1 Introduction

This paper deals with the derivation of homogenized bending–torsion theory for rods, starting from 3D nonlinear elasticity by means of \(\Gamma \)-convergence. The main novelty is that we do not presuppose any kind of periodicity, but work in a general framework.

There is vast literature on deriving rod, plate and shell equations from 3D elasticity. The first work in deriving lower-dimensional models by \(\Gamma \)-convergence techniques was [1], where the authors derived the string model. It was well known that the obtained models depend on the relation of the external loads (i.e., the energy) with respect to the thickness of the body h. The first rigorous derivation of higher-order models was done in [8, 9] for the case of bending and von Kármán plate. The key mathematical ingredient in these cases was the theorem on geometric rigidity.

After these pioneering works, there were many papers devoted to the rigorous derivation of lower-dimensional models from 3D elasticity by means of \(\Gamma \)-convergence. We mention only those works that refer to the derivation of rod theories.

In [17], the authors derived the bending–torsion rod theory by assuming that the stored energy density function is fixed, i.e., that there are no oscillations of the material. As customary to derivations of bending theories, they assume that the energy is of the order \(h^2\), where h is the thickness of the body (after division with the order of vanishing volume, which is \(h^2\)). In [18] the authors derived the rod model in the so-called von Kármán regime, where the order of energy was assumed to be \(h^4\). In [19] the authors analyzed the stationary points (i.e., the equations) in the case of bending rod and show that the limit equation is the one corresponding to the limit energy obtained by \(\Gamma \)-convergence. However, due to nonlinearity, it is not generally true that the global minimizers of 3D problem (even if we have their existence) satisfy the Euler–Lagrange equations from which they start the derivation (see [7, 20] for details). We emphasize the fact that the techniques used here can also be adapted to the approach in [19].

It is important to notice that, although the bending theory is small strain theory, deformations are large, in contrast to von Kármán theory, where the limit deformation is a rigid deformation and the energy depends on the correctors. Thus, we can say that bending theory carries more nonlinearity. We also mention the work [24], where the author gave the full asymptotic (higher-order) theory for curved rods.

This paper deals with the effects of simultaneous homogenization and dimensional reduction. There is vast literature on the effects of simultaneous homogenization and dimensional reduction on limit equations, in different context. In [11], the authors study these effects for a linear elasticity system without periodicity assumption, adapting H-convergence to dimensional reduction. In [4], the authors study the same effects for nonlinear systems (membrane plate) by means of \(\Gamma \)-convergence, also without periodicity assumptions. In [6], the authors study nonlinear monotone operators in the context of simultaneous homogenization and dimensional reduction in a general framework. Much earlier, in [15], the authors study the same effects in the case of a linearized rod model, where it was assumed that the rod is homogeneous along its central line, but the microstructure is given in the cross section. We also mention the work of Arrieta on the Laplace equation and thin domains with oscillatory boundaries (see, e.g., [3]). Finally we emphasize the work [21], where the author presented the systematic approach which consisted in combining the techniques from [8, 9] and two-scale convergence methods [2] to obtain the model of homogenized bending rod.

Recently, the techniques from [8, 9] were combined with two-scale convergence to obtain the models of homogenized von Kármán plate (see [23, 26]), homogenized von Kármán shell (see [14]) and homogenized bending plate (see [13, 27]). Most of these models were derived under the assumption of periodic oscillations of material, where it was assumed that the material oscillates on the scale \(\varepsilon (h)\), while the thickness of the body is h. The obtained models depend on the parameter \(\gamma =\lim _{h \rightarrow 0} \tfrac{h}{{\varepsilon {\scriptstyle (h)}}}\). In the case of the von Kármán plate, the situation when \(\gamma =0\) corresponds to the case in which dimensional reduction is dominated and the obtained model is the model of homogenized von Kármán plate and can be obtained as the limit case when \(\gamma \rightarrow 0\). Analogously, the situation when \(\gamma =\infty \) corresponds to the case when homogenization dominates and can again be obtained as the limit when \(\gamma \rightarrow \infty \); this is the model of the von Kármán plate, obtained starting from homogenized energy. In the case of the von Kármán shell and the bending plate, the situation \(\gamma =0\) was more subtle and the derived models depend on the further assumption of the relation between \(\varepsilon (h)\) and h. We obtained different models for the cases \({\varepsilon {\scriptstyle (h)}}^2 \ll h \ll {\varepsilon {\scriptstyle (h)}}\) and \(h \sim {\varepsilon {\scriptstyle (h)}}^2\).

In this paper, we derive the bending–torsion rod model by simultaneous homogenization and dimensional reduction without any periodicity assumption. This is a generalization of the work [21], where the author derived the bending–torsion rod theory via two-scale convergence techniques, assuming that the material oscillates periodically along the central line of the rod. Our result can be interpreted as a form of a stability result: We obtain the same type of equations starting with any kind of oscillating or non-oscillating material, and the oscillations can happen in any direction (even in the cross section). Moreover, we derive an abstract variational formula for the limit energy density which covers all the possible cases and can also be used to obtain all the regimes in the periodic case. We use slight variations of standard \(\Gamma \)-convergence techniques for homogenization, adapted to the special case of dimensional reduction for higher-order models in elasticity. This approach has already been used in [25] to derive the model of the von Kármán plate via simultaneous homogenization and dimensional reduction techniques. Let us emphasize the fact that this kind of stability result is not expected to be valid in the case of the bending plate or even the von Kármán shell due to more complex phenomenology in the periodic case as explained above (see also the explanations in [27] and the model obtained in [22]).

The main results in this paper are given in Theorem 2.13 and Theorem 2.14, where the “lower bound” and the “upper bound” are proven, respectively. Together with Lemma 2.6 and Lemma 2.10 they imply standard \(\Gamma \)-compactness results, because these lemmas imply that the Assumption 2.11 is valid on a subsequence. We prove that, on a subsequence, the limit energy density is a quadratic form in the strain of the limit deformation. The limit deformation and the strain itself are standard ones for the case of the bending rod.

1.1 Notation

  • By B(x, r) we denote the open ball of radius \(r>0\) around \(x \in \mathbb {R}^n\) in Euclidean norm;

  • for \(x \in \mathbb {R}\), by \(\lfloor x \rfloor \) we denote the greatest integer less or equal to x;

  • \(e_1,e_2,e_3\) denotes the canonical basis in \(\mathbb {R}^3\);

  • \(\nabla _h\) is the scaled gradient \(\nabla _h=(\partial _1, \tfrac{1}{h} \partial _2, \tfrac{1}{h} \partial _3)\);

  • \(\mathbb {M}^{m \times n}\) is the space of matrices with m rows and n columns, while \(\mathbb {M}^n\) is the space of quadratic matrices of order n. \(\mathbb {M}^n_{{\text {sym}}\,}\) denotes the space of symmetric matrices of order n, while \(\mathbb {M}^n_{{\text {skw}}\,}\) denotes the space of skew symmetric matrices of order n;

  • for \(A \in \mathbb {M}^n\) by \({\text {sym}}\,A\) we denote the symmetric part of A; \({\text {sym}}\,A=\tfrac{1}{2} (A+A^t)\), while by \({\text {skw}}\,A\) we denote the skew symmetric part of A; \({\text {skw}}\,A=\tfrac{1}{2}(A-A^t)\);

  • for \(A, B \in \mathbb {M}^n\) by \(A \cdot B\) we denote the scalar product \({\text {tr}}(AB^t)\);

  • \(\iota :\mathbb {\mathbb {R}}^{3} \rightarrow \mathbb {M}^3\) is the natural inclusion

    $$\begin{aligned} \iota (m)=\sum _{i=1}^3 m_{i} e_i \otimes e_1; \end{aligned}$$

  • we denote the projection of \(\mathbb {R}^3\) on the \(x_2 x_3\)-plane by \(d_{\omega }\), \(d_{\omega }(x)=(0,x_2,x_3)^t\);

  • if \(O \subset \mathbb {R}^n\) open, by \(W^{1,p}(O;M)\) we denote the subset of Sobolev space of functions taking values in \(M \subset \mathbb {R}^m\) for a.e. \(x \in O\). It is easy to see if M is a subspace of \(\mathbb {R}^m\) then \(W^{1,p}(O;M)\) is a subspace of \(W^{1,p}(O;\mathbb {R}^m)\). If M is closed subset of \(\mathbb {R}^m\), then \(W^{1,p}(O;M)\) is a closed subset of \(W^{1,p}(O;\mathbb {R}^m)\) in weak and strong topology;

  • for \(S \subset \mathbb {R}^n\), by \(\chi _S\) we denote the characteristic function of S; \(\chi _S: \mathbb {R}^n \rightarrow \{0,1\}\);

  • by |S| we denote the Lebesgue measure of S;

  • when writing sequences \((h_n)\), \((W^h)\) etc., we usually omit the subscripts denoting where there the indexes live (\(n\in \mathbb {N}, \ h>0\)).

1.2 General framework

Let \(\omega \subset \mathbb {R}^2\) be an open connected set with Lipschitz boundary. We define by \(\Omega ^h =[0,L] \times h\omega \), the reference configuration of a rod-like body with thickness h. When \(h=1\) we omit the superscript and write \(\Omega =\Omega ^1\). We may assume that the coordinate axes are chosen such that

$$\begin{aligned} \int _{\omega } x_2 \,\mathrm {d}x_2 \,\mathrm {d}x_3= \int _{\omega } x_3 \,\mathrm {d}x_2 \,\mathrm {d}x_3= \int _{\omega } x_2 x_3 \,\mathrm {d}x_2 \,\mathrm {d}x_3=0. \end{aligned}$$
(1)

We denote the moments of inertia by \(\mu _{i}=\int _{\omega } x_i^2 \,\mathrm {d}x_2 \,\mathrm {d}x_3\) for \(i=2,3\).

For each \(h>0\), the elastic energy functional on the canonical domain \(\Omega \) is given by

$$\begin{aligned} \int _{\Omega } W^h(x, \nabla _h y^h) \,\mathrm {d}x, \end{aligned}$$

where \(W^h\) is an elastic energy density function and \(\nabla _h y^h = (\partial _1{y^h}, \tfrac{1}{h} \partial _2{y^h}, \tfrac{1}{h} \partial _3{y^h}) \) is the scaled gradient of a deformation \(y^h:\Omega \rightarrow \mathbb {R}^3\). We will assume that we are in the bending regime, i.e., that there is a positive constant C independent of h such that the energy of a minimizing sequence \((y^h)\) satisfies the inequality:

$$\begin{aligned} \int _{\Omega } W^h(x, \nabla _h y^h) \,\mathrm {d}x \le Ch^2. \end{aligned}$$
(2)

This assumption can be replaced by the assumption on the scaling of external loads, see [9] for details. Here we state the standard assumptions on the energy densities \(W^h\) of a composite material.

Definition 1.1

(Nonlinear material law) Let \(\eta _1,\eta _2\) and \(\rho \) be any positive constants such that \(\eta _1 \le \eta _2\). The class \(\mathcal {W}(\eta _1,\eta _2,\rho )\) consists of all measurable functions \(W\,:\,\mathbb {R}^{3 \times 3}\rightarrow [0,+\infty ]\) that satisfy the following properties:

$$\begin{aligned}&\text {frame indifference}\\&W(RF)=W(F)\quad \text {for all}\quad F\in \mathbb {M}^3, R\in {\text {SO}}(3); \end{aligned}$$
(W1)
$$\begin{aligned}&\text {non-degeneracy}\\&W(F)\ge \eta _1 {\text {dist}}^2(F,{\text {SO}}(3))\quad \text {for all}\quad F\in \mathbb {M}^3;\\&W(F)\le \eta _2{\text {dist}}^2(F,{\text {SO}}(3))\quad \text {for all}\quad F\in \mathbb {M}^3 \text { with } {\text {dist}}^2(F,{\text {SO}}(3))\le \rho ; \end{aligned}$$
(W2)
$$\begin{aligned}&W\text { is minimal at }I\\&W(I)=0; \end{aligned}$$
(W3)
$$\begin{aligned}&W\text { admits a quadratic expansion at }I\\&W(I+G)=Q(G)+o(|G|^2),\quad \text {as }G \rightarrow 0, \ G \in \mathbb {M}^3\\&\text {where }Q\,:\,\mathbb {M}^3\rightarrow \mathbb {R}\text { is a quadratic form.} \end{aligned}$$
(W4)

In the following definition, we state our assumptions on the family \((W^h)\).

