in the PhD thesis [@B9_ChanDiss] and, in the simplified case $\alpha\sim 1$ and $H=L$, are presented in [@ChanContiSFBBericht]. Here we focus for simplicity on the case that $\Omega$ is a rectangle, keeping the two lengths separated to emphasize, in the case of Theorem \[Theorem:K1\], the competition between the two orientations of the microstructure. Although the detailed comparison between the different patterns is geometry-dependent, the leading-order scaling in $\eps$ in the small-$\eps$ limit is not expected to depend on the shape of $\Omega$. This is addressed (for similar models) in [@B9_KohnMuller94] for parallelograms and in [@B9_Diermeier] for general polygons. The case of piecewise smooth domains is treated [@B9_BCDM2002] for a model of thin-film blistering, without obtaining significant differences with the case of a rectangle.

In closing this introduction, we briefly review the literature on energy scaling in martensitic phase transformations. The scalar model $$E_\mathrm{KM}[u]=
  \begin{cases}\displaystyle
    \int_{\Omega} (\partial_1 u)^2 dx + \eps |\partial_2\partial_2 u|(\Omega) & \text{ if } \partial_2 u \in\{-1,1\}\text{ a.e.,}\\
\infty & \text{ otherwise,}
  \end{cases}$$ corresponding to a rigid version of our $K_1$ case, was proposed by Kohn and Müller in 1992. They have shown that, if the Dirichlet boundary data $u(0,\cdot)=0$ are imposed, for small $\eps$, the energy scales 