account that 1 B j ∇u j ⇀∇u weakly in L 2 (, R3×3 ), we then obtain 1 B j D j ⇀E(u) weakly in L 2 (, R3×3 ). Let now c j ∈ R3 such that u j − c j ⇀u weakly in W 1, p (, R3×3 ). By taking into account that, up to 0 , we get subsequences, Lemma 4.1 entails R j → R ∈ SL lim L(−R j u j ) = lim L(−R j (u j − c j )) = −L(Ru). j→+∞ j→+∞ Hence, by (W 3), (4.7) and by the weak L 2 (, R3×3 ) lower semicontinuity of the map F  →  FT D 2 W (x, I) F dx, we conclude  1 E(u) D 2 W (x, I)E(u) dx − L(Ru) ≥ G I (u) lim inf GhI j (v j ) ≥ j→+∞ 2  ⊔ ⊓ which ends the proof. We next provide the construction for the recovery sequence, taking advantage of the following approximation result from [12]. Lemma 4.3 ( [12, Lemma 6.2]) Suppose that ∂ has a finite number of connected compo1 (, R3 ). There exists a nents. Let (h j ) j∈N ⊂ (0, 1) be a vanishing sequence. Let u ∈ Hdiv 2,∞ 3 sequence (u j ) j∈N ⊂ W (, R ) such that i) det(I + h j ∇u j ) = 1 for any j ∈ N, ii) h j ∇u j  L ∞ (,R3×3 ) → 0 as j → +∞, iii) u j → u strongly in H 1 (, R3 ) as j → +∞. Lemma 4.4 (Upper bound). Suppose that ∂ has a finite number of connected components. Assume (W 1), (W 2), (W 3), (W 4). Let (h j ) j∈N ⊂ (0, 1) be a vanishing sequence. For every u ∈ W 1, p (, R3 ) there exists a sequence (u j ) j∈N ⊂ W 1, p (, R3 ) such 0 such that by setting that u j ⇀u weakly in W 1, p (, R3 ) as j → +∞ and R∗ ∈ SL y j := R∗ (x + h j u j ) we have lim sup GhI j (y j ) ≤ G I (u). j→+∞ 1 (, R3 ). We take the sequence (u ) Proof It is enough to prove the result in case u ∈ Hdiv j j∈N from Lemma 4.3 so that u j → u strongly in H 1 (, R3 ) as j → +∞, and we take   0 . Q I (x, E(u)) dx − L(Ru) : R ∈ SL R∗ ∈ argmin  123 Sharp conditions for the linearization of finite elasticity Page 19 of 31 164 We set y j := R∗ (x + h j u j ) and F (u j ) := 21  ∇uTj D 2 W (x, I)∇u j dx − L(R∗ u j ) for any j ∈ N. Property ii) of Lemma 4.3 yields I + h j ∇u j ∈ U for a.e. x in  if j is large enough, where U is the neighbor of S O(3) that appears in (W 3). In particular, D 2 W (x, ·) ∈ C 2 (U ) for a.e. x ∈  0 to obtain and we make use of (3.2) together with det(I + h j ∇u j ) = 1 and R∗ ∈ SL     1  1 I T 2 I lim sup |Gh j (y j ) − F (u j )| ≤ lim sup  2 W (x, I + h j ∇u j ) − ∇u j D W (x, I)∇u j  dx  2 j→+∞ j→+∞   h j      1  1 = lim sup  2 W (x, I + h j ∇u j ) − ∇uTj D 2 W (x, I)∇u j  dx  2 j→+∞   h j  ≤ lim sup ω(h j |∇u j |) |∇u j |2 dx j→+∞  ≤ lim sup ω(h j |∇u j |) L ∞ () j→+∞   |∇u j |2 dx = 0. The limit in the last line is zero since h j ∇u j → 0 in L ∞ (, R3×3 ), since ω is increasing with limt→0+ ω(t) → 0 and since u j → u in H 1 (, R3 ) as j → +∞. Then, lim sup |GhI j (y j ) − G I (u)| ≤ lim sup |GhI j (y j ) − F (u j )| + lim sup |F (u j ) − G I (u)| j→+∞ j→+∞ j→+∞      1 1 ∇uTj D 2 W (x, I)∇u j − ∇uT D 2 W (x, I)∇u ≤ lim sup  2  j→+∞ 2  + lim sup |L(R∗ u j ) − L(R∗ u)| = 0 j→+∞ where the limit is zero since u j → u strongly in H 1 (, R3 ) as j → +∞. ⊔ ⊓ We next conclude the proof of Theorem 2.5 after having recalled that functionals uniformly bounded from below, which is a result that is shown in [12]. GhI are Lemma 4.5 ( [12, Lemma 4.1]) Assume (W 1),(W 2),(W 3),(W 4), (L1) and (L2). There exists a constant C > 0 (only depending on , p, f, g) such that GhI (y) ≥ −C for any h ∈ (0, 1) and any y ∈ W 1, p (, R3 ). Proof of Theorem 2.5. We obtain (2.6) from Lemma 4.5. If (y j ) j∈N ⊂ W 1, p (, R3 ) is a sequence of quasi-minimizers of GhI j , then by Lemma 4.1 there exists u∗ ∈ H 1 (, R3 ) T such that if R j ∈ A p (y j ) and u j (x) := h −1 j R j (y