Definition 1.2

(Admissible composite material) Let \(\eta _1,\eta _2\) and \(\rho \) be positive constants such that \(\eta _1\le \eta _2\). We say that a family \((W^h)\)

$$\begin{aligned} W^h:\Omega \times \mathbb {M}^{3} \rightarrow [0,+\infty ] \end{aligned}$$

describes an admissible composite material of class \(\mathcal W(\eta _1,\eta _2,\rho )\) if

  1. (i)

    for each \(h>0\), \(W^h\) is almost everywhere equal to a Borel function on \(\Omega \times \mathbb {M}^3\),

  2. (ii)

    \(W^h(x,\cdot )\in \mathcal W(\eta _1,\eta _2,\rho )\) for every \(h>0\) and almost every \(x\in \Omega \),

  3. (iii)

    there exists a monotone function \(r:\mathbb {R}^+\rightarrow (0,+\infty ]\), such that \(r(\delta )\rightarrow 0\) as \(\delta \rightarrow 0\) and

    $$\begin{aligned} \forall G\in \mathbb {M}^3\, ,\forall h>0:\, \mathop {{\text {ess sup}}}_{x\in \Omega }|W^h(x,I+G)-Q^h(x,G)|\le r(|G|) |G|^2, \end{aligned}$$
    (3)

    where \(Q^h(x,\cdot )\) are quadratic forms defined in (W4).

Notice that \(Q^h\) inherits the measurability properties of \(W^h\), since for each \(h>0\), it can be written as the pointwise limit

$$\begin{aligned} (x,G) \rightarrow Q^h(x,G) := \lim _{\varepsilon \rightarrow 0} \tfrac{1}{\varepsilon ^2} W^h(x, Id+\varepsilon G). \end{aligned}$$
(4)

Lemma 1.3

Let \((W^h)\) be as in Definition 1.2 and let \((Q^h)\) be the family of the quadratic forms associated with \((W^h)\) through the expansion (W4). Then for all \(h>0\) and almost all \(x\in \Omega \) the map \(Q^h(x,\cdot )\) is quadratic and satisfies

  1. (Q1)

    \(\displaystyle \eta _1|{\text {sym}}\,G|^2\le Q^h (x,G)=Q^h(x,{\text {sym}}\,G)\le \eta _2|{\text {sym}}\,G|^2\), for all \( G\in \mathbb {M}^3\);

  2. (Q2)

    \( |Q^h(x,G_1)-Q^h(x,G_2)| \le \eta _2 |{\text {sym}}\,G_1-{\text {sym}}\,G_2|\cdot |{\text {sym}}\,G_1+{\text {sym}}\,G_2|,\)    for all    \(G_1,G_2 \in \mathbb {M}^3\).

Proof

The property (Q1) is a direct consequence of (W2), while (Q2) follows from (Q1) and quadraticity. \(\square \)

1.3 The strategy of the proofs

Here we briefly summarize the main steps of the proof of the lower bound and comment on the proof of the upper bound. The approach taken here is analogous to the one used by the second author in [25], where the equations of the von Kármán plate were derived. However, for the proof of the upper bound we found a simpler argument in this case.

Compactness. We use the rigidity estimate which was proved in [19, Proposition 4.1] (see Theorem 2.15 in the “Appendix”). For an arbitrary sequence \((y^{h})\), there is a sequence of rotations \((R^{h}) \subset C^{\infty }([0,L];\mathbb {R}^{3 \times 3})\) which approximate the scaled gradients \(\nabla _h y^h\). We define the approximate strains by

$$\begin{aligned} G^{h}= \frac{(R^{h})^t\nabla _{h}y^{h}-I}{h}. \end{aligned}$$

Identifying the relaxation field. In the proof of the Theorem 2.13, we show that \({\text {sym}}\,G^h\) is equal to

$$\begin{aligned} \underbrace{{\text {sym}}\,\iota (Ad_{\omega })+a e_1 \otimes e_1}_{\mathrm{limiting \ part}}+\underbrace{{\text {sym}}\,\iota \left( ({A}^{h})'d_{\omega } \right) +{\text {sym}}\,\nabla _{h} v^{h}}_{\mathrm{relaxation \ field}}, \end{aligned}$$

on a large set and up to a term which converges to zero in \(L^2\). Here \(A \in L^2([0,L];\mathbb {M}^3_{{\text {skw}}\,})\) is the limiting strain, a is the limiting variable that additionally appears, and \(d_\omega \) is defined in the notation paragraph. The relaxation field consists of sequences \((A^h)\subset L^2([0,L];\mathbb {M}^3_{{\text {skw}}\,})\) and \((v^h) \subset L^2(\Omega ;\mathbb {R}^3)\), which are such that \(A^h \rightarrow 0\) and \(v^h \rightarrow 0\) strongly in \(L^2\) and \(({A}^{h})'\), \((\nabla _h v^h)\) are bounded in \(L^2\). To enable the use of the truncation argument, we replace the above-mentioned relaxation fields with equi-integrable ones (these are well-known results given in [10] and adapted for the case of dimension reduction in [5]).

Equivalence of the relaxation fields. In Sect. 2.1 by using the modified Griso’s decomposition ([12]), we proved that the above relaxation field is essentially equivalent to the relaxation field \({\text {sym}}\,\nabla _h \psi ^h\), where \(\psi ^h\) are such that \(({\text {sym}}\,\nabla _h \psi ^h)\) is bounded in \(L^2\) and \((\psi ^h_1,h\psi ^h_2,h\psi ^h_3) \rightarrow 0\) and \(\int _{\omega } x_3 \psi ^{h}_2 \rightarrow 0\) strongly in \(L^2\) (or \(\int _{\omega } x_2 \psi ^{h}_3 \rightarrow 0\) strongly in \(L^2\)). Notice also that the relaxation field \({\text {sym}}\,\nabla _h \psi ^h\) is equivalent to the relaxation field \(\nabla _h \psi ^h\), due to the property (Q1) given above.

Construction of the limit energy density function. Section 2.2 is devoted to the definition of the limit energy density function. This construction is similar to the one in [25]. The decomposition of the limiting strain and the relaxation field naturally imposes the definition of the functional

$$\begin{aligned} K(Ad_{\omega }+ae_1,B(\bar{x}_1,r)\times \omega )= & {} \inf \Big \{\liminf _{h \rightarrow 0} \int _{B(\bar{x}_1,r)\times \omega } Q^{h}\left( x,\iota (Ad_{\omega }+ae_1)+\nabla _{h} \psi ^{h}\right) \,\mathrm {d}x: \\&\qquad (\psi _1^{h},h\psi _2^{h},h\psi _3^{h}) \rightarrow 0\, \mathrm{strongly\, in }\, L^2(B(\bar{x}_1,r) \times \omega ;\mathbb {R}^3), \\&\qquad \int _{\omega } x_3 \psi ^{h}_2 \rightarrow 0\, \mathrm{strongly\, in }\, L^2(B(\bar{x}_1,r)) \Big \}. \end{aligned}$$

Then we derive the integral representation of K through the quadratic density

$$\begin{aligned} Q(\bar{x}_1,A,a) =\lim _{r\rightarrow 0}\frac{1}{2r}K(Ad_{\omega }+ae_1,B(\bar{x}_1,r)\times \omega ). \end{aligned}$$

The natural candidate for the limit energy density is the function \(Q_0\)

$$\begin{aligned} Q_0(\bar{x}_1,A) =\min _{a \in \mathbb {R}} Q_0(\bar{x}_1,A,a). \end{aligned}$$

We finish the proof by invoking the truncation argument to establish the lower bound.

Construction of the recovery sequence. To prove the upper bound, we start from the representation formula for K given in Lemma 2.8. As in the proof of the lower bound, we use the equivalence of the relaxation fields to construct the sequences \((A^h)\) and \((v^h)\) appearing in the decomposition of the limiting strains. It is important to notice that for the proof of the upper bound we do not use any kind of additional regularity or higher integrability of the minimizing sequence (this would be non-trivial, since we have scaled gradients), but only their equi-integrability.

2 Derivation of the model

2.1 Characterization of relaxation field

As announced in the introduction, here we establish the equivalence of the relaxation field appearing in the approximating strain. First we recall a result by G.Griso, which is a special case of [12, Theorem 2.1] (given for general curved rod) obtained by applying it to the straight rod and by rescaling argument.

Lemma 2.1

Let \(\Omega ,\omega , L\) be as above and \(1<q<+\infty \). There exists a constant \(C(\omega )\) and a number \(h_0>0\) such that the following holds: For an arbitrary \(0<h<Lh_0\) and any given \(u \in W^{1,q}(\Omega ;\mathbb {R}^3)\), there are functions \(U_e\) and \(\bar{u}\) such that \(u=U_e+\bar{u}\), where \(\bar{u} \in W^{1,q}(\Omega ;\mathbb {R}^3)\) and \(U_e\) is an elementary displacement, i.e., there exist \(\mathcal {U} \in W^{1,q}([0,L];\mathbb {R}^3)\) and \(\mathcal {R} \in W^{1,q}([0,L];\mathbb {R}^3)\) such that

$$\begin{aligned} U_e(x) =\mathcal {U}(x_1) + \mathcal {R}(x_1) \times (h x_2e_2 + h x_3e_3), \end{aligned}$$

and the following estimates hold

$$\begin{aligned} \left\| \bar{u}\right\| _{L^q(\Omega ;\mathbb {R}^3)}&\le C(\omega ) h \left\| {\text {sym}}\,\nabla _h u\right\| _{L^q(\Omega ;\mathbb {R}^{3 \times 3})},\end{aligned}$$
(5)
$$\begin{aligned} \left\| \nabla _h \bar{ u}\right\| _{L^q(\Omega ;\mathbb {R}^{3 \times 3})}&\le C(\omega ) \left\| {\text {sym}}\,\nabla _h u\right\| _{L^q(\Omega ;\mathbb {R}^{3 \times 3})}, \end{aligned}$$
(6)
$$\begin{aligned} h\left\| \mathcal {R}' \right\| _{L^q([0,L];\mathbb {R}^3)}+\left\| \mathcal {U}_1' \right\| _{L^q([0,L])}&+\left\| \mathcal {U}_2' -\mathcal {R}_3 \right\| _{L^q([0,L])} \nonumber \\ +\left\| \mathcal {U}_3' +\mathcal {R}_2 \right\| _{L^q([0,L])}&\le C(\omega ) \left\| {\text {sym}}\,\nabla _h u\right\| _{L^q(\Omega ;\mathbb {R}^{3 \times 3})}. \end{aligned}$$
(7)

We slightly alter this claim.

Lemma 2.2

Let \(1<q<+\infty \). There exist a constant \(C(\omega )>0\) and a number \(h_0>0\) such that the following is true: For an arbitrary \(0<h<Lh_0\) and any given \(u \in W^{1,q}(\Omega ;\mathbb {R}^3)\), there are \(a \in \mathbb {R}^3\), \(B \in \mathbb {M}^3_{\mathrm{skw}}\), \(\varphi _\alpha \in W^{2,q}([0,L])\) for \(\alpha =1,2\), \(w \in W^{1,q}([0,L])\) and \(z \in W^{1,q}(\Omega ;\mathbb {R}^3)\) such that

$$\begin{aligned} u(x)=a+B(x_1,hx_2,hx_3)^t+\left( \begin{array}{c}-(\varphi _{1})'(x_1)x_2-(\varphi _{2})'(x_1)x_3+z_{1}(x)\\ \tfrac{1}{h} \varphi _{1}(x_1) +w(x_1) x_3+z_{2}(x)\\ \tfrac{1}{h} \varphi _{2}(x_1)-w(x_1)x_2+z_{3}(x) \end{array} \right) , \end{aligned}$$
(8)

and the following inequalities hold

$$\begin{aligned}&\Vert \varphi _{1}\Vert _{W^{2,q}([0,L])}+\Vert \varphi _{2}\Vert _{W^{2,q}([0,L])}+\Vert w\Vert _{W^{1,q}([0,L])} \le C(\omega )\Vert {\text {sym}}\,\nabla _h u\Vert _{L^q(\Omega ;\mathbb {R}^3)}, \end{aligned}$$
(9)
$$\begin{aligned}&\Vert z\Vert _{L^q(\Omega ;\mathbb {R}^3)} + \Vert \nabla _h z\Vert _{L^q(\Omega ;\mathbb {R}^3)} \le C(\omega ) \Vert {\text {sym}}\,\nabla _h u\Vert _{L^q(\Omega ;\mathbb {R}^3)}. \end{aligned}$$
(10)

