such that, under the assumptions (\[assumption:K1\]), (\[assumption:1\]), (\[assumption:2\]) and (\[assumption:E\]), one has $$\frac{1}{c}f(\alpha,\varepsilon,L,H) \leq \min_{u\in\calM} E_1^\eps[u,\Omega] \leq c
f(\alpha,\varepsilon,L,H)\,,$$ where $$\begin{aligned}
1\nonumber
    f(\alpha,\varepsilon,L,H)=\min&\Bigl\{
\alpha^{4/3}\varepsilon^{2/3}L^{1/3}H+\alpha\varepsilon L,\\
\nonumber
&\phantom{\Bigl\{}
\alpha^{4/3}\varepsilon^{2/3}LH^{1/3}+\alpha^4LH+\alpha\varepsilon H,\\
&\phantom{\Bigl\{}
\alpha^2 LH\Bigr\}\,.
\label{eqdefFtheo1}
  \end{aligned}$$ The constant $c$ does not depend on $\alpha$, $\varepsilon$, $L$ and $H$.

Follows from Lemma \[lemmaUBK1\] and Lemma \[lemmaLBK1\].

![Sketches of the deformation $u$ in the different phases appearing in Theorem \[Theorem:K1\].[]{data-label="fig:PD2"}](figPD2-crop){width="11cm"}

Theorem \[Theorem:K1\] shows that, depending on the values of the four parameters, different types of microstructure give the optimal energy scaling. In Figure \[fig:PD2\] we sketch the different microstructures (using the constructions from the upper bound) and in Figure \[fig:PD1\] the corresponding phase diagram. We now briefly discuss the different phases.

![Schematic phase diagram in case $K_1$, in the $L/\eps$, $H/\eps$ plane.[]{data-label="fig:PD1"}](figPD1-crop){width="8cm"}

1.  If the domain is very small, then the boundary data and the regularization term dominate the picture and the transformation to martensite is not energetically convenient. The energy scales as $\alpha^2 LH$, which is easily attained by the deformation $u(x)=x$. We call this austenitic (A) phase. From (\[eqdefFtheo1\]), one easily sees that this phase is the optimal one if $L/\eps<\alpha^{-1}$ or $H/\eps<\alpha^{-1}$.

2.  In the branching regime (BR) the deformation gradient 