have attracted a large interest in the mechanics literature, as discussed for example in [@B9_Bhatta; @OtsukaWayman1999], starting with the works of Ball and James [@B9_BallJames87; @B9_BallJames92] also in the vectorial calculus of variations, see also [@ChipotKinderlehrer88; @Dacorogna1989; @MuellerLectureNotes] and the references therein. The typical model, based on nonlinear elasticity, takes the form $$\label{eqfctnonlineaelast}
\int_\Omega W(Du) dx\,,$$ where $u:\Omega\to\R^n$ is the elastic deformation. The energy density $W:\R^{n\times n}\to\R$ is minimized by several copies of the set of proper rotations $\operatorname{SO}(n)$, which correspond to the several phases of the material. The macroscopic material behavior can be analyzed using the theory of relaxation, which – under appropriate continuity and growth conditions – leads to the study of the quasiconvex envelope of $W$ [@Dacorogna1989; @MuellerLectureNotes]. Partial results in this direction have been obtained for example in [@B9_BallJames92; @Sverak1993; @DolzmannLN; @DolzmannKirchheim2003; @ContiDolzmannKirchheim2007; @ContiDolzmanntwowell]. The theory of relaxation does not, however, make any prediction on the length scale of the microstructure, as the functional (\[eqfctnonlineaelast\]) is scale invariant.

A more precise analysis of the microstructure is possible if one includes a singular perturbation, typically of the form $\varepsilon|D^2u|(\Omega)$ (in the sense of the total variation of the gradient of the $BV$ function $Du$) or $\varepsilon^2 \|D^2u\|_{L^2}^2$, 