Proof

We define the functions:

$$\begin{aligned} a&= \left( \mathcal {U}_1(0), \mathcal {U}_2(0),\mathcal {U}_3(0)\right) ,\\ B&=\left( \begin{array}{c@{\quad }c@{\quad }c} 0 &{} -\mathcal {R}_3(0) &{} \mathcal {R}_2(0) \\ \mathcal {R}_3(0) &{} 0 &{} - \mathcal {R}_1(0) \\ -\mathcal {R}_2(0) &{} \mathcal {R}_1(0) &{} 0 \end{array} \right) , \\ \varphi _1(x_1)&=h\left( \int _0^{x_1} \mathcal {R}_3(t) dt - x_1 \mathcal {R}_3(0)\right) ,\\ \varphi _2(x_1)&=h\left( - \int _0^{x_1} \mathcal {R}_2(t) dt + x_1 \mathcal {R}_2(0)\right) ,\\ w(x_1)&=-h ({\mathcal {R}}_1(x_1) -\mathcal {R}_1(0)), \\ z_1(x)&= \mathcal {U}_1(x_1) -\mathcal {U}_1(0) + \bar{u}_1(x),\\ z_2(x)&= \mathcal {U}_2 (x_1) - \mathcal {U}_2(0)- \int _0^{x_1} \mathcal {R}_3(t) dt +\bar{u}_2(x),\\ z_3(x)&= \mathcal {U}_3(x_1) -\mathcal {U}_3(0)+ \int _0^{x_1} \mathcal {R}_2(t) dt +\bar{u}_3(x). \end{aligned}$$

It is now straightforward to check that (8) holds. Taking into account that \(\varphi _{\alpha }(0)=(\varphi _{\alpha })'(0)=w(0)=0\) and by using the Poincaré inequality and (6), we obtain that

$$\begin{aligned} \Vert \varphi _{1}\Vert _{W^{2,q}}+\Vert \varphi _{2}\Vert _{W^{2,q}}+\Vert w\Vert _{W^{1,q}} \le C_P h \Vert \mathcal {R}'\Vert _{L^q} \le C(\omega )\Vert {\text {sym}}\,\nabla _h u\Vert _{L^q}. \end{aligned}$$

By using the same arguments and (5 and 6), we also derive the second estimate

$$\begin{aligned} \Vert z\Vert _{L^q} +\Vert \nabla _h z \Vert _{L^q}\le & {} C_P\left( \left\| \mathcal {U}_1'\right\| _{L^q}+ \left\| \mathcal {U}_2' -\mathcal {R}_3 \right\| _{L^q}+\left\| \mathcal {U}_3' +\mathcal {R}_2 \right\| _{L^q}\right) \\&+\Vert \bar{u}\Vert _{L^{q}}+\Vert \nabla _h \bar{u}\Vert _{L^{q}} +h \Vert \mathcal {R}'\Vert _{L^q} \le C(\omega ) \left\| {\text {sym}}\,\nabla _h u\right\| _{L^q} .\end{aligned}$$

\(\square \)

The following corollary gives us the full characterization of sequences with bounded symmetrized gradients.

Corollary 2.3

Let \(1<q<+\infty \), \(C(\omega )\) and \(h_0\) the constants from Lemma 2.2 and let the sequence \((u^h) \subset W^{1,q}(\Omega ,\mathbb {R}^3)\) (for \(0<h<Lh_0\)) be such that \((\Vert {\text {sym}}\,\nabla _h u^h\Vert _{L^q})\) is bounded, \((u^h_1,hu^h_{2},hu^h_{3})\) converges to zero strongly in \(L^q\), and \(\int _{\omega } x_3 u^h_2\) (or \(\int _{\omega } x_2 u^h_3\)) converges to zero strongly in \(L^q\). Take the sequences \((a^h) \subset \mathbb {R}^3\), \((B^h) \subset \mathbb {M}^3_{{\text {skw}}\,}\), \((z^h) \subset W^{1,q}(\Omega ;\mathbb {R}^3)\), \((\varphi ^h_{\alpha }) \subset W^{2,q}([0,L];\mathbb {R}^3)\), for \(\alpha =1,2\), and \((w^h) \subset W^{1,q}([0,L];\mathbb {R}^3)\) from Lemma 2.2. Then:

  1. a.

    \((a^h_1,ha^h_2,ha^h_3) \rightarrow 0\), \(hB^h \rightarrow 0\), \(z^h_{1} \rightarrow 0\) strongly in \(L^q\), \(w^h \rightarrow 0\) strongly in \(L^q\) and for \(\alpha =1,2\), \(\varphi ^h_{\alpha } \rightarrow 0\) strongly in \(W^{1,q}\) as \(h \rightarrow 0\).

  2. b.

    For the following decomposition of \(z^h\), \(z^h=\overline{z}^h+\tilde{z}^h\), where \(\overline{z}^h=\int _{\omega } z^h\), we have that \(\bar{z}^h_1 \rightarrow 0\) strongly in \(L^q\) and \(\Vert \tilde{z}^h\Vert _{L^{q}} \le C(\omega )h\Vert {\text {sym}}\,\nabla _h u^h\Vert _{L^q}\), for some \(C(\omega )>0\).

  3. c.

    There are sequences \((A^h) \subset W^{1,q}([0,L];\mathbb {M}^3_{{\text {skw}}\,})\) and \((v^h) \subset W^{1,q}(\Omega ;\mathbb {R}^3)\), such that \(A^h \rightarrow 0\) and \(v^h \rightarrow 0\) strongly in \(L^{q}\) and the following decomposition holds

    $$\begin{aligned} {\text {sym}}\,\nabla _h u^h={\text {sym}}\,\iota ((A^h)' d_{\omega })+{\text {sym}}\,\nabla _h z^h={\text {sym}}\,\iota ((A^h)' d_{\omega })+{\text {sym}}\,\nabla _h v^h+O(h), \end{aligned}$$

    where \(\Vert O(h)\Vert _{L^q} <Ch\), for some \(C>0\). Moreover, we have that

    $$\begin{aligned} \Vert A^h\Vert _{W^{1,q}}+\Vert v^h \Vert _{L^{q}}+\Vert \nabla _h v^h \Vert _{L^{q}}\le C(\omega ) \Vert {\text {sym}}\,\nabla _h u^h \Vert _{L^q}. \end{aligned}$$
    (11)

Proof

Since \(\int _{\omega }\tilde{z}^h=0\) we conclude from the Poincaré inequality that

$$\begin{aligned} \Vert \tilde{z}^h\Vert _{L^q} \le C(\omega ) h\Vert \nabla _h z^h\Vert _{L^q} \le C(\omega )h \Vert {\text {sym}}\,\nabla _h u^h\Vert _{L^q}. \end{aligned}$$

Thus, \(\tilde{z}^h \rightarrow 0\) strongly in \(L^q\). After redefining \(a^h\) and \(B^h\), we can assume that

$$\begin{aligned} \int _{\Omega } z^h=\int _0^L \bar{z}^h=\int _0^L w^h=\int _0^L \varphi ^h_{\alpha }=\int _0^L x_1\varphi ^h_{\alpha }=0,\, \mathrm{for }\, \alpha =1,2. \end{aligned}$$
(12)

Integrating the first equation in (8) over \(\omega \) and taking into account the choice of coordinate axes (1), we conclude that \(a^h_1 |\omega |+\bar{z}^h_{1} \rightarrow 0\) strongly in \(L^q(\omega )\). From this, by integration over [0, L], we obtain that \(a^h_1 \rightarrow 0\) and, consequently, \(\bar{z}^h_{1} \rightarrow 0\) strongly in \(L^q(\Omega )\). By taking into account (9) and (10), we obtain that \(hB^h_{12}\) and \(h B^h_{13}\) are bounded sequences.

We multiply the second and third equations of (8) by \(h(x_1-\tfrac{L}{2})\), integrate over \(\Omega \) and take the limit as \(h\rightarrow 0\) to obtain that \(hB^h_{12} \rightarrow 0\) and \(h B^h_{13} \rightarrow 0\). Again, multiplying the second and third equations of (8) with h and then integrating over \(\Omega \) and taking the limit as \(h \rightarrow 0\), we deduce that \(ha^h_2 \rightarrow 0\) and \(ha^h_3 \rightarrow 0\). We also obtain that \(\varphi ^h_{\alpha } \rightarrow 0\) strongly in \(W^{1,q}\), since it is bounded in \(W^{2,q}\).

We multiply the second equation in (8) by \(x_3\) and integrate over \(\omega \). Using the decomposition of \(z^h\), we conclude that \(hB^h_{23}+w^h \rightarrow 0\) strongly in \(L^q\). From this, using (12), it follows that \(hB^h_{23} \rightarrow 0\) and \(w^h \rightarrow 0\) strongly in \(L^q\). This finishes the proof of (a) and (b).

To prove (c) we take a sequence \(p^h = (p^h_2,p^h_3) \subset C^\infty ((0,L);\mathbb {R}^2)\) such that

$$\begin{aligned} \Vert p^h-(\bar{z}^h_2,\bar{z}^h_3)\Vert _{L^q} \rightarrow 0,\quad \Vert p^h\Vert _{W^{1,q}} \le C\Vert (\bar{z}^h_2,\bar{z}^h_3)\Vert _{W^{1,q}},\quad h\Vert p^h\Vert _{W^{2,q}} \rightarrow 0, \end{aligned}$$

for some \(C>0\). The sequence \(p^h\) can be constructed by mollification of \((\bar{z}^h_2,\bar{z}^h_3)\) such that the mollifiers are on a scale \(r_h \gg h\). We define

$$\begin{aligned} v^h=z^h-(0,p^h_2,p^h_3)^t+(hx_2 p^h_2+hx_3 p^h_3,0,0)^t,\quad O(h)=(-hx_2 (p^h_2)'-hx_3 (p^h_3)') e_1 \otimes e_1, \end{aligned}$$

and conclude the proof. \(\square \)

It is easy to prove the other implication.

Lemma 2.4

Let \(q\ge 1\), \(h>0\) and let \(A \in W^{1,q}(\Omega ;\mathbb {M}^3_{{\text {skw}}\,})\) and \(v \in W^{1,q}(\Omega ;\mathbb {R}^3)\). Then there exists \(u^h \in W^{1,q}(\Omega ;\mathbb {R}^3)\) such that

$$\begin{aligned} {\text {sym}}\,\nabla _h u^h={\text {sym}}\,\left( \iota (A' d_{\omega })\right) + {\text {sym}}\,\nabla _h v. \end{aligned}$$

If, in addition, \(A=0\) and \(v=0\) in the neighborhood of \(\{0,L\} \times \omega \), then \(u^h=0\) in a neighborhood of \(\{0\} \times \omega \) and \(u^h\) is constant in a neighborhood of \(\{L\} \times \omega \). If \((A^{h}) \subset W^{1,q}([0,L];\mathbb {M}^3_{{\text {skw}}\,})\) and \((v^h) \subset W^{1,q}(\Omega ;\mathbb {R}^3)\) are such that \(A^h \rightarrow 0\) and \(v^h \rightarrow 0\) strongly in \(L^q\), then \((u_1^h, hu_2^h,hu_3^h) \rightarrow 0\) and \(\int _{\omega } x_3 u^h_2 \rightarrow 0\) and \(\int _{\omega } x_2 u^h_3 \rightarrow 0\) strongly in \(L^q\).

Proof

The proof is easily obtained by defining

$$\begin{aligned} u^h= & {} \Big (A_{12}(x_1) x_2+A_{13}(x_1) x_3 , \tfrac{1}{h} \int _0^{x_1} A_{21}(t) \,\mathrm {d}t+A_{23}(x_1)x_3 , \tfrac{1}{h}\\&\quad \int _0^{x_1} A_{31}(t) \,\mathrm {d}t+A_{32}(x_1)x_2 \Big )^t+v. \end{aligned}$$

\(\square \)

2.2 Definition of limit energy density

The goal here is to derive the integral representation of the limit functional. The approach and techniques here are analogous to the one used in [25] to derive von Kármán equations. Hence, we only state the necessary results and refer the reader to [25] for details.

One of the main points is to establish the claim in Lemma 2.6, which tells us that Assumption 2.7 is satisfied on a subsequence.

