the sharp condition $\partial_2 u\in\{-1,1\}$ is replaced by a penalization and using different norms for the singular perturbation, were studied by Zwicknagl [@B9_Zwicknagl], leading for small $\eps$ to a result similar to (\[eqscalingKMbeta\]). The geometrically nonlinear version was addressed by Dolzmann and Müller [@B9_DolzmannMueller1995], who proved a rigidity result that implies that the energy scaling of the present problem is superlinear in $\eps$. Lorent [@Lorent2006] has shown that the energy scaling of the singularly perturbed problem and the one of a finite-element approximation are the same, $\Gamma$-convergence to a sharp-interface model (after dividing the energy by $\eps$) was derived in [@B9_ContiSchweizer2006b], quantitative rigidity estimates for low-energy states of singularly perturbed multiwell problems were obtained in [@B9_Lorent2005; @B9_ContiSchweizer2006b; @B9_ChermisiConti2010; @JerrardLorent2013]. The energy scaling for the geometrically nonlinear model was obtained in [@B9_Chan; @B9_ContiOWJun2007], a version of Theorem \[Theorem:K1\] with a geometrically linear model and without the explicit dependence on $H$ and $L$ was given in [@B9_Diermeier]. A somewhat different approach, in which not only the deformation but also the shape of the inclusion was optimized, was followed in [@KnuepferKohnOtto2013] in a geometrically linear setting, mimicking nucleation of martensite and leading to different scalings.

The rest of this paper discusses the proofs 