j (x) − R j x) then, up to subsequences, p ∇u j ⇀∇u∗ weakly in L (). Hence by Lemma 4.2 lim inf GhI j (y j ) ≥ G I (u∗ ). j→+∞ On the other hand, by Lemma 4.4, for every u ∈ W 1, p (, R3 ) there exist a sequence (u j ) j∈N ⊂ W 1, p (, R3 ) satisfying u j ⇀u weakly in W 1, p (, R3 ) as j → +∞ and R∗ ∈ 0 such that by setting ỹ := R (x + h u ) we have SL j ∗ j j lim sup GhI j (ỹ j ) ≤ G I (u). j→+∞ Since GhI j (y j ) + o(1) = inf W 1, p (,R3 ) GhI j ≤ GhI j (ỹ j ) as j → +∞, 123 164 Page 20 of 31 E. Mainini, D. Percivale by passing to the limit as j → +∞ we get G I (u∗ ) ≤ G I (u) for every u ∈ H 1 (, R3 ) thus completing the proof. 4.2 The compressible case Here we briefly show how to adapt the previous arguments to obtain the proof of Theorem 2.4. Lemma 4.6 (Compactness). Assume (W 1),(W 2),(W 3),(W 4), (L1) and (L2). Let (h j ) j⊂N ⊂ (0, 1) be a vanishing a sequence, let M > 0 and let (y j ) j∈N ⊂ W 1, p (, R3 ) be a sequence such that Gh j (y j ) ≤ M ∀ j ∈ N. (4.8) T Let R j ∈ A p (y j ) and u j (x) := h −1 j (R j y j (x)−x). Then, the sequence (∇u j ) j∈N is bounded p 3×3 in L (, R ) and any of its weak L p (, R3×3 ) limit points is of the form ∇u∗ for some u∗ ∈ H 1 (, R3 ). Moreover, any limit point of the sequence (R j ) j∈N ⊂ S O(3) belongs to 0. SL Proof It is readily seen that inequalities (4.2) and (4.3) holds true with W in place of W I hence by arguing as in Lemma 4.1 we get that ∇u j are equibounded in L p . Moreover (4.8) entails the analogous of (4.5) hence L((R j − I)x) → 0 as j → +∞ and if R j → R∗ along 0 . The remaining part of a suitable subsequence, we have L((R∗ − I)x) = 0 that is R∗ ∈ SL the proof is identical to that of Lemma 4.1. ⊔ ⊓ Lemma 4.7 (Lower bound). Assume (L1), (L2), (W 1), (W 2), (W 3), (W 4). Let (y j ) j∈N ⊂ T W 1, p (, R3 ) be a sequence. For any j ∈ N, let R j ∈ A p (y j ) and u j (x) := h −1 j R j (y j (x) − 1, p 3 R j x). Suppose that there exists u ∈ W (, R ) such that ∇u j ⇀∇u weakly in L p (, R3 ). Then lim inf Gh j (y j ) ≥ G (u). j→+∞ Proof It is enough to notice that inequality (4.7) in the proof of Lemma 4.2 holds true with Gh j in place of GhI j . The proof follows by means of the same arguments therein. ⊔ ⊓ Lemma 4.8 (Upper bound) Assume (W 1), (W 2), (W 3), (W 4). Let (h j ) j∈N ⊂ (0, 1) be a vanishing sequence. For every u ∈ W 1, p (, R3 ) there exists a sequence (u j ) j∈N ⊂ 0 such that W 1, p (, R3 ) such that u j ⇀u weakly in W 1, p (, R3 ) as j → +∞ and R∗ ∈ SL by setting y j := R∗ (x + h j u j ) we have lim sup Gh j (y j ) ≤ G (u). j→+∞ Proof We assume wlog that u ∈ H 1 (, R3 ). If we let (u j ) j∈N be a sequence obtained by a standard mollification of u, then properties ii) and iii) of Lemma 4.3 hold true. We also let   0 R∗ ∈ argmin Q(x, E(u)) dx − L(Ru) : R ∈ SL ,  so that by letting y j := R∗ (x + h j u j ) we obtain lim sup |Gh j (y j ) − G (u j )| j→+∞ 123     1  1 ≤ lim sup  2 W (x, I + h j ∇u j ) − ∇uTj D 2 W (x, I)∇u j  dx = 0   2 h j→+∞  j Sharp conditions for the linearization of finite elasticity Page 21 of 31 164 where the limit is zero by the same argument used in the proof of Lemma 4.4. Therefore lim sup |Gh j (y j ) − G (u)| ≤ lim sup |Gh j (y j ) − G (u j )| + lim sup |G (u j ) − G (u)| j→+∞ j→+∞ j→+∞      1 1 ≤ lim sup  ∇uTj D 2 W (x, I)∇u j − ∇uT D 2 W (x, I)∇u 2  j→+∞ 2  + lim sup |L(R∗ u j ) − L(R∗ u)| = 0 j→+∞ where the limit is zero thanks to the strong convergence of u j to u in H 1 (, R3 ). ⊔ ⊓ Proof of Theorem 2.4. By arguing as in Lemma 3.1 of [10] it is readily seen that there exists a constant