For any open set \(O \subset [0,L]\), function m in \(L^2(\Omega ;\mathbb {R}^3)\) and sequence \((h_n)\) monotonically decreasing to zero, we define

$$\begin{aligned}&K^{-}_{(h_n)}(m,O)=\inf \Big \{\liminf _{n \rightarrow \infty } \int _{O \times \omega } Q^{h_n}\left( x,\iota (m)+\nabla _{h_n} \psi ^{h_n}\right) \,\mathrm {d}x: \\&(\psi _1^{h_n},h_n\psi _2^{h_n},h_n\psi _3^{h_n}) \rightarrow 0\, \mathrm{strongly\, in }\, L^2(O \times \omega ;\mathbb {R}^3), \int _{\omega } x_3 \psi ^{h_n}_2 \rightarrow 0 \, \mathrm{strongly\, in }\, L^2(O) \Big \},\\&K^{+}_{(h_n)}(m,O)=\inf \Big \{\limsup _{n \rightarrow \infty } \int _{O \times \omega } Q^{h_n}\left( x,\iota (m)+\nabla _{h_n} \psi ^{h_n}\right) \,\mathrm {d}x:\\&(\psi _1^{h_n},h_n\psi _2^{h_n},h_n \psi _3^{h_n}) \rightarrow 0 \, \mathrm{strongly\, in }\, L^2(O \times \omega ;\mathbb {R}^3), \ \int _{\omega } x_3 \psi ^{h_n}_2 \rightarrow 0 \mathrm{strongly in } L^2(O) \Big \}. \end{aligned}$$

Remark 1

By using standard diagonalization argument, it can be shown that for any \((h_n)\) monotonically decreasing to 0 the infima are attained.

The following lemma, together with Lemma 2.8, is a key to establish the properties of the functional K.

Lemma 2.5

(continuity in m) There exists a constant \(C>0\) dependent only on \(\eta _1\) and \(\eta _2\) such that for every sequence \((h_n)\) monotonically decreasing to 0 and \(A \subset [0,L]\) open set the following inequality holds

$$\begin{aligned} \left| K^{-}_{(h_n)}(m_1,A)-K^{-}_{(h_n)}(m_2,A)\right|\le & {} C \Vert m_1-m_2\Vert _{L^2}\left( \Vert m_1\Vert _{L^2}+\Vert m_2\Vert _{L^2}\right) ,\\\nonumber&\ \forall m_1,m_2 \in L^2(\Omega ,\mathbb {R}^3), \end{aligned}$$
(13)

The analogous claim holds for \(K^{+}_{(h_n)}\).

Proof

The proof is identical as the one in [25, Lemma 3.4]. The only difference is that not only \((\psi _1^{h_n},h_n\psi _2^{h_n},\) \(h_n \psi _3^{h_n}) \rightarrow 0\), but also \(\int _{\omega } x_3 \psi ^{h_n}_2 \rightarrow 0\), but this condition can be handled in an analogous way. \(\square \)

If A and B are subsets of [0, L], we denote by \(A\ll B\) if \(\bar{A}\) is compact and contained in B. We recall the property of density of family of sets (see [16]). We say that a family \(\mathcal {D}\) of subsets is dense in a family \(\mathcal {A}\), if for every \(A,B \in \mathcal {A}\), with \(A \ll B\), there exists \(D \in \mathcal {D}\), such that \(A\ll D \ll B\).

Let \(\mathcal {D}\) denote a countable family of open subsets of [0, L] which is dense in the class \(\mathcal {A}\) of all open subsets of [0, L] and such that every \(D \in \mathcal {D}\) is a finite union of open intervals which are subsets of [0, L].

By using Lemma 2.5 and diagonal procedure, we can easily argument the following claim (see [25, Lemma 2.6]).

Lemma 2.6

For every sequence \((h_n)\) monotonically decreasing to zero, there exists a subsequence, still denoted by \((h_n)\), such that

$$\begin{aligned} K^+_{(h_n)}(m, D)=K^{-}_{(h_n)} (m,D),\quad \forall m \in L^2(\Omega ,\mathbb {R}^{3}), \ \forall D \in \mathcal {D}. \end{aligned}$$

We will now make an assumption on the sequence \((h_n)\) and family \((Q^{h_n})\).

Assumption 2.7

For a given sequence \((h_n)\) monotonically decreasing to zero, we suppose that

$$\begin{aligned} K^{+}_{(h_n)}(m,D)=K^{-}_{(h_n)}(m,D)=:K(m,D),\quad \forall m \in L^2(\Omega ,\mathbb {R}^{3}), \ \forall D \in \mathcal {D}. \end{aligned}$$

Although the numbers K(m, D) also depend on the sequence, we will not write it, since it will be clear from the context on which sequence we are referring to.

Remark 2

As in [25, Lemma 3.8] we can see that if a sequence \((h_n)\) satisfies the Assumption 2.7 then we have that

Lemma 2.8

Let \((h_n)\) be a sequence monotonically decreasing to 0 which satisfies Assumption 2.7. For \(m \in L^2(\Omega ,\mathbb {R}^3)\) and \(O \subset \omega \) open, there exist a subsequence \((h_{n(k)})\) and \((\vartheta _k) \subset W^{1,2}(O \times \omega ,\mathbb {R}^3)\) such that

  1. (a)

    \(((\vartheta _{k})_1,h_{n(k)} (\vartheta _{k})_2,h_{n(k)} (\vartheta _{k})_3) \rightarrow 0\), \(\int _{\omega } x_3 (\vartheta _{k})_2 \rightarrow 0\) strongly in \(L^2\),

  2. (b)

    \((|{\text {sym}}\,\nabla _{h_{n(k)}} \vartheta _k|^2 )\) is equi-integrable and there are sequences \((A_k)\) \( \subset \) \( W^{1,2}([0,L];\mathbb {M}^3_{{\text {skw}}\,})\), \(A_k \rightarrow 0\) strongly in \(L^2\) and \((v_k)\subset W^{1,2}(\Omega ;\mathbb {R}^3)\), \(v_k \rightarrow 0\) strongly in \(L^{2}\) such that

    $$\begin{aligned} {\text {sym}}\,\nabla _{h_{n(k)}} \vartheta _k={\text {sym}}\,\iota ((A_k)' d_{\omega })+{\text {sym}}\,\nabla _{h_{n(k)}} v_{k}. \end{aligned}$$

    Moreover, we have that \(\left( |(A_k)'|^2\right) _{k\in \mathbb {N}}\) and \(\left( | \nabla _{h_{n(k)}} v_{k}|^2\right) \) are equi-integrable. Also the following is valid

    $$\begin{aligned} \limsup _{k \rightarrow \infty } \left( \Vert A_k\Vert _{W^{1,2}(O)}+\Vert \nabla _{h_{n(k)}}v_k \Vert _{L^2(O \times \omega )}\right) \le C \left( \eta _2 \Vert m\Vert ^2_{L^2}+1 \right) , \end{aligned}$$

    where C is independent of the domain O. For each \(k \in \mathbb {N}\) we have that \(A_k=0\) in a neighborhood of \(\partial O\) and \(v_k=0\) in a neighborhood of \(\partial O \times \omega \).

  3. (c)

    \(\displaystyle K(m,O)=\lim _{k\rightarrow \infty } \int _{O \times \omega } Q^{h_{n(k)}}(x,\iota (m)+\nabla _{h_{n(k)}} \vartheta _k) \,\mathrm {d}x.\)

Proof

The proof is analogous to the one in [25, Lemma 3.10]. Therefore, we just state the main arguments.

Firstly, it is easy to establish the claim of lemma if the set \(O\subset [0,L]\) is nice, e.g., a finite union of intervals. Then we use Corollary 2.3 to write the relaxation field in the form given there. Next, by using Lemma 2.16 and Lemma 2.17, we replace this relaxation field with the equi-integrabile one (this is possible since the relaxation field is a minimizing sequence). The truncation argument on the equi-integrabile sequence can easily be made to obtain the relaxation field which is zero near the boundary.

Then this claim, together with Lemma 2.5, is used to establish some properties of the functional K ([25, Lemma 3.7]). Using the localization property, we obtain the claim of the lemma for an arbitrary \(O\subset [0,L]\) open. \(\square \)

The following lemma gives us the important claim that if we know the relaxation sequence for [0, L], we obtain by restriction the relaxation sequence for arbitrary \(O\subset [0,L]\) open (if m is fixed). Moreover, it gives a kind of uniqueness of relaxation field, up to a term converging to zero in \(L^2\).

Lemma 2.9

Take a sequence \((h_n)\) monotonically decreasing to 0 that satisfies Assumption 2.7 and \(m \in L^2(\Omega ;\mathbb {R}^{3})\). Let \((\vartheta _n) \subset W^{1,2}(\Omega ,\mathbb {R}^3)\) be such that

  1. (a)

    \(((\vartheta _{n})_1,h_{n} (\vartheta _{n})_2,h_{n} (\vartheta _{n})_3) \rightarrow 0\), \(\int _{\omega } x_3 \vartheta _{n,2} \rightarrow 0\) strongly in \(L^2\);

  2. (b)

    \(\displaystyle K(m,[0,L])=\lim _{n\rightarrow \infty } \int _{\Omega } Q^{h_{n}}(x,\iota (m)+\nabla _{h_{n}} \vartheta _n) \,\mathrm {d}x.\)

Then we have that:

  1. (I)

    \((|{\text {sym}}\,\nabla _{h_n} \vartheta _n|^2)\) is equi-integrable;

  2. (II)

    for every O open subset of [0, L], we have that

    $$\begin{aligned} K(m,O)=\lim _{n\rightarrow \infty } \int _{O \times \omega } Q^{h_{n}}(x,\iota (m)+\nabla _{h_{n}} \vartheta _n) \,\mathrm {d}x; \end{aligned}$$
    (14)
  3. (III)

    if \((\psi _n) \subset W^{1,2}(\Omega ;\mathbb {R}^3)\) is any other sequence that satisfies (a) and (b), then

    $$\begin{aligned} \Vert {\text {sym}}\,\nabla _{h_n} \psi _n - {\text {sym}}\,\nabla _{h_n} \vartheta _n\Vert _{L^2} \rightarrow 0, \end{aligned}$$

    and \((|{\text {sym}}\,\nabla _{h_n} \psi _n|^2)\) is equi-integrable.

Proof

From (Q1) and by taking the zero subsequence, we obtain the bound

$$\begin{aligned} \limsup _{n \rightarrow \infty } \Vert {\text {sym}}\,\nabla _{h_{n}}\vartheta _n \Vert _{L^2(\Omega )}\le C \left( \eta _2 \Vert m\Vert ^2_{L^2}+1 \right) . \end{aligned}$$
(15)

From Corollary 2.3 there are sequences \((A_n) \subset W^{1,2}((0,L);\mathbb {M}^3_{{\text {skw}}\,})\) and \((v_n) \subset W^{1,2}(\Omega ;\mathbb {R}^3)\) such that \(A_n \rightarrow 0\) and \(v_n \rightarrow 0\) strongly in \(L^{2}\) and

$$\begin{aligned} \left\| {\text {sym}}\,\nabla _{h_{n}} \vartheta _n-{\text {sym}}\,\iota ((A_n)' d_{\omega })-{\text {sym}}\,\nabla _{h_{n}} v_{n}\right\| _{L^2} \rightarrow 0. \end{aligned}$$

From (11) we obtain that

$$\begin{aligned} \limsup _{k \rightarrow \infty } \left( \Vert A_n\Vert _{W^{1,2}(\Omega )}+\Vert \nabla _{h_{n}}v_n \Vert _{L^2(\Omega )}\right) \le C(\omega ) \left( \eta _2 \Vert m\Vert ^2_{L^2}+1 \right) . \end{aligned}$$

To prove that \((|{\text {sym}}\,\nabla _{h_{n}} \vartheta _n|^2 )\) is equi-integrable, let us assume the opposite, i.e., that there is \(\varepsilon >0\) such that for every \(k>0\) there are measurable sets \((S_k)\) such that \(\vert S_k\vert < \frac{1}{k} \) and there is an increasing function \(n:\mathbb {N}\rightarrow \mathbb {N}\) such that

$$\begin{aligned} \int _{S_k} | {\text {sym}}\,\nabla _{h_{n(k)}} \vartheta _{n(k)}|^2 \,\mathrm {d}x \ge \varepsilon . \end{aligned}$$

On the other hand, by Lemma 2.16 and 2.17 there is a further subsequence, still denoted by n(k) and sequences \((\tilde{A}_{k}) \subset W^{1,2}((0,L);\mathbb {M}^3_{{\text {skw}}\,})\) and \((\tilde{v}_{k}) \subset W^{1,2}(\Omega ;\mathbb {R}^3)\) such that

  1. (i)

    \(\displaystyle \lim _{k \rightarrow \infty } \left| \Omega \cap \left\{ \tilde{A}_{k} \ne A_{n(k)} \text{ or } \tilde{A}_{k}' \ne A_{n(k)}' \right\} \right| =0;\)

  2. (ii)

    \( \lim _{k \rightarrow \infty } \left| \Omega \cap \left\{ \tilde{v}_{k} \ne v_{n(k)} \text{ or } \nabla \tilde{v}_{k} \ne \nabla v_{n(k)} \right\} \right| =0;\)

  3. (iii)

    \(\tilde{A}_{k}'\) and \(\nabla _{h_{n(k)}}\tilde{v}_{k}\) are equi-integrabile.

