where $\varepsilon$ represents a (scaled) surface energy. A scalar simplification of this type of model was proposed by Kohn and Müller in 1992 [@B9_KohnMuller92; @B9_KohnMuller94]. They have shown that the minimum energy scales as $\varepsilon^{2/3}$, and that this scaling is achieved by a self-similar microstructure, which refines close to the boundary. Finer results were later obtained in [@B9_Conti00; @B9_Conti2006; @B9_Zwicknagl], including in particular the statement of asymptotic self-similarity for the minimizers of the Kohn-Müller functional. Their approach was extended to many other physical problems, including for example micromagnetism [@CK98; @CKO99; @Viehmanndiss], magnetic structures in superconductors [@B9_CKO2004; @B9_CCKO2008], dislocation structures in plasticity [@B9_ContiOrtiz05], coarsening in thin film growth [@ContiOrtiz2008].

We consider in this work the full vectorial problem of nonlinear elasticity with surface energy, namely, the functional $$\label{intro:1}
  \int_{\Omega}W(Du)\,dx + \varepsilon |D^2u|(\Omega)\,.$$ Here, $u:\Omega\subset\R^n\rightarrow\R^n$ represents the elastic deformation, $W:\R^{n\times n}\rightarrow\R$ the energy density, which is minimized on a set $K\subset\R^{n\times n}$ and grows quadratically. For definiteness, we shall focus on $$\label{intro:2}
 W(F) = \operatorname{dist}^2(F,K)= \inf\{|F-G|^2: \ G\in K\}\,,$$ although all results hold for any energy with the same set of minimizers and the same growth. The small positive parameter $\varepsilon$ in (\[intro:1\]) represents the surface energy, and the admissible deformations are 