C > 0 (only depending on , p, f, g) such that Gh (y) ≥ −C for any h ∈ (0, 1) and any y ∈ W 1, p (, R3 ). Therefore the proof can be achieved by repeating the argument of the proof of Theorem 2.5. ⊔ ⊓ 5 The gap with linear elasticity: Proof of Theorem 2.7 0 is not reduced to the identity matrix, the minimization In this section we show that if SL problem in the definition of functional G I (u) (resp. G (u)) is not solved in general by R = I if u minimizes G I (resp. G ) over W 1, p (, R3 ). In particular, the minimal value of G I (resp. G ) can be strictly below the minimum value of the standard functional of linearized elasticity E I (resp. E ). Here and in the following of this section,  is the set defined in (2.10), B denotes the unit ball centered at the origin in the x y plane, while ∇ and  shall denote the gradient and the Laplacian in the x, y variables, respectively. Moreover, the form of external forces is that of (2.11), and the conditions (f1)–(f2) are assumed to hold. We also introduce the auxiliary volume force field f̃(x, y, z) := (ϕ y (x, y), −ϕx (x, y), ψ(z)), and we notice that f̃ = R̃f, where R̃ is the rotation matrix ⎞ ⎛ 0 10 R̃ := ⎝ −1 0 0 ⎠. 0 01 (5.1) Exploiting (f1) and (f2), it is not difficult to check that L from (2.11) satisfies (L1). Concerning (L2), in view of the general form of W ∈ R3×3 Skew , i.e., ⎞ ⎛ 0 a b a, b, c ∈ R, W := ⎝ −a 0 c ⎠ −b −c 0 under assumptions (f1) and (f2) we have   f(x) · Wx dx = f̃(x) · Wx dx = 0   and   f(x) · W2 x dx =   f̃(x) · W2 x dx = −π(b2 + c2 )  1 0 zψ(z) dz ≤ 0, (5.2) 123 164 Page 22 of 31 E. Mainini, D. Percivale so that by invoking the Euler-Rodrigues formula (2.1) we see that L satisfies (L2) as well. 1 From (5.2) we see that if 0 zψ(z) dz > 0, then  f(x) · W2 x dx = 0  0 coincides with the set of rotation = 0, so that in view of (2.1) the set SL matrices around the z axis, i.e., ⎞ ⎛ cos θ sin θ 0 0 (5.3) SL = {Rθ : θ ∈ [−π, π]}, where Rθ = ⎝ − sin θ cos θ 0 ⎠, 0 0 1 if and only if b2 + c2 0 is not reduced to the identity matrix and it is a strict subset of S O(3). in particular SL 1 0 ≡ S O(3). On the other hand, if 0 zψ(z) dz = 0, then SL In both cases we clearly have 0 R̃ ∈ SL and 0 R̃ T ∈ SL . (5.4) Concerning the strain energy density, in this section we assume that W satisfies (W 1), (W 2), (W 3), (W 4) and (2.12), and we let W I (x, F) = W I (F) be equal to W (F) if det F = 1 and equal to +∞ otherwise. With these assumptions on W , the functional of linearized elasticity is reduced to  E (u) := 4 |E(u)|2 dx − L(u), u ∈ H 1 (, R3 ),  while the limit functional G becomes  G (u) = 4 |E(u)|2 dx − L(u) − max L((R − I)x), 0 R∈SL  u ∈ H 1 (, R3 ). For the following arguments, it is also convenient to introduce the auxiliary functional  G̃ (u) := 4 |E(u)|2 dx − L(R̃ T u), u ∈ H 1 (, R3 ) (5.5)  where R̃ is given by (5.1). The above functionals E , G , G̃ are extended as usual to W 1, p (, R3 ) \ H 1 (, R3 ) with value +∞. Due to (L1) and Korn inequality, it follows from standard arguments that the functionals E , G , G̃ admit minimizers over H 1 (, R3 ). We shall also consider the incompressible case, by considering functionals G I , E I , as defined in Sect. 2, under the assumption (2.12) for W , and with external loads given by functional L from (2.3) with g ≡ 0 and f of the form (2.11). Again, we introduce the auxiliary functional ⎧  1 (, R3 ) ⎨4 |E(u)|2 dx − L(R̃ T u) if u ∈ Hdiv G̃ I (u) :=  ⎩ +∞ otherwise in W 1, p (, R3 ). where R̃ is given by (5.1). Existence of minimizers also holds for E I , G I , G̃ I , since the divergence-free constraint is weakly closed in H 1 (, R3 ). Before proving Theorem 2.7, we provide an auxiliary statement. 