We have

$$\begin{aligned} K(m,[0,L])&=\liminf _{k\rightarrow \infty } \int _{\Omega } Q^{h_{n(k)}}\left( x,\iota (m) + \nabla _{h_{n(k)}}\vartheta _{n(k)}\right) \,\mathrm {d}x \\&> \liminf _{k\rightarrow \infty } \int _{\Omega } \chi _{\Omega \setminus S_k} Q^{h_{n(k)}}\left( x,\iota (m) + \nabla _{h_{n(k)}}\vartheta _{n_k}\right) \,\mathrm {d}x \\&= \liminf _{k\rightarrow \infty } \int _{\Omega } \chi _{\Omega \setminus S_{k}} Q^{h_{n(k)}}\left( x,\iota (m) + {\text {sym}}\,\iota ((\tilde{A}_k)'d_{\omega }) + {\text {sym}}\,{\nabla _{h_{n(k)}} \tilde{v}_k}\right) \,\mathrm {d}x \\&= \liminf _{k\rightarrow \infty } \int _{\Omega } Q^{h_{n(k)}} \left( x,\iota (m) + {\text {sym}}\,\iota ((\tilde{A}_k)'d_{\omega }) + {\text {sym}}\,{\nabla _{h_{n(k)}} \tilde{v}_k}\right) \,\mathrm {d}x \\&=K(m,[0,L]), \end{aligned}$$

which is a contradiction. Therefore, \((|{\text {sym}}\,\nabla _{h_{n}} \vartheta _n|^2 )\) is equi-integrabile.

We now show that \((\vartheta _n)\) is optimal for any open set \(O \in \mathcal {D}\) which is a finite union of disjoint open intervals. Otherwise, there would be a subsequence, still denoted by \((h_n)\), such that there is a sequence \((\psi _n^1) \subset W^{1,2}(O\times \omega ,\mathbb {R}^3)\) satisfying the conditions of Lemma 2.8 and

$$\begin{aligned} K(m,O)= & {} \lim _{n \rightarrow \infty } \int \limits _{O \times \omega } Q^{h_{n}}\left( x,\iota (m)+\nabla _{h_{n}}\psi _n^1\right) \,\mathrm {d}x \\< & {} \lim _{n \rightarrow \infty } \int \limits _{O\times \omega } Q^{h_{n}}\left( x,\iota (m)+\nabla _{h_{n}}\vartheta _n \right) \,\mathrm {d}x. \end{aligned}$$

On the other hand, on a further subsequence, still denoted by \((h_n)\) we take the sequence \((\psi _n^2) \subset W^{1,2}([0,L]\setminus \bar{O},\mathbb {R}^3)\) satisfying the conditions of Lemma 2.8 and

$$\begin{aligned} K(m,(0,L) \backslash \bar{O} )= & {} \lim _{n \rightarrow \infty } \int \limits _{((0,L)\setminus \bar{O}) \times \omega } Q^{h_n}\left( x,\iota (m)+\nabla _{h_n}\psi _n^2\right) \,\mathrm {d}x\\\le & {} \lim _{n \rightarrow \infty } \int \limits _{([0,L]\setminus \bar{O})\times \omega } Q^{h_n}\left( x,\iota (m)+\nabla _{h_{n}}\vartheta _n\right) \,\mathrm {d}x. \end{aligned}$$

By using Lemma 2.4 we define \((\psi _n) \subset W^{1,2}(\Omega ;\mathbb {R}^3)\) such that

$$\begin{aligned} {\text {sym}}\,\nabla _{h_n} \psi _n=\chi _{O} {\text {sym}}\,\nabla _{h_n} \psi _n^1+\chi _{[0,L] \backslash \bar{O}} {\text {sym}}\,\nabla _{h_n} \psi _n^2. \end{aligned}$$

We conclude that

$$\begin{aligned} \lim _{n \rightarrow \infty }\int \limits _{[0,L]\times \omega } Q^{h_n}\left( x,\iota (m)+\nabla _{h_n}\psi _n\right) \,\mathrm {d}x&< \lim _{n \rightarrow \infty }\int \limits _{[0,L]\times \omega } Q^{h_n}\left( x,\iota (m)+\nabla _{h_n}\vartheta _n\right) \,\mathrm {d}x\\&=K(m,[0,L]), \end{aligned}$$

which yields a contradiction with the optimality of the sequence \((\vartheta _n)\).

For any open \(O\subset [0,L]\), by density, there is an increasing family of sets \((D_k) \subset \mathcal {D}\) which exhausts O. Since \((\vartheta _n)\) is optimal on each \(D_k\) and since \(K(m,O) \ge K(m,D_k)\) (this can be easily seen from Lemma 2.8), we deduce from equi-integrability of \(\left( |{\text {sym}}\,{\nabla _{h_n}\vartheta _{n}}|^2\right) \) that

$$\begin{aligned} K(m,O)\ge \lim _{k \rightarrow \infty } K(m,D_k)= \lim _{n \rightarrow \infty } \int _{O \times \omega } Q^{h_n}\left( x,\iota (m)+\nabla _{h_n}\vartheta _n\right) \,\mathrm {d}x. \end{aligned}$$

Hence \((\vartheta _k)\) is also optimal for K(m, O), and (II) is proved.

Notice that, for every \(h>0\), there is a measurable mapping \(\mathcal {L}^h:\Omega \times \mathbb {M}^3 \rightarrow \mathbb {M}^3_{{\text {sym}}\,}\) such that for a.e. \(x \in \Omega \), \(\mathcal {L}^h(x, \cdot )\) is a positive semidefinite linear operator and

$$\begin{aligned} Q^h (x,M)=\mathcal {L}^h(x, M) \cdot M \end{aligned}$$

holds for all \(M \in \mathbb {M}^3\). Notice also that

$$\begin{aligned} \mathcal {L}^h (x, M)= \mathcal {L}^h (x, {\text {sym}}\,M), \quad \Vert \mathcal {L}^h\Vert _{L^\infty } \le \eta _2. \end{aligned}$$
(16)

To prove (III) we first show that

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{\Omega } \mathcal {L}^h (x, \iota (m)+\nabla _{h_n} \vartheta _n) \cdot \nabla _{h_n} \tilde{\psi }_n =0, \end{aligned}$$
(17)

for every \((\tilde{\psi _n}) \subset W^{1,2}(\Omega ;\mathbb {R}^3)\) that satisfies (a) and such that \((\vert {\text {sym}}\,{\nabla _{h_n} \tilde{\psi }_n}\vert )\) is bounded in \(L^2\).

To prove this we take \(\varepsilon > 0\), and for n large enough we derive:

$$\begin{aligned} 0&\le \int _{\Omega } Q^{h_n} (x,\iota (m)+\nabla _{h_n} \vartheta _n + \varepsilon \nabla _{h_n} \tilde{\psi }_n ) \,\mathrm {d}x - \int _{\Omega } Q^{h_n} (x,\iota (m)+\nabla _{h_n} \vartheta _n) \,\mathrm {d}x\\&= \int _{\Omega } \mathcal {L}^h (x, \iota (m)+\nabla _{h_n} \vartheta _n+ \varepsilon \nabla _{h_n} \tilde{\psi }_n) \cdot (\iota (m)+\nabla _{h_n} \vartheta _n+ \varepsilon \nabla _{h_n} \tilde{\psi }_n) \,\mathrm {d}x \\&\quad - \int _{\Omega } \mathcal {L}^h (x, \iota (m)+\nabla _{h_n} \vartheta _n) \cdot (\iota (m)+\nabla _{h_n} \vartheta _n ) \,\mathrm {d}x \\&= 2\varepsilon \int _{\Omega } \mathcal {L}^h (x, \iota (m)+\nabla _{h_n} \vartheta _n) \cdot ( \nabla _{h_n} \tilde{\psi }_n) \,\mathrm {d}x +\varepsilon ^2 \int _{\Omega } \mathcal {L}^h (x, \nabla _{h_n} \tilde{\psi }_n) \cdot ( \nabla _{h_n} \tilde{\psi }_n) \,\mathrm {d}x \\&\le 2\varepsilon \int _{\Omega } \mathcal {L}^h (x, \iota (m)+\nabla _{h_n} \vartheta _n) \cdot ( \nabla _{h_n} \tilde{\psi }_n) \,\mathrm {d}x +\varepsilon ^2 \eta _2 \vert {\text {sym}}\,{\nabla _{h_n} \tilde{\psi }_n}\vert ^2\\&= 2\varepsilon \int _{\Omega } \mathcal {L}^h (x, \iota (m)+{\text {sym}}\,\nabla _{h_n} \vartheta _n) \cdot ( {\text {sym}}\,\nabla _{h_n} \tilde{\psi }_n) \,\mathrm {d}x +\varepsilon ^2 \eta _2 \vert {\text {sym}}\,{\nabla _{h_n} \tilde{\psi }_n}\vert ^2. \end{aligned}$$

If (17) did not hold, we would choose \(\varepsilon \) (by taking the appropriate sign) such that the linear term dominates and the inequality is violated. Thus, we deduce (17), by the contradiction. To finish the proof, we take two sequences \((\vartheta _n) \subset W^{1,2}(\Omega ;\mathbb {R}^3)\), \((\psi _n) \subset W^{1,2}(\Omega ;\mathbb {R}^3)\) that satisfy (a) and (b). We have, using (17)

$$\begin{aligned} \eta _1\Vert {\text {sym}}\,\nabla _{h_n} (\psi _n- \vartheta _n)\Vert ^2_{L^2}\le & {} \int _{\Omega } \mathcal {L}^{h_n}(x, \nabla _{h_n} (\psi _n- \vartheta _n)) \cdot \nabla _{h_n} (\psi _n- \vartheta _n) \,\mathrm {d}x \\= & {} \int _{\Omega } \mathcal {L}^{h_n}(x,\iota (m)+ \nabla _{h_n} \psi _n) \cdot \nabla _{h_n} (\psi _n-\vartheta _n) \,\mathrm {d}x\\&-\int _{\Omega } \mathcal {L}^{h_n}(x,\iota (m)+ \nabla _{h_n} \vartheta _n) \cdot \nabla _{h_n} (\psi _n-\vartheta _n) \,\mathrm {d}x\rightarrow 0. \end{aligned}$$

\(\square \)

The following lemma proves the compactness result we need.

Lemma 2.10

For every sequence \((h_n)\) that satisfy the Assumption 2.7, there exists a subsequence, still denoted by \((h_n)\) such that for each \(m \in L^2(\Omega ;\mathbb {R}^3)\) there exists \(\left( \vartheta _n^m\right) \subset W^{1,2}(\Omega ;\mathbb {R}^3)\) which satisfies

  1. (a)

    \(\left( (\vartheta _{n}^m)_1,h_{n} (\vartheta _{n}^m)_2,h_{n} (\vartheta _{n}^m)_3\right) \rightarrow 0\), \(\int _{\omega }x_3(\vartheta _{n}^m)_2 \rightarrow 0\) strongly in \(L^2\),

  2. (b)
    $$\begin{aligned} K(m,[0,L])=\lim _{n\rightarrow \infty } \int _{\Omega } Q^{h_{n}}(x,\iota (m)+\nabla _{h_{n}} \vartheta _n^m)\, \,\mathrm {d}x. \end{aligned}$$

Proof

Let \(\mathcal {M} \subset L^2(\Omega ;\mathbb {R}^3)\) be a countable dense family. By diagonalization procedure, it is possible to construct the subsequence, still denoted by \((h_n)\), such that for each \(m \in \mathcal {M}\) there is a sequence \((\vartheta (m)_n)\) for which (a) and (b) holds. Now we take the sequence \((m_{n}) \subset \mathcal {M}\) such that \(m_n \rightarrow m\) in \(L^2\) as \(n \rightarrow \infty \) and define the strictly increasing function \(k:\mathbb {N}\rightarrow \mathbb {N}\) in a way that for every \(n_0 \in \mathbb {N}\) we have

$$\begin{aligned} \left| K(m_{n_0},[0,L]) - \int _{\Omega } Q^{h_n}\left( x,\iota (m_{n_0}) + \nabla _{h_n} \vartheta _{n}^{m_{n_0}} \right) \,\mathrm {d}x \right|< & {} \frac{1}{n_0}, \quad \text{ for } \text{ every } n \ge k(n_0),\\ \left\| \left( (\vartheta _{n}^{m_{n_0}})_1,h_{n} (\vartheta _{n}^{m_{n_0}})_2,h_{n} (\vartheta _{n}^{m_{n_0}})_3\right) \right\| _{L^2}< & {} \frac{1}{n_0}, \quad \text{ for } \text{ every } n \ge k(n_0), \\ \left\| \int _{\omega }x_3(\vartheta _{n}^{m_{n_0}})_2\right\| _{L^2}< & {} \frac{1}{n_0}, \quad \text{ for } \text{ every } n \ge k(n_0). \end{aligned}$$

For every \(i \in \mathbb {N}\) and \(j \in [k(i),k(i+1))\) take \(\vartheta _j^m:= \vartheta _j^{m_{k(i)}}\) and use Lemma 2.5 to show (b). \(\square \)

We make the assumption on the sequence \((Q^{h_n})\) of quadratic forms.

