which we now present.

In order to fix the boundary conditions on the internal boundaries, we fix a continuous $1$-periodic function $\vartheta:\R\to\R$ such that $\vartheta(0)=0$, $\vartheta'=1$ on $(-1/4,1/4)$, $\vartheta'=-1$ on $(1/4,3/4)$ (equivalently, $\vartheta(t)=\operatorname{dist}(t+\frac14,\Z)-\frac14$) and scale it according to $\vartheta_h(t)=h\vartheta(t/h)$. Then $\vartheta_h$ is $h$-periodic and has derivative $\pm1$ almost everywhere. We observe that the function $$w_h(x,y)=
  \begin{pmatrix}
    x\\y +\alpha\vartheta_h(y)
  \end{pmatrix}$$ fulfills $Dw_h\in\{A_2,B_2\}$ almost everywhere. Further, on each line $\{(x,y): y\in h\Z\}$ we have $w_h(x,y)=(x,y)$, corresponding to the prescribed boundary conditions. The construction will be based on using $w_h(x,y)$ for selected values of $x$, with different values of $h$, larger in the center of the domain and smaller close to $x=0$ and $x=L$. The key ingredient in the proof is the following period-doubling construction, which permits to join different values of $h$ with small energetic cost.

\[eqconstronerecttheo2\] Let $0<h\le\ell$, $\omega=(x_0,x_0+\ell)\times(y_0,y_0+h)$, $\alpha\in(0,1)$, $\eps>0$. Then there is a function $v:\omega\to\R^2$ such that $$E_2^\eps[v, \omega]\le 
    c \left(\alpha^2\frac{h^5}{\ell^3} +  \alpha \eps \ell\right)\,,$$ with $\|Dv-\Id\|_{L^\infty}\le c\alpha$ and fulfilling the boundary conditions $v(x,y)=(x,y)$ for $y\in\{y_0,y_0+h\}$, $v(x_0,y)=(x_0,y+\alpha \vartheta_{h/2}(y-y_0))$, $v(x_0+\ell,y)=(x_0+\ell,y+\alpha\vartheta_{h}(y-y_0))$. The constant $c$ is universal.

We assume without loss of generality $x_0=y_0=0$. We fix a smooth interpolation function $\gamma:[0,1]\rightarrow [0,1]$ with $\gamma(0)=0$, $\gamma(1)=1$ and the first two derivatives vanishing at 