of the $\omega_i$ we see by inspection that $|D^2v|\le c \alpha
h/\ell^2$ (we use repeatedly that $h\le\ell$ to simplify expressions). The jump contributions can be estimated by $2\|Dv-\Id\|_{L^\infty}$ times the length of the jump set; since $\gamma$ is monotone, this is bounded by $4(h+\ell)$. We conclude that $$\varepsilon |D^2v|(\omega)\leq c\varepsilon\left(\alpha (\ell+h) + \alpha
  h\ell \frac{h}{\ell^2}\right)\leq c\varepsilon \alpha \ell\,.$$ This completes the proof.

We now turn to the boundary layer. One could use a linear interpolation as in [@ChanContiSFBBericht]. For the sake of variety, we present an alternative construction, which is based on a variant of the construction of Lemma \[eqconstronerecttheo2\] and does not use gradients far from $A$, $B$ and $\Id$.

\[eqconstronerecttheo2bl\] 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 $$E_2^\eps[v, \omega]\le 
 c \left(\alpha^2h\ell +  \alpha \eps \ell\right)\,,$$ with the boundary conditions $v(x,y)=(x,y)$ for $y\in\{y_0,y_0+h\}$ or $x=x_0$, $v(x_0+\ell,y)=(x_0+\ell,y+\alpha\vartheta_{h}(y-y_0))$ and $\|Dv-\Id\|_{L^\infty}\le c\alpha$. The constant $c$ is universal.

The construction is similar to the one in Lemma \[eqconstronerecttheo2\]. Again, it suffices to treat the case $x_0=y_0=0$. With $\gamma$ as in Lemma \[eqconstronerecttheo2\], we set $$\begin{split}
 \omega_B' &= \left\{ x\in (0,\ell), \ 0<y<\frac{h}{4}\gamma
   \left(\frac{x}{\ell}\right)\right\}, \\
 \omega_M' &= \left\{ x\in(0,\ell), \
   \frac{h}{4}\gamma\left(\frac{x}{\ell}\right)<y<\frac{h}{2}-\frac{h}{4}\gamma\left(\frac{x}{\ell}\right)
\right\}, 
 \\ 
 \omega_A &= \left\{x\in (0,\ell), \
   