Assumption 2.11

For given \((h_n)\) monotonically decreasing to zero, we suppose that for every \(m \in L^2(\Omega ;\mathbb {R}^3)\) there exists \((\vartheta _n) \subset W^{1,2}(\Omega ;\mathbb {R}^3)\) (we omit the superscript m) such that

  1. (a)

    \(\left( (\vartheta _{n})_1,h_n (\vartheta _{n})_2,h_n (\vartheta _{n})_3\right) \rightarrow 0\), \(\int _{\omega } x_3 (\vartheta _{n})_2 \rightarrow 0\) strongly in \(L^2\),

  2. (b)

    for every \(O \subset [0,L]\) open we have

    $$\begin{aligned} K(m,O)= \lim _{n \rightarrow \infty } \int _{O \times \omega } Q^{h_n}(x,\iota (m)+\nabla _{h_n} \vartheta _n)\,\mathrm {d}x. \end{aligned}$$

    Here

    $$\begin{aligned} K(m,O)= & {} \min \Big \{\liminf _{n \rightarrow \infty } \int _{O \times \omega } Q^{h_n}\left( x,\iota (m)+\nabla _{h_n} \psi ^{h_n}\right) \,\mathrm {d}x: \nonumber \\&\qquad \quad (\psi _1^{h_n},h_n\psi _2^{h_n},h_n\psi _3^{h_n}) \rightarrow 0\, \mathrm{strongly\, in }\, L^2(O \times \omega ;\mathbb {R}^3)\\&\qquad \quad \int _{\omega } x_3 \psi ^{h_n}_2 \rightarrow 0 \, \mathrm{strongly\, in }\, L^2(O) \Big \}\\= & {} \min \Big \{\limsup _{n \rightarrow \infty } \int _{O \times \omega } Q^{h_n}\left( x,\iota (m)+\nabla _{h_n} \psi ^{h_n}\right) \,\mathrm {d}x:\\&\quad \qquad (\psi _1^{h_n},h_n\psi _2^{h_n},h_n \psi _3^{h_n}) \rightarrow 0 \, \mathrm{strongly\, in }\, L^2(O \times \omega ;\mathbb {R}^3), \\&\qquad \quad \int _{\omega } x_3 \psi ^{h_n}_2 \rightarrow 0 \, \mathrm{strongly\, in }\, L^2(O) \Big \}. \end{aligned}$$

We define the mapping \(m: L^2([0,L];\mathbb {M}^3_{{\text {skw}}\,}) \times L^2([0,L]) \rightarrow L^2([0,L];\mathbb {R}^3)\) by \(m(A,a)=A(0,x_2,x_3)^t+a e_1\). Finally, we derive the integral representation of K.

Proposition 2.12

Let \((h_n)\) be a sequence monotonically decreasing to zero for which the Assumption 2.11 is valid. Then there exists a measurable function \(Q:[0,L] \times \mathbb {M}^3_{{\text {skw}}\,}\times \mathbb {R}\rightarrow [0,+\infty \rangle \) possibly depending on this sequence such that for every \(O \subset [0,L]\) open and every \(A \in L^2([0,L];\mathbb {M}^3_{{\text {skw}}\,})\) we have

$$\begin{aligned} K(m(A,a),O)=\int _{O} Q(x_1,A(x_1),a(x_1)) \,\mathrm {d}x_1. \end{aligned}$$
(18)

Moreover, Q satisfies the following property

(Q’1):

for almost all \(x_1\in [0,L]\) the map \(Q(x_1,\cdot ,\cdot )\) is a quadratic form and there is a positive constant \(C_{\omega }\), independent of \(x_1\), such that

$$\begin{aligned}&C_{\omega }(|A|^2+|a|^2) \le Q (x_1,A,a)\nonumber \\&\quad \le \eta _2\left( \max \{\mu _2,\mu _3\}|A|^2+|a|^2\right) \ \text {for all }(A,a) \in \mathbb {M}^3_{{\text {skw}}\,} \times \mathbb {R}. \end{aligned}$$
(19)

Proof

The existence of Q and the proof of (18) is identical as in [25, Proposition 2.9]. Therefore, we will only prove the boundedness and coercivity property. The function Q is defined via:

$$\begin{aligned} Q(\bar{x}_1,A,a) =\lim _{r \rightarrow 0}\frac{1}{2r}K\left( m(A,a),B(\bar{x}_1,r)\right) ,\, \mathrm{for \quad a.e. }\ \bar{x}_1 \in [0,L]. \end{aligned}$$
(20)

The upper bound in (19) is easily obtained by taking the zero subsequence \(\vartheta _n=0\) and by using (Q1) and (1) to deduce

$$\begin{aligned} \vert Q(\bar{x}_1,A,a) \vert \le \eta _2 \left( \vert {\text {sym}}\,\iota (ae_1 + Ad_{\omega }) \vert ^2 \right) \le \eta _2 \left( \max { \left\{ \mu _2,\mu _3\right\} } \vert A \vert ^2 + \vert a \vert ^2 \right) , \end{aligned}$$

for a.e. \(\bar{x}_1 \in (0,L)\).

From Assumption 2.11 and Corollary 2.3 we deduce that there are bounded sequences \((A^{h_n}) \subset W^{1,2}([0,L];\mathbb {M}^3_{{\text {skw}}\,})\) and \((v^{h_n}) \subset W^{1,2}(\Omega ;\mathbb {R}^3)\) such that \(A^{h_n} \rightarrow 0\) and \(v^{h_n} \rightarrow 0\) strongly in \(L^{2}\) and

$$\begin{aligned}&K(m(A,a),B(\bar{x}_1,r))\\&\quad =\lim _{n \rightarrow \infty }\int _{B(\bar{x}_1,r) \times \omega } Q^{{h_n} } \left( x,\iota (m) + {\text {sym}}\,\iota ((A^{h_n} )'d_{\omega }) + {\text {sym}}\,{\nabla _{{h_n}} v^{h_n}}\right) \,\mathrm {d}x. \end{aligned}$$

for some \(C>0\). We can assume, by the density argument, that \(v^{h_n}\) and \(A^{h_n}\) are smooth functions. Using the property (Q1), we have

$$\begin{aligned} K(m(A,a),B(\bar{x}_1,r)) \ge \eta _1 (I_1 + I_2), \end{aligned}$$

where \(I_1\) and \(I_2\) are defined by:

$$\begin{aligned} I_1 =&\lim _{h\rightarrow 0} \int \limits _{B(\bar{x}_1,r)\times \omega } \left( a + A_{12}x_2 + A_{13}x_3 + (A^{h_n}_{12})'x_2 + (A^{h_n}_{13})'x_3 +\partial _1{v_1^{h_n}} \right) ^2 dx \\ I_2 =&\frac{1}{2}\lim _{h\rightarrow 0} \left\{ \int \limits _{B(\bar{x}_1,r)\times \omega } \left( A_{23}x_3 + (A_{23}^{h_n})' x_3 + \partial _1{v_2^{h_n}}+ \frac{\partial _{2}{v_1^{h_n}}}{h} \right) ^2 dx \right. \\&\qquad \qquad \left. + \int \limits _{B(\bar{x}_1,r)\times \omega } \left( - A_{23}x_2 -(A_{23}^{h_n})' x_2 +\partial _1{v_3^{h_n}} + \frac{\partial _{3}{v_1^{h_n}}}{{h_n}} \right) ^2 dx \right\} \end{aligned}$$

From the choice of the coordinate axis (1), we have that for every \(x_1 \in B(\bar{x}_1,r)\)

$$\begin{aligned} \int \limits _{\{x_1\} \times \omega } a A_{12}x_2 \,\mathrm {d}x_2 \,\mathrm {d}x_3 = \int \limits _{\{x_1\} \times \omega } a A_{13}x_3 \,\mathrm {d}x_2 \,\mathrm {d}x_3 = \int \limits _{\{x_1\} \times \omega } A_{13}A_{12}x_2 x_3 \,\mathrm {d}x_2 \,\mathrm {d}x_3 =0. \end{aligned}$$

Thus, we derive that

$$\begin{aligned} I_1 \ge&\int _{B(\bar{x}_1,r)\times \omega }\left( \vert a \vert ^2 + x_2^2 A_{12}^2 + x_3^2 A_{13}^2\right) \,\mathrm {d}x \\&+ 2\lim _{n \rightarrow \infty } \int _{B(\bar{x}_1,r)\times \omega } \left( a + A_{12}x_2 + A_{13}x_3\right) \left( (A^{h_n}_{12})'x_2 + (A^{h_n}_{13})'x_3 +\partial _1{v_1^{h_n}} \right) \,\mathrm {d}x . \end{aligned}$$

Since \((A_{h_n})' \rightharpoonup 0\) and \(\partial _1{v_1}^{h_n} \rightharpoonup 0\) weakly in \(L^2\), the mixed term vanishes as \(n \rightarrow \infty \). Hence, we obtain that

$$\begin{aligned} I_1 \ge 2r \left( \vert a \vert ^2 + \mu _2 \vert A_{12}\vert ^2 +\mu _3 \vert A_{13}\vert ^2 \right) . \end{aligned}$$
(21)

To obtain the lower bound for \(I_2\), we look for a solution of the minimum problem

$$\begin{aligned} \min _{\psi \in H^1(\omega )} \int _{\omega } \vert u - \nabla \psi \vert ^2 \,\mathrm {d}x. \end{aligned}$$

The solution of this problem is unique up to constant and satisfies the variational equation

$$\begin{aligned} \int _{\omega } \left( \nabla \varphi _u - u\right) \cdot \nabla \psi \ \,\mathrm {d}x = 0, \end{aligned}$$
(22)

for every \(\psi \in H^1(\omega ) \). The solution corresponds to \(L^2\) projection on the space

$$\begin{aligned} G(\omega )= \left\{ w \in L^2(\omega ;\mathbb {R}^2): w=\nabla p, \text{ for } \text{ some } p \in H^1(\omega ) \right\} , \end{aligned}$$

which is a closed subspace in \(L^2(\omega ;\mathbb {R}^2)\). We denote by \(Pu = u - \nabla \varphi _u \).

Denote also by

$$\begin{aligned} \Psi ^h(x)= \left( A _{23}+(A_{23}^h)' \right) \left( \begin{array}{c} x_3 \\ -x_2 \end{array} \right) + \left( \begin{array}{c} \partial _1{v_2^{h_n}}\\ \partial _1{v_3^{h_n}} \end{array} \right) + \frac{1}{{h_n}} \left( \begin{array}{c} \partial _2{v_1^{h_n}}\\ \partial _3{v_1^{h_n}} \end{array} \right) . \end{aligned}$$

Since P is a projection, we conclude

$$\begin{aligned} I_2 =\int \limits _{B(\bar{x}_1,r)\times \omega } \vert \Psi ^{h_n} \vert ^2 dx \ge \int \limits _{B(\bar{x}_1,r)\times \omega } \vert P\Psi ^{h_n} \vert ^2 dx, \end{aligned}$$

where \(P \Psi ^h\) equals

$$\begin{aligned} P(\Psi ^{h_n}(x))= \left( A _{23}+(A_{23}^{h_n})' \right) P\left( \begin{array}{c} x_3 \\ -x_2 \end{array} \right) + P \left( \begin{array}{c} \partial _1{v_2^{h_n}}\\ \partial _1{v_3^{h_n}} \end{array} \right) . \end{aligned}$$

Notice that the projection is taken for every \(x_1 \in [0,L]\). This yields that:

$$\begin{aligned} I_2 \ge&\bar{C}_{\omega } \lim _{n \rightarrow \infty } \left( \int _{B(\bar{x}_1,r) } \vert A_{23} \vert ^2 +2 \int _{B(x_1,r)} A_{23} (A_{23}^{h_n})' dx_1\right) \\&+2 \lim _{n \rightarrow \infty } \int _{B(\bar{x}_1,r) \times \omega } A_{23} P \left( \begin{array}{c} x_3 \\ -x_2 \end{array} \right) \cdot P\left( \begin{array}{c} \partial _1{v_2^{h_n}}\\ \partial _1{v_3^{h_n}} \end{array} \right) dx, \end{aligned}$$

where the constant \(\bar{C}_{\omega }\) equals

$$\begin{aligned} \bar{C}_{\omega }=\int _{\omega } \left| P\left( \begin{array}{c} x_3 \\ -x_2 \end{array} \right) \right| ^2 \,\mathrm {d}x_2 \,\mathrm {d}x_3. \end{aligned}$$
(23)