123 Sharp conditions for the linearization of finite elasticity Page 23 of 31 Lemma 5.1 Assume (2.10), (2.11), (2.12), (f1), (f2). Then there hold   8u 2x y + 2(u yy − u x x )2 + uϕ + min min G̃ ≤ min H 1 (,R3 ) u∈H 2 (B) B  1 −π H 1 (,R3 )  1 w′ 2 0 wψ < 0 0 and min w∈H 1 (0,1) B 4π 164 G̃ I ≤ min  u∈H 2 (B) B 8u 2x y + 2(u yy − u x x )2 +  uϕ < 0. B Proof Let us prove the first statement. Let K ⊂ H 1 (, R3 ) be the class of displacement fields v ∈ H 1 (, R3 ) of the form v(x, y, z) = (u y (x, y), −u x (x, y), w(z)), (5.6) for some u ∈ H 2 () and some w ∈ H 1 (0, 1). It is easily seen that K is weakly closed in H 1 (, R3 ). In particular, there are minimizers of G̃ over K. We notice that by means of (5.6) any couple u ∈ H 2 (B), w ∈ H 1 (0, 1) uniquely determines v ∈ K. Conversely, any v ∈ K uniquely determines u ∈ H 2 (B), up to an additive constant, and w ∈ H 1 (0, 1). Moreover a computation shows that the energy functional G̃ takes the following form for any v ∈ K:   G̃ (v) = G̃ ((u y , −u x , w)) = 8u 2x y + 2(u yy − u x x )2 − (u y , −u x ) · (ϕ y , −ϕx ) + 4π =  B  0 B 1 w ′ 2 dz − π  B 1 wψ dz 0 8u 2x y + 2(u yy − u x x )2 +  B uϕ + 4π  1 0 w ′ 2 dz − π  1 wψ dz, 0 having exploited the fact that ∇ϕ · n = 0 on ∂ B, where n is the outer unit normal to ∂ B. We deduce   min G̃ ≤ min G̃ = min 8u 2x y + 2(u yy − u x x )2 + uϕ H 1 (,R3 ) K u∈H 2 (B) B + min w∈H 1 (0,1) 4π  0 1 w′ 2 − π  B 1 (5.7) wψ, 0 where the last minimization problem in (5.7) has a solution as well, thanks to  ψ = 0 from (f2) and to Poincaré inequality. If we introduce the perturbation u + εζ , where ε > 0 and ζ ∈ H02 (B), we get the first order optimality condition for the first minimization problem in the right hand side of (5.7)   4 4u x y ζx y + u yy ζ yy + u x x ζx x − u yy ζx x − u yy ζx x = − ζ ϕ, B B and after integration by parts we obtain the Euler-Lagrange equation 42 u + ϕ = 0 in D′ (B), where 2 denotes the planar biharmonic operator. In view of (5.7), and since G̃ (0) = 0, in order to conclude it is enough to show that minK G̃  = 0. Assume by contradiction that minK G̃ = 0: then u ≡ 0 on B, w ≡ 0 on (0, 1) are solutions to the minimization problems in the right hand side of (5.7), so that 123 164 Page 24 of 31 E. Mainini, D. Percivale 2 u ≡ 0 in B and from above the Euler-Lagrange equation we deduce ϕ ≡ 0 in B. This is a contradiction with assumption (f1). 1 (, R3 ) made In order to prove the second statement, we consider the subset Kdiv of Hdiv of vector fields of the form v(x, y, z) = (u y (x, y), −u x (x, y), 0) for some u ∈ H 2 (B), and for any v ∈ Kdiv the energy functional G̃ I has the expression   G̃ I (v) = G̃ I ((u y , −u x , 0)) = 8u 2x y + 2(u yy − u x x )2 + uϕ, B so that min H 1 (,R3 ) G̃ I ≤ min G̃ I = Kdiv  min u∈H 2 (B) B B 8u 2x y + 2(u yy − u x x )2 +  uϕ B ⊔ ⊓ and the proof concludes with the same argument as above. We proceed to the proof of Theorem 2.7. Proof of Theorem 2.7. We will prove that there holds min H 1 (,R3 ) G≤ min H 1 (,R3 ) G̃ < min E, min EI. H 1 (,R3 ) and that if ψ L 2 () is small enough there also holds min H 1 (,R3 ) GI ≤ min H 1 (,R3 ) G̃ I < H 1 (,R3 ) We start by noticing that the first inequality in both statements is trivial, due to (5.4). Therefore, we are left to prove the second inequality in both cases. Let v = (v1 , v2 , v3 ) ∈ H 1 (, R3 ). Let ṽ := (v1 , v2 ), so that ṽ ∈ H 1 (, R2 ). Let 1 ũ ∈ H 1 (B, R2 ) be defined by ũ(x, y) := 0 ṽ(x, y, z) dz. Let moreover w̃ ∈ H 1 (0, 1) be 1 defined by w̃(z) := π B v3 (x, y, z) d x d y. Since |E(v)| ≥ | E(v)| = | E(ṽ)|, where  E(·) is the upper-left 2 × 2 submatrix of E(·), by Jensen inequality we have   E I (v) ≥ E (v) ≥ 4 |E(v)|2 − (ϕx , ϕ y , ψ) · v   2 2   1  1     1     v3,z d x d y  dz E(ṽ) dz  d x d y + 4π ≥4  π B 0 B 0    1  1 ψ(z) − (ϕx , ϕ y ) · v3 (x, y, z) d x d y ṽ dz d x d y − B ≥ J (ũ) + 4π  0 0 1 w̃ ′ 2 − π  0 dz B 1 w̃ψ, 0 (5.8) where we have used the notation vi,a := ∂a vi , i ∈ {1, 2, 3}, a ∈ {x, y, z}, and where   J ( u) := 4 | E(ũ)|2 − ∇ϕ · ũ. B (5.9) B We claim that functional J admits a minimizer over H 1 (B, R2 ) which is the gradient of a H 2 (B) function. Indeed, thanks to (f1) and Korn inequality, a minimizer exists and it is unique up to planar infinitesimal rigid displacements. Moreover, thanks to a first variation argument, it is a solution to the boundary value problem  −8 div  E(ũ) = ∇ϕ in B (5.10)  E(ũ)n = 0 on ∂ B. 123 Sharp conditions for the linearization of finite elasticity Page 25 of 31 164 1 2 We claim that problem (5.10)  admits a solution ũ ∈ H (B, R ) of the form ũ(x, y) := −1 1 2 2 2 r η(r )(x, y), where r := x + y (since ũ ∈ H (B, R ), we necessarily have η(0) = 0). Indeed, a direct calculation shows that ũ(x, y) = r −1 η(r )(x, y) is a H 1 (B, R2 ) solution to problem (5.10) if and only if ⎧ 2 ′′ ⎨ r η + r η′ − η = − 81 r 2 φ ′ (r ) in (0, 1) η(0) = 0 ⎩ ′ η (1) = 0 (5.11) where φ is the radial profile of ϕ. Problem (5.11) has indeed a solution, whose explicit form is  r 1 1 η∗ (r ) = − r φ(r ) + t 2 φ ′ (t) dt. (5.12) 16 16r 0 r Therefore, letting  : B → R be the radial function defined by (x, y) := 0 η∗ (t) dt, we obtain  ∈ H 2 (B) and moreover ∇ = r −1 η(r )(x, y) solves (5.10), hence it minimizes J over H 1 (, R2 ). The claim is proved. Therefore, the estimate (5.8) rewrites as E I (v) ≥ E (v) ≥ J (ũ) + 4π ≥ min ∈H 2 (B) 4   1 0 B w̃ ′ 2 − π  1 w̃ψ 0 | E(∇)|2 −  B  ∇ · ∇ϕ + 4π 1 w̃ ′2 −π 0  1 w̃ψ. 0 Integrating by parts, since (f1) yields φ = 0 on ∂ B, we get E I (v) ≥ E (v) ≥ min ∈H 2 (B) 4  B |D 2 |2 +  B  ϕ + 4π  1 0 w̃ ′ 2 − π  1 w̃ψ, 0 where D 2 denotes the Hessian in the x, y variables, hence   min E I ≥ min E ≥ min 82x y + 42x x + 42yy + ϕ H 1 (,R3 ) H 1 (,R3 ) ∈H 2 (B) B + min w∈H 1 (0,1) 4π  0 1 w′ 2 − π  B (5.13) wψ.  ˜ ∈ H 2 (B) is a solution to the first minimization problem on the right hand Suppose that  ˜ solves (5.10), and taking the divergence side of (5.13). We have already proven that ∇  ˜ solves the biharmonic equation 82  = −ϕ in B. As (f1) requires therein shows that  ˜ is not identically zero as well. This that ϕ is not identically zero on B, we deduce that  implies by Young inequality  B ˜ 2x y + 2( ˜ yy −  ˜ x x )2 + 8  B ˜ ϕ <  B ˜ 2x y + 4 ˜ 2x x + 4 ˜ 2yy + 8  B ˜ ϕ. (5.14) ˜ yy −  ˜ x x )2 ≤ 4 ˜ 2x x +4 ˜ 2yy Notice that the inequality is strict, since the Young inequality 2( ˜ x x = − ˜ yy , and we have just checked that  ˜ does not holds with equality if and only if  ˜ yy −  ˜ x x )2 < 4 ˜ 2x x + 4 ˜ 2yy on a set of positive vanish identically on B. In particular, 2( 123 Page 26 of 31 164 E. Mainini, D. Percivale measure in B. From Lemma 5.1, from (5.14) and (5.13) we infer min G̃ H 1 (,R3 ) ≤ ≤ < min  u∈H 2 (B) B   8u 2x y + 2(u yy − u x x )2 + B ˜ 2x y + 2( ˜ yy −  ˜ x x )2 + 8 B ˜ 2x y + 4 ˜ 2x x + 4 ˜ 2yy + 8  = ∈H 2 (B) B ≤ H 1 (,R3 ) min min   B B  ˜ ϕ + ˜ ϕ + 82x y + 42x x + 42yy + E≤ min EI B uϕ +  B min w∈H 1 (0,1) min w∈H 1 (0,1) min w∈H 1 (0,1) ϕ + 