connection is present, then the energy is much larger, and typically no microstructure is formed.

Before stating the main results we introduce the key parameters and the set of admissibile functions. We focus on a rectangle, and assume throughout $$\label{assumption:1}
\Omega= (0,L)\times (0,H), \quad H,L>0, \quad \alpha\in(0,1),\quad \varepsilon>0\,,$$ in some parts additionally $\alpha< 1/2$. We shall impose Dirichlet boundary conditions corresponding to the average of $A_j$ and $B_j$, which, with the present choices, is in both cases the identity, $\frac12(A_j+B_j)=\Id$. Precisely, the admissible deformations are in the set $$\label{assumption:2}
\calM=\{u\in W^{1,2}(\Omega;\R^2): \ Du\in BV(\Omega;\R^{2\times 2}), \ u(x)=x \ \forall x\in \partial\Omega \}.$$ For $j\in\{1,2\}$, we are interested in the scaling of the minimum of $$\label{assumption:E}
E_j^\varepsilon[u,\Omega] = \int_{\Omega}\operatorname{dist}^2(Du,K_j)dx + \varepsilon
|D^2u|(\Omega)$$ over all $u\in\calM$. Existence of minimizers follows immediately from the direct method of the calculus of variations and will not be further addressed here. In the following, $c$ will denote a generic positive constant, which may change its value from line to line but in particular does not depend on the four parameters $\alpha$, $\varepsilon$, $H$ and $L$. We shall denote by $E_j^\eps[v,\omega]$ the integral of the expression in (\[assumption:E\]) over a subset $\omega\subset \Omega$.

\[Theorem:K1\] There is $c>0$ 