Since \(A_h' \rightharpoonup 0\) in \(L^2\), the second term converges to zero. Since P is the projection, we have that

$$\begin{aligned} \int \limits _{B(\bar{x}_1,r) \times \omega } A_{23}P\left( \begin{array}{c} x_3 \\ -x_2 \end{array} \right) \cdot P\left( \begin{array}{c} \partial _1{v_2^{h_n}}\\ \partial _1{v_3^{h_n}} \end{array} \right) \,\mathrm {d}x = \int \limits _{B(\bar{x}_1,r) \times \omega } A_{23}P\left( \begin{array}{c} x_3 \\ -x_2 \end{array} \right) \cdot \left( \begin{array}{c} \partial _1{v_2^{h_n}}\\ \partial _1{v_3^{h_n}} \end{array} \right) \,\mathrm {d}x \rightarrow 0, \end{aligned}$$

since \(\partial _1{v}^{h_n} \rightharpoonup 0\) weakly in \(L^2\). We obtain that

$$\begin{aligned} I_2 \ge 2r \bar{C}_{\omega } A_{23}^2. \end{aligned}$$

Combing this with (20) and (21) and taking the limit as \(r \rightarrow 0\) yield the coercivity of Q. \(\square \)

For a given sequence \((h_n)\) monotonically decreasing to zero for which the Assumption 2.11 is satisfied, we also define the function \(Q_0: [0,L] \times \mathbb {M}^3_{{\text {skw}}\,} \rightarrow \mathbb {R}\) such that

$$\begin{aligned} Q_0(x_1,A) =\min _{a \in \mathbb {R}} Q(x_1,A,a), \end{aligned}$$
(24)

and mapping \(a_{\min }:[0,L] \times \mathbb {M}^3_{{\text {skw}}\,} \rightarrow \mathbb {R}\) that satisfies

$$\begin{aligned} Q_0(x_1,A)=Q(x_1,A,a_{\min }(x_1, A)). \end{aligned}$$
(25)

It is easy to see that \(Q_0\) satisfies the following property.

(\(Q'_{0}1\)):

For almost all \(x_1\in [0,L]\) the map \(Q_0(x_1,\cdot )\) is a quadratic form and satisfies

$$\begin{aligned}&C_1(\eta _1,\eta _2,\mu _1,\mu _2, \bar{C}_{\omega })|A|^2\\&\quad \le Q_0 (x_1,A)\le C_2(\eta _1,\eta _2,\mu _1,\mu _2, \bar{C}_{\omega })|A|^2, \quad \text {for all} \quad A \in \mathbb {M}^3_{{\text {skw}}\,}, \end{aligned}$$

where \(\bar{C}_{\omega }\) is defined in (23) and \(C_1,C_2\) depend only on the constants in the bracket.

The mapping \(a_{\min }\) is well defined, linear in A and for some \(C_a= C_a(\eta _1,\eta _2,\mu _1,\mu _2, \bar{C}_{\omega })>0\) we have

$$\begin{aligned} |a_{\min }(x_1,A)| \le C_a |A|, \quad \mathrm{for \quad a.e. } x_1 \in [0,L]. \end{aligned}$$

The function \(Q_0\) will be the energy density of the limit functional.

2.3 Identification of limit equations

We will state and prove liminf and limsup inequality.

Theorem 2.13

Let the family \((W^h)\) describe an admissible composite material in the sense of the Definition 1.2. Let \((h_n)\) be a sequence monotonically decreasing to zero such that the Assumption 2.11 is valid. Let \((y^{h_n}) \subset W^{1,2}(\Omega ;\mathbb {R}^3)\) be a sequence of deformations such that

$$\begin{aligned} \int _{\Omega } W^{h_n}(x, \nabla _{h_n} y^{h_n}) \,\mathrm {d}x \le Ch_n^2, \mathrm{for\, some }\, C>0. \end{aligned}$$

Then, there is a subsequence (still denoted by \((h_n)\)) such that \(\nabla _{h_n} y^{h_n} \rightarrow R\) strongly in \(L^2\), where \(R \in W^{1,2}([0,L];{\text {SO}}(3))\). Moreover we have

$$\begin{aligned} \displaystyle \liminf _{n \rightarrow \infty } \tfrac{1}{h_n^2} \int _{\Omega } W^{h_n}(x,\nabla _{h_n} y^{h_n}) \ge \int _{[0,L]} Q_0(R^t R')dx_1. \end{aligned}$$

Proof

By Theorem 2.15 there is a sequence \((R^{h_n}) \subset C^{\infty }([0,L]; \mathbb {R}^{3 \times 3})\) such that \(R^{h_n}(x_1) \in {\text {SO}}(3)\) for a.e. \(x_1 \in [0,L]\) and \(R^h\) satisfies (41) and (42). From (42) we conclude that on a subsequence \(R^{h_n} \rightharpoonup R\) weakly in \(W^{1,2}([0,L];\mathbb {R}^3)\) and thus also in \(C([0,L];\mathbb {R}^3)\). We take a further subsequence, without relabeling, on which \(\liminf \) is accomplished. We define the sequence \(v^h\) by the following decomposition

$$\begin{aligned} y^{h_n}=\tfrac{1}{|\omega |} \int _{\{x_1\} \times \omega } y^{h_n} +h_nx_2 R^{h_n} e_2+h_nx_3R^{h_n} e_3+h_n v^{h_n}. \end{aligned}$$
(26)

Integrating over \(\omega \) and using (1) yield:

$$\begin{aligned} \int _{\{x_1\} \times \omega } v^{h_n} dx=0, \end{aligned}$$
(27)

for a.e. \(x_1 \in [0,L]\). Note that

$$\begin{aligned} \nabla _{h_n} v^{h_n}=\tfrac{1}{h_n}(\nabla _{h_n} y^{h_n}-R^{h_n})-\left( p^{h_n}+x_2(R^{h_n})'e_2+x_3 (R^{h_n})'e_3|0|0 \right) , \end{aligned}$$

where

$$\begin{aligned} p^{h_n}= \tfrac{1}{{h_n}|\omega |} \int _{\{x_1\} \times \omega } \left( \partial _1 y^{h_n}-R^{h_n} e_1\right) . \end{aligned}$$

From (41) we obtain that \(\Vert p^{h_n}\Vert _{L^2}\) is bounded. Hence, there is \(p \in L^2([0,L];\mathbb {R}^3)\) such that \(p^{h_n} \rightharpoonup p\) weakly in \(L^2\) (on a subsequence). By using (41) and (42), we deduce that there exists a constant \(C>0\) such that

$$\begin{aligned} \Vert \nabla _{h_n} v^{h_n}\Vert _{L^2} \le C. \end{aligned}$$
(28)

Using (27) and the Poincaré inequality, we conclude that for some \(C>0\)

$$\begin{aligned} \Vert v^{h_n}\Vert _{L^2} \le Ch_n. \end{aligned}$$
(29)

Define the approximate strain by

$$\begin{aligned} G^{{h_n}} =\frac{(R^{h_n})^T \nabla _{{h_n}} y^{{h_n}} -I}{{h_n}}. \end{aligned}$$
(30)

From (41) we conclude that \((G^{h_n})\) is bounded in \(L^2\). It can be easily checked that

$$\begin{aligned} G^{h_n}=\iota \left( (A+A^{h_n}) d_{\omega }\right) +((R^{h_n})^t p^{h_n}|0|0)+(R^{h_n})^t \nabla _{h_n} v^{h_n}, \end{aligned}$$
(31)

where \(A=R^tR'\), \(A^{h_n}=(R^{h_n})^t (R^{h_n})'-R^tR'\). Take any sequence \((r_n) \subset C^1([0,L];\mathbb {R}^3)\) such that

$$\begin{aligned} r_n \rightarrow R^t p \qquad \text{ and } \qquad h_n (r_n)' \rightarrow 0, \end{aligned}$$

strongly in \(L^2\) and define the functions:

$$\begin{aligned} \tilde{p}^{h_n}= & {} \int _0^{x_1} \left( (R^{h_n})^tp^{h_n}-R^tp\right) ,\\ \tilde{v}^{h_n}= & {} (R^{h_n})^t v^{h_n}+(h_nx_2 (r_{n})_2+h_nx_3 (r_{n})_3,0,0)^t+\tilde{p}^{h_n}, \\ o^{h_n}= & {} (R^{h_n})^t \nabla _{h_n} v^{h_n}-\nabla _{h_n} ((R^{h_n})^t v^{h_n})-(h_n x_2 (r_{n})_2'+h_n x_3 (r_{n})_3') e_1 \otimes e_1\\&+\sum _{i=2,3}\left( (R^t p)_i-(r_{n})_i\right) e_i \otimes e_1. \\ \tilde{A}^{h_n}= & {} \int _0^{x_1} A^{h_n}. \end{aligned}$$

It is straightforward to check that

$$\begin{aligned} {\text {sym}}\,G^{h_n}= {\text {sym}}\,\iota (Ad_{\omega })+(R^tp)_1 e_1 \otimes e_1+{\text {sym}}\,\iota ( (\tilde{A}^{h_n})'d_{\omega })+{\text {sym}}\,\nabla _{h_n} \tilde{v}^{h_n}+{\text {sym}}\,o^{h_n}. \end{aligned}$$
(32)

By the Sobolev embedding theorem, we deduce from (42) that \(\Vert h_n (R^{h_n})'\Vert _{L^\infty } \rightarrow 0\). By combining this and (29), we obtain that

$$\begin{aligned} (R^h)^t \nabla _{h_n} v^{h_n}-\nabla _{h_n} ((R^{h_n})^t v^{h_n})=-((R^{h_n})'v^{h_n}|0|0) \longrightarrow 0, \mathrm{strongly\, in } L^2. \end{aligned}$$

From that it follows that

$$\begin{aligned} o^{h_n} \rightarrow 0 \, \mathrm{strongly\, in }\, L^2. \end{aligned}$$
(33)

It also easily follows that

$$\begin{aligned} \tilde{A}^{h_n} \rightarrow 0, \ \tilde{v}^{h_n} \rightarrow 0, \, \mathrm{strongly\, in }\, L^2, \quad \Vert \nabla _{h_n} \tilde{v}^{h_n}\Vert _{L^2} \le C, \end{aligned}$$
(34)

for some \(C>0\).

Using Lemma 2.16 and Lemma 2.17, we take a subsequence \((h_{n(k)})\) such that there exist sequences \((\bar{A}_k) \subset W^{1,2}([0,L]; M^3_{{\text {skw}}\,})\), \(\bar{v}_k \subset W^{1,2}(\Omega ; \mathbb {R}^3)\) which satisfy

  1. (i)

    \( \lim _{k \rightarrow \infty } |\{\bar{v}_k \ne \tilde{v}^{h_{n(k)}} \ \mathrm{or } \ \nabla \bar{v}_k \ne \nabla \tilde{v}^{h_{n(k)}} \}|=0,\) \( \lim _{k \rightarrow \infty } |\{\bar{A}_k \ne \tilde{A}^{h_{n(k)}} \ \mathrm{or } \ \bar{A}_k' \ne (\tilde{A}^{h_{n(k)}})' \}|=0;\)

  2. (ii)

    \(|\bar{A}_k'|^2\) and \((|\nabla _{h_{n(k)}} \bar{v}_k|^2)\) are equi-integrable.