4π 4π   0 min 0 1 4π  1 0 1 w ′2 w′ 2 − π −π w′ 2 − π w∈H 1 (0,1) 4π  1 0   1  1 wψ 0 wψ 0 1 wψ 0 w′ 2 − π  1 wψ 0 H 1 (,R3 ) thus concluding the proof of the first statement. Let us now prove the second statement, concerning the incompressible case. Let  1/2 . C(ϕ, ψ) := ψ L 2 () |∇ϕ|2 + |ψ|2  If v∗ = (v1∗ , v2∗ , v3∗ ) minimizes E I over H 1 (, R3 ), then we have   d  I ∗ 0= E ((1 + ε)v ) = 8 |E(v∗ )|2 − L(v∗ ), dε ε=0  and then by applying (L1) and Hölder inequality  8 |E(v∗ )|2 d x = L(v∗ ) = L(v∗ − Pv∗ ) ≤ f L 6/5 (,R3 ) v∗ − Pv∗  L 6 (,R3 ) .  By (3.5) and Hölder inequality again we deduce therefore   1/2 |∇ϕ|2 + |ψ|2 ≤ K 1 f L 2 (,R3 ) ≤ K 1 |E(v∗ )|2  1/2 (5.15)  for some suitable constant K 1 (only depending on ). By taking into account (5.15), still by Korn and Hölder inequality and by (f2) we get          ψv ∗  =  ψ(v∗ − Pv∗ )3  ≤ ψ L 2 () v∗ − Pv∗  L 2 () 3    (5.16)   ∗ ≤ K 2 ψ L 2 () E(v ) L 2 () ≤ K C(ϕ, ψ), where K 2 is another constant that depends only on  and K = K 1 K 2 . By taking into account (5.16), we get   1 ∗ ∗ 2 ∗ 2 ∗ 2 + v2,x ) − (v1∗ ϕx + v2∗ ϕ y ) − K C(ϕ, ψ) min E I ≥ 4 |v1,x | + |v2,y | + (v1,y 2 H 1 (,R3 )   and by Jensen inequality   1 ∗ ∗ 2 ∗ 2 ∗ 2 + ṽ2,x ) − (ṽ1∗ ϕx + ṽ2∗ ϕ y ) − K C(ϕ, ψ) min E I ≥ 4 (|ṽ1,x | + |ṽ2,y | + (ṽ1,y 2 H 1 (,R3 ) B B = J (ṽ∗ ) − K C(ϕ, ψ) 123 Sharp conditions for the linearization of finite elasticity Page 27 of 31 164 1 where we have set ṽ∗ (x, y) := 0 v∗ (x, y, z) dz and where J is defined by (5.9). By repeating ˜ be defined in the same way) we the argument used in the compressible case (and letting  obtain   min E I ≥ min ϕ − K C(ϕ, ψ) 82x y + 42x x + 42yy + H 1 (,R3 ) ∈H 2 (B) B B   ˜ 2yy + ˜ 2x x + 4 ˜ 2x y + 4 ˜ (5.17) = 8 ϕ − K C(ϕ, ψ) B B  ˜ 2x y + 2( ˜ yy −  ˜ x x )2 + 2|| ˜ 2+ ˜ = 8 ϕ − K C(ϕ, ψ). B B By (5.17) and by the second statement of Lemma 5.1 we deduce  I ˜ 2 − K C(ϕ, ψ) + min 2|| min E ≥ H 1 (,R3 ) H 1 (,R3 ) B G̃ I . ˜ 2 > 0 as previously observed, we may choose ψ L 2 () so small that Since B || ˜ 2 and deduce K C(ϕ, ψ) < B 2|| min H 1 (,R3 ) EI > min G̃ I H 1 (,R3 ) ⊔ ⊓ thus completing the proof. We conclude by providing some more properties for the compressible case that are directly deduced by refining the arguments in the proof of Theorem 2.7, under the further assumption 1 0 zψ(z) dz > 0. First we check that minimizers of the limit functional G are not unique up to infinitesimal rigid displacements, in the sense that there are two minimizers whose difference is not an infinitesimal rigid displacement. In a second statement we obtain a solution of the problem min G , thus showing more explicitly the gap between min G and min E . 1 Proposition 5.2 Assume (2.10), (2.11), (2.12), (f1), (f2) and 0 zψ(z) dz > 0. If u ∈ argmin H 1 (,R3 ) G , then u is of the form u = w0 +w, where E(w0 ) ≡ 0 and w = (w1 , w2 , w3 ) is such that w1 , w2 do not depend on z and w3 does not depend on x, y. Moreover, minimizers of G over H 1 (, R3 ) are not unique up to infinitesimal rigid displacements. Proof Suppose that u = (u 1 , u 2 , u 3 ) ∈ H 1 (, R3 ) is a minimizer for functional G and let ū := (ū 1 , ū 2 , ū 3 ), where  1  1  1 ū 1 := u 2 (x, y, z) dz, ū 3 := u 1 (x, y, z) dz, ū 2 := u 3 (x, y, z) d x d y. π B 0 0 1 0 is given by (5.3), and then it is immediate to check that, Since 0 zψ(z) dz > 0, the set SL due to the specific form of f from (2.11), there holds   0 f · Ru dx = f · Rū dx ∀ R ∈ SL . (5.18)   Moreover, by applying Jensen inequality, similarly to the proof of Theorem 2.7, we obtain     u 21,x ≥ ū 21,x , u 22,y ≥ ū 22,y ,         (5.19) u 23,z ≥ ū 23,z , (u 1,y + u 2,x )2 ≥ (ū 1,y + ū 2,x )2     123 164 Page 28 of 31 E. Mainini, D. Percivale so that  4 |E(u)|2 dx     = 4u 21,x + 4u 22,y + 4u 23,z + (u 1,y + u 2,x )2 + (u 1,z + u 3,x )2 + (u 2,z + u 3,y )2     ≥ 4ū 21,x + 4ū 22,y + 4ū 23,z + (ū 1,y + ū 2,x )2 = 4 |E(ū)|2 dx.   