those $u\in
W^{1,2}(\Omega;\R^n)$ with $Du\in BV(\Omega;\R^{n\times n})$ which obey specific boundary conditions. By $|D^2u|(\Omega)$ we denote the total variation of the measure $D^2u$ over the set $\Omega$. The set $K\subset\R^{n\times n}$ of energy-minimizing deformation gradients, as used in (\[intro:2\]), depends on the crystallography of the phase transformation considered.

In this paper we focus on the two-dimensional case, $n=2$, and study two different crystallographic settings. The first situation we address, denoted case $K_1$, is $$\label{assumption:K1}
 A_1= {\ensuremath{\begin{pmatrix}1&-\alpha \\ 0&1 \end{pmatrix}}}, \quad B_1={\ensuremath{\begin{pmatrix}1&\alpha \\ 0&1 \end{pmatrix}}}, \quad K_1=\operatorname{SO}(2)A_1\cup\operatorname{SO}(2)B_1$$ for some $\alpha>0$. These matrices have the same determinant and are representative of volume-preserving phase transformations in which the two wells have two rank-one connections, in the sense that $\det(A_1-QB_1)=0$ has two solutions $Q\in
\operatorname{SO}(2)$. This is the vectorial situation that was represented by the scalar model of Kohn and Müller; indeed, one can obtain a model similar to the one they used if $K_1$ is reduced to the two matrices $\{A_1, B_1\}$, see [@B9_Zwicknagl]. The two components of $u$ are in this case decoupled, only the first one gives a nontrivial energy contribution.

We show that for this problem the energy is proportional to $\varepsilon^{2/3}$ if the aspect ratio of the domain is 