It can be easily seen that \(\bar{A}_k \rightarrow 0\), \(\bar{v}_k \rightarrow 0\), strongly in \(L^2\) (i.e., weakly in \(W^{1,2}\)). By using Lemma 2.4, we obtain a sequence \((\psi _k) \subset W^{1,2}(\Omega ; \mathbb {R}^3)\) such that

$$\begin{aligned}&{\text {sym}}\,\nabla _{h_{n(k)}} \psi _k = {\text {sym}}\,\iota ((\bar{A}_k)'d_{\omega })+{\text {sym}}\,\nabla _{h_{n(k)}} \bar{v}_{k}, \nonumber \\&((\psi _{k})_1,h_{n(k)} (\psi _{k})_2, h_{n(k)} (\psi _{k})_3) \rightarrow 0, \ \int _{\omega } x_3 (\psi _{k})_2 \rightarrow 0\, \mathrm{strongly\, in } L^2, \end{aligned}$$
(35)

and the sequence \((| {\text {sym}}\,\nabla _{h_{n(k)}} \psi _k|^2)\) is equi-integrable. We define the sets

$$\begin{aligned} C^{h_n}=\left\{ x \in \Omega : |G^{h_n}| \le \tfrac{1}{\sqrt{{h_n}}} \right\} . \end{aligned}$$

From the boundedness of the sequence \((G^{h_n})\) in \(L^2\), we conclude that \(|\Omega \backslash C^{h_n}| \rightarrow 0\) as \(h \rightarrow 0\). Using (Q1), the decomposition of \(G^{h_n}\), (35) and the equi-integrability of \((| {\text {sym}}\,\nabla _{h_{n(k)}} \psi _k|^2)\), we deduce that

$$\begin{aligned}&\liminf _{k \rightarrow \infty } \int _{\Omega } Q^{h_{n(k)}}(x,\chi _{C^{h_{n(k)}}} G^{h_n}) \nonumber \\&\quad = \liminf _{k \rightarrow \infty } \int _{\Omega } Q^{h_{n(k)}} \left( x,\chi _{C^{h_{n(k)}}} \left( \iota (Ad_{\omega })+(R^tp)_1 e_1 \otimes e_1+ \iota ( (\tilde{A}^{h_n})'d_{\omega } )+ \nabla _{h_n} \tilde{v}^{h_n} \right) \right) \nonumber \\&= \liminf _{k \rightarrow \infty } \int _{\Omega } Q^{h_{n(k)}} \left( x, \iota (Ad_{\omega })+(R^tp)_1e_1 \otimes e_1+\nabla _{h_{n(k)}} \psi ^{h_{n(k)}} \right) . \end{aligned}$$
(36)

Using frame indifference property (W1), we have that \(W^{h_n}(x, \nabla _h y^h)=W^{h_n}(x,I+hG^h)\). From (3), by integrating, we conclude that

$$\begin{aligned}&\limsup _{n \rightarrow \infty } \left| \tfrac{1}{{h_n}^2}\int _{\Omega } W^{h_n} (\cdot ,I+{h_n}\chi _{C^{h_n}} G^{h_n})-\int _{\Omega } Q^{h_n}(\cdot ,\chi _{C^{h_n}} G^{h_n}) \right| \nonumber \\&\quad \le r(\sqrt{{h_n}}) \int _{\Omega }|\chi _{C^{h_n}} G^{h_n}|^2\rightarrow 0. \end{aligned}$$
(37)

Finally, we conclude, using (36), (37) and the definition of K and \(Q_0\)

$$\begin{aligned} \liminf _{k \rightarrow \infty } \tfrac{1}{h_{n(k)} ^2} \int _{\Omega } W^{h_{n(k)}}(x,\nabla _{h_{n(k)}} y^{h_{n(k)}})\ge & {} \liminf _{k \rightarrow \infty } \tfrac{1}{h_{n(k)} ^2} \int _{\Omega }\chi _{C^{h_{n(k)}}} W^{h_{n(k)}}(x,\nabla _{h_{n(k)}} y^{h_{n(k)}})\\= & {} \liminf _{k \rightarrow \infty } \int _{\Omega } Q^{h_{n(k)}}(x,\chi _{C^{h_{n(k)}}} G^{h_n}) \\\ge & {} K(m(A,(R^tp)_1,[0,L]) \\\ge & {} \int _{[0,L]} Q_0\left( R^t R'(x_1)\right) dx_1. \end{aligned}$$

\(\square \)

The next theorem gives the construction of the recovery sequence. Note that the statement of the Theorem 2.14 is not the classical upper bound statement, since the subsequence depends on R, but we can find one subsequence on which the convergences are true for all R, by using the density argument. However, Theorem 2.13 and Theorem 2.14 do imply in their form the convergence of minimizers on every subsequence of \((h_n)\), on which they are converging. Thus, Theorem 2.14 can also be seen as an upper bound statement.

Theorem 2.14

Let the family \((W^h)\) describe an admissible composite material in the sense of the Definition 1.2. Let \((h_n)\) be a sequence monotonically decreasing to zero for which the Assumption 2.11 is valid. Then for every \(R \in W^{1,2}([0,L];{\text {SO}}(3))\) there exists a subsequence, still denoted by \((h_n)\), such that

  1. a.

    there exists \((y_n) \subset W^{1,2}(\Omega ;\mathbb {R}^3)\) such that \(y_n \rightarrow \int _0^{x_1} Re_1\) strongly in \(W^{1,2}\) and \(\nabla _{h_n} y_n \rightarrow R\) strongly in \(L^2\).

  2. b.

    \(\lim _{n \rightarrow \infty } \frac{1}{h_n^2} \int _{\Omega } W^{h_n} (x, \nabla _{h_n} y_n)=\int _{[0,L]} Q_0\left( R^t R'(x_1)\right) \,\mathrm {d}x_1\).

Proof

It is easy to see that smooth rotations are dense in \(W^{1,2}([0,L];{\text {SO}}(3))\). This can be seen by approximating with smooth maps taking values in \(\mathbb {M}^3\) and then projecting on \({\text {SO}}(3)\) (by Sobolev embedding weak \(W^{1,2}\) implies strong convergence in \(L^{\infty }\) and we can project from tubular neighborhood of \({\text {SO}}(3)\)). Therefore, without the loss of generality we may assume that \(R \in C^2([0,L]; {\text {SO}}(3))\), since in the general case we can use the diagonal procedure.

Take any \(a \in C([0,L])\) and define \(A \in C^1([0,L];\mathbb {M}^3_{{\text {skw}}\,})\) by \(A=R^t R'\). Now we take \(m=m(R^t R',a)\) and the sequence \((\vartheta _n) \subset W^{1,2}(\Omega ;\mathbb {R}^3)\) which satisfies (a) and (b) of the Assumption 2.11. From Lemma 2.9 we have that the sequence \((|{\text {sym}}\,\nabla _{h_n} \vartheta _n|^2)\) is bounded (see (15)) and equi-integrabile. By Corollary 2.3, there are sequences \((A_n) \subset W^{1,2}([0,L];\mathbb {M}^3_{{\text {skw}}\,})\) and \((v_n) \subset W^{1,2}(\Omega ;\mathbb {R}^3)\) such that \(A_n\rightarrow 0\), \(v_n \rightarrow 0\) strongly in \(L^2\) and

$$\begin{aligned} \Vert {\text {sym}}\,\nabla _{h_n} \vartheta _n-{\text {sym}}\,\iota (A'_n d_{\omega })-{\text {sym}}\,\nabla _{h_n} v_n\Vert _{L^2} \rightarrow 0. \end{aligned}$$

Moreover, we have that

$$\begin{aligned} \sup \Vert A_n\Vert _{W^{1,2}} + \sup \left( \Vert v_n\Vert _{L^2}+\Vert \nabla _{h_n} v_n\Vert _{L^2}\right) <\infty . \end{aligned}$$
(38)

Choose a subsequence \((h_{n(k)})\) such that \(k h_{n(k)} \rightarrow 0\). Using Lemma 2.16 and Lemma 2.17, we conclude that there exist sequences \((\tilde{A}_k) \subset W^{1,\infty }([0,L];\mathbb {M}^3_{{\text {skw}}\,})\) and \((\tilde{v}_k) \subset W^{1,\infty }(\Omega ;\mathbb {R}^3)\) such that for some \(C>0\) we have (on a further subsequence; not relabeled)

  1. (i)

    \(|\bar{A}_k'| \le Ck\), for a.e. \(x_1 \in [0,L]\), \(|\nabla _{h_{n(k)}} \tilde{v}_{k} | \le Ck\) for a.e. \(x \in \Omega \);

  2. (ii)

    \( \lim _{k \rightarrow \infty } |\{\tilde{A}_k \ne A_{n(k)} \ \mathrm{or } \ \tilde{A}'_k \ne A'_{n(k)}\}|=0\); \( \lim _{k \rightarrow \infty } |\{\tilde{v}_k \ne v_{n(k)} \ \mathrm{or } \ \nabla \tilde{v}_k \ne \nabla v_{n(k)} \}|=0.\)

  3. (iii)

    the sequences \((|\tilde{A}_k'|^2)\), \((|\nabla _{h_{n(k)}} \tilde{v}_k|^2)\) are equi-integrable.

It is easy to argument that \(\tilde{A}_k \rightarrow 0\), \(\tilde{v}_k \rightarrow 0\) strongly in \(L^2\) (i.e., weakly in \(W^{1,2}\)). We define the sequence \((R_k) \subset W^{1,\infty }([0,L];\mathbb {M}^3)\) as the solutions of the following Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{lll}R_k'&{}=&{} R_k(A+\tilde{A}'_k), \\ R_k(0) &{}=&{} R(0). \end{array}\right. \end{aligned}$$
(39)

Since the right-hand side of the first equation in (39) is Lipschitz function, this system has a unique solution. Moreover, since it is tangential to \({\text {SO}}(3)\) it can be easily argumented that we have \(R_k(x_1) \in SO(3)\) for every \(x_1 \in [0,L]\) (this can be done, e.g., by approximating \(A_k\) with smooth fields and then using the standard theorem for the solutions of ODE system whose right-hand side is tangential to some smooth manifold). Notice also that \(R_k \rightharpoonup R\) weakly in \(W^{1,2}\) and thus, by Sobolev embedding, strongly in \(L^\infty \). Define for every \(k \in \mathbb {N}\); \(\bar{v}_k=\tilde{v}_k-\int _{\Omega } \tilde{v}_k\) to accomplish \(\Vert \bar{v}_k\Vert _{W^{1,\infty }} \le Ck\), which follows by the Poincaré inequality. By the equi-integrability property, we obtain that

$$\begin{aligned} \Vert {\text {sym}}\,\nabla _{h_{n(k)}} \vartheta _{n(k)}-{\text {sym}}\,\iota (\tilde{A}'_k d_{\omega })-{\text {sym}}\,\nabla _{h_{n(k)}} \bar{v}_k\Vert _{L^2} \rightarrow 0. \end{aligned}$$
(40)

Define the recovery sequence with the formulae

$$\begin{aligned} y_k= & {} \int _0^{x_1} R_k e_1 +h_{n(k)} x_2 R_k e_2+h_{n(k)} x_3 R_k e_3+h_{n(k)} R \bar{v}_k+h_{n(k)}\int _0^{x_1}aRe_1. \end{aligned}$$

Define also the approximate strain by

$$\begin{aligned} G_k=\frac{R_k^t \nabla _{h_{n(k)}} y_k-I}{h_{n(k)}}. \end{aligned}$$

The following properties can easily be verified:

  1. (i)

    \(\Vert y_k-\int _0^{x_1} Re_1 \Vert _{L^{\infty }} \rightarrow 0\), \(\Vert \nabla _{h_{n(k)}} y_k -R_k\Vert _{L^{\infty }} \rightarrow 0\);

  2. (ii)

    \(\Vert h_{n(k)} G_k\Vert _{L^{\infty }} \rightarrow 0\);

  3. (iii)

    \(\left\| {\text {sym}}\,G_k-a e_1 \otimes e_1-{\text {sym}}\,\iota ((A+\tilde{A}_k') d_{\omega })-{\text {sym}}\,\nabla _{h_{n(k)}} \bar{v}_k\right\| _{L^2} \rightarrow 0\).

This proves (a). To prove (b) notice that from the frame indifference property (W1) of Definition 1.1, we have that \(W^{h_{n(k)}}(x,\nabla _{h_{n(k)}} y_k)= W^{h_{n(k)}}(x,I+h_{n(k)}G_k)\), for a.e. \(x \in \Omega \). Using property (iii) of Definition 1.2 and the property (ii) of \(G_k\), we conclude that

$$\begin{aligned} \left| \tfrac{1}{h^2_{n(k)}} \int _{\Omega } W^{h_{n(k)}}(x,\nabla _{h_{n(k)}}y_k)- \int _{\Omega } Q^{h_{n(k)}}(x, G_k)\right| \rightarrow 0. \end{aligned}$$

From (40) we have that

$$\begin{aligned} \left\| {\text {sym}}\,G_k-a e_1 \otimes e_1-{\text {sym}}\,\iota \left( A d_{\omega } \right) -{\text {sym}}\,\nabla _{h_{n(k)}} \vartheta _{n(k)} \right\| _{L^2} \rightarrow 0. \end{aligned}$$

Using this we obtain that

$$\begin{aligned} \left| \lim _{k \rightarrow \infty } \int _{\Omega } Q^{h_{n(k)}}(x, G_k)-\int _{[0,L]} Q(x_1,A,a)\right| \rightarrow 0. \end{aligned}$$

Finally, we approximate \(a_{\min } (\cdot ,A(\cdot ))\in L^{\infty } ([0,L])\), defined in (25), with continuous maps in \(L^2\) norm and use the diagonalizing procedure to prove (b). \(\square \)