But the latter inequality and the inequalities (5.19) are necessarily equalities, otherwise in view of (5.18) we would deduce G (ū) < G (u), contradicting minimality of u. This implies u 1,z + u 3,x = u 2,z + u 3,y ≡ 0 (5.20) along with the fact that u 3,z does not depend on x, y, since the Jensen inequality 1 π  B (u 3,z )2 d x d y ≥ 1 π  2 u 3,z d x d y B = ū 23,z is strict unless u 3,z is independent of x, y. Similarly, we deduce that u 1,x , u 2,y and u 1,y +u 2,x do not depend on z, therefore there exist functions H = H (z), T = T (x, y), A = A(y, z), B = B(x, y), C = C(x, z) and D = D(x, y) such that u 1 , u 2 , u 3 have the form u 1 (x, y, z) = A(y, z) + B(x, y), u 2 (x, y, z) = C(x, z) + D(x, y), (5.21) ∂z (u 1,y + u 2,x ) = A yz (y, z) + C x z (x, z) ≡ 0. (5.22) u 3 (x, y, z) = H (z) + T (x, y), and there holds Taking (5.20) into account we deduce 0 ≡ A z (y, z) + Tx (x, y) = C z (x, z) + Ty (x, y) (5.23) so that A zz = C zz = Tx x = Tyy ≡ 0 and 0 ≡ A yz (y, z) + Tx y (x, y) = C x z (x, z) + Tx y (x, y). The latter entails, thanks to (5.22), A yz = C x z = Tx y ≡ 0. We conclude that T is a linear function of x, y, i.e., T (x, y) = ax + by + c for some real constants a, b, c, and then from (5.23) we deduce that there are functions Q = Q(y) and S = S(x) such that A(y, z) = −az + Q(y) and C(x, z) = −bx + S(x). Substituting in (5.21) we have u 1 (x, y, z) = −az + B(x, y) + Q(y), u 2 (x, y, z) = −bz + D(x, y) + S(x), u 3 (x, y, z) = H (z) + c + ax + by, where E(−az, −bz, c + ax + by) = 0. This shows that if u ∈ H 1 (, R3 ) minimizes G , then up to adding an infinitesimal rigid displacement u 1 , u 2 depend only on x, y and u 3 depends only on z. Let now u∗ = (u ∗1 , u ∗2 , u ∗3 ) ∈ H 1 (, R)3 be a minimizer of G such that u ∗1 , u ∗2 do not depend on z and u ∗3 does not depend on x, y, so that by defining û := (−u ∗1 , −u ∗2 , u ∗3 ) we 123 Sharp conditions for the linearization of finite elasticity Page 29 of 31 164 get |E(u∗ )|2 ≡ |E(û)|2 . Let R∗ ∈ argmaxS 0 L(Ru∗ ). By letting R̂ := diag(−1, −1, 1) R∗ , L 0 and we get we have R̂ ∈ SL   ∗ 2 ∗ ∗ min G = |E(u )| dx − L(R u ) = |E(û)|2 dx − L(R̂û), H 1 (,R3 )   thus showing that û is also a minimizer of G . However, û − u∗ is not an infinitesimal rigid displacement. Indeed, assume by contradiction that E(u∗ − û) ≡ 0. Then u ∗1,x ≡ u ∗2,y ≡ u ∗1,y + u ∗2,x ≡ 0, implying the existence of real constants ā, b̄, c̄ such that u ∗1 (x, y) = ā + c̄y and u ∗2 (x, y) = b̄ − c̄x. Therefore, (0, 0, u ∗3 ) differs from u∗ by an infinitesimal rigid displacements, and since G is invariant under the addition of infinitesimal rigid displacements, 0 and the fact that we obtain the minimality of (0, 0, u ∗3 ) for G . But then the form (5.3) of SL ∗ u 3 depends only on z directly imply  min G = G (0, 0, u ∗3 ) = 4 (u ∗3,z )2 dx H 1 (,R3 )  u ∗3,z so that needs to be identically zero and we deduce that the trivial displacement field minimizes G , so that 0 = G (0) = min H 1 (,R3 ) G≤ min H 1 (,R3 ) G̃ , ⊔ ⊓ where G̃ is defined by (5.5). This contradicts Lemma 5.1 and concludes the proof. In the next statement, for every θ ∈ [−π, π] and for Rθ as in 5.3, we use the notation ⎧  ⎪ ⎪ |E(u)|2 dx − LRθ (u) if u ∈ H 1 (, R3 ) ⎨4  Gθ (u) = ⎪ ⎪ ⎩ +∞ otherwise in W 1, p (, R3 ), where LR is defined by (2.14). With this notation we clearly have G0 ≡ E and G−π/2 ≡ G̃ , where G̃ is defined by (5.5). Proposition 5.3 Under the same assumptions of Proposition 5.2, let u0 ∈ argmin W 1, p (,R3 ) G0 and let u−π/2 ∈ argmin W 1, p (,R3 ) G−π/2 . Then, min W 1, p (,R3 ) Gθ = cos2 θ G0 (u0 ) + sin2 θ G−π/2 (u−π/2 ) (5.24) and min W 1, p (,R3 ) G= min min θ ∈[−π,π ] u∈W 1, p (,R3 ) Gθ (u) = G−π/2 (u−π/2 ) < G0 (u0 ) = min W 1, p (,R3 ) E. (5.25) Proof It is  possible to check with a computation that the vector field 2r −1 η∗ (r )(y, −x), where r = x 2 + y 2 and η∗ is defined by (5.12), solves the problem  −8 div  E(ũ) = (ϕ y , −ϕx ) in B  E(ũ)n = 0 on ∂ B, where  E(·) denotes the upper-left 2 × 2 submatrix of E(·). Moreover, thanks to the very same argument of the proof of Proposition 5.2, it is possible to find a minimizer of functional 123 164 Page 30 of 31 E. Mainini, D. Percivale G−π/2 in which the first two components do not depend on z and the third component does not depend on x, y. Therefore, it is possible to find such a minimizer by decoupling the corresponding Euler–Lagrange equation for u = (u 1 , u 2 , u 3 ), i.e.,  −8 div E(u) = Rπ/2 (ϕx , ϕ y , ψ) in  E(u)n = 0 on ∂, in the above problem on B for ũ = (u 1 , u 2 ), and in the ordinary differential equation −8u ′′3 = ψ in the interval (0, 1), complemented by the conditions u ′3 (0) = u ′3 (1) = 0, that gets solved, recalling (f2), by the function   1 z s (z) = − ψ(t) dt ds. 8 0 0 Therefore, a minimizer of G−π/2 over W 1, p (, R3 ) is given by (2r −1 η∗ (r ) y, −2r −1 η∗ (r ) x, (z)). Similarly, (r −1 η∗ (r ) x, r −1 η∗ (r ) y, (z)) is a minimizer of G0 ≡ E , recalling that r −1 η∗ (r )(x, y) solves (5.10) as seen in the proof of Theorem 2.7. Hence, given u0 ∈ argmin W 1, p (,R3 ) G0 and u−π/2 ∈ argmin W 1, p (,R3 ) Gπ/2 , we may assume w.l.o.g. that u−π/2 = (2r −1 η∗ (r ) y, −2r −1 η∗ (r ) x, (z)), u0 = (r −1 η∗ (r ) x, r −1 η∗ (r ) y, (z)), and more generally we let uθ := cos θ (r −1 η∗ (r ) x, r −1 η∗ (r ) y, 0) − sin θ (2r −1 η∗ (r ) y, −2r −1 η∗ (r ) x, 0) + (0, 0, (z)), so that uθ solves  −8 div E(u) = RθT (ϕx , ϕ y , ψ) in  E(u)n = 0 on ∂, for any θ ∈ [−π, π], and then uθ is indeed a minimizer of Gθ . Taking advantage of radiality and of the form of uθ , it is easy to check that  E(u0 ) : E(u−π/2 ) dx = 0  and that (5.24) holds. But as shown in the proof of Theorem 2.7 we have G0 (u0 ) = min W 1, p (,R3 ) G0 = min W 1, p (,R3 ) E> min W 1, p (,R3 ) G̃ = min W 1, p (,R3 ) G−π/2 = G−π/2 (u−π/2 ), so that min θ ∈[−π,π ] G (uθ ) = min θ ∈[−π,π ] cos2 θ G0 (u0 ) + sin2 θ G−π/2 (u−π/2 ) = G−π/2 (u−π/2 ). We conclude that the optimal rotation realizing the maximum in the definition of G (u−π/2 ) is given by R−π/2 and that (5.25) holds true. ⊔ ⊓ We eventually remark that in view of the latter propositions (and in the same assumptions) and in view of Theorem 2.10, by taking θ = ±π/2, there is no gap between the minimal value of functional GRθ and that of functional ERθ . Acknowledgements The authors acknowledge support from the MIUR-PRIN project No 2017TEXA3H. The authors are members of the GNAMPA group of the Istituto Nazionale di Alta Matematica (INdAM). Funding Open access funding provided by Università degli Studi di Genova within the CRUI-CARE Agreement. 123 Sharp conditions for the linearization of finite elasticity Page 31 of 31 164 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. 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