\frac{h}{2}-\frac{h}{4}\gamma\left(\frac{x}{\ell}\right)<y<
   \frac{h}{2}+\frac{h}{4}\gamma\left(\frac{x}{\ell}\right)
\right\}\,,\\
 \omega_M'' &= \left\{ x\in(0,\ell), \
\frac{h}2+\frac{h}{4}\gamma\left(\frac{x}{\ell}\right)<y<h-\frac{h}{4}\gamma\left(\frac{x}{\ell}\right)
\right\}, 
 \\ 
 \omega_B'' &= \left\{ x\in (0,\ell), \ h-\frac{h}{4}\gamma \left(\frac{x}{\ell}\right)<y<h\right\}, 
 \end{split}$$ see Figure \[fig:2\]. We set $v_1(x,y)=x$ and $$  v_2(x,y)=
 \begin{cases}
 y+\alpha y & \text{on $\omega_B'$}, \\
 y+\frac{1}{4} \alpha h\gamma\left(\frac{x}{\ell}\right) & \text{on
   $\omega_M'$}, \\ 
 y+\alpha(\frac{h}{2}-y) & \text{on $\omega_A$},\\
 y-\frac{1}{4}\alpha h\gamma\left(\frac{x}{\ell}\right) & \text{on
   $\omega_M''$}, \\ 
 y+\alpha (y-h) & \text{on $\omega_B''$}.
 \end{cases}$$ One then checks that $v_2$ is continuous, $|D^2v_2|\le c\alpha h/\ell^2$ inside each of the five sets, $Dv=B_2$ on $\omega_B'\cup\omega_B''$, $Dv=A_2$ in $\omega_A$, and $|Dv-\Id|\le 
\alpha h \|\gamma'\|_{L^\infty}/\ell$ in $\omega_M'\cup\omega_M''$. Therefore, $\|Dv-\Id\|_{L^\infty}\le c\alpha$ and, treating the jump terms as above, $$E_2^{\varepsilon}[v,(0,\ell)\times (0,h)] \leq c\alpha^2
h\ell+c\eps\alpha \ell \,.$$

![Global pattern used in Lemma \[lemmaUBK2\] for the construction of the upper bound of Theorem \[Theorem:K2\].[]{data-label="fig:1"}](fig1c-crop){width="12cm"}

We are now ready to present the full construction.

\[lemmaUBK2\] Under the assumptions of Theorem \[Theorem:K2\], there is a deformation $u\in \calM$ such that $$E_2^\eps[u,\Omega]\le  c \min\Bigl\{ \alpha^{6/5}\eps^{4/5} L^{1/5}H + \alpha
  \eps L, \alpha^2 LH\Bigr\}\,.$$ The constant $c$ is universal.

Inserting $u(x)=x$, we easily obtain a construction with energy $\alpha^2
LH$. Therefore, we only need to consider the case in which the first term in the minimum is smaller than the second. Then, necessarily, $\eps^{4/5}\alpha^{6/5} L^{1/5}H \le \alpha^2 LH$, 