MiS Preprint Repository

We address some properties of a scalar 2D model which has been proposed to describe microstructure in martensitic phase transformations, consisting in minimizing the bulk energy $$I[u]={\int }_0^1{\int }_0^h{u}_x^2+\sigma |u_{yy}|$$ where $|u_y|=1$ a.e. and $u(0,\cdot )=0$. Kohn and Müller [R. V. Kohn and S. Müller, Comm. Pure and Appl. Math. 47, 405 (1994)] proved the existence of a minimizer for $\sigma >0$, and obtained bounds on the total energy which suggested self-similarity of the minimizer. Building upon their work, we derive a local upper bound on the energy and on the minimizer itself, and show that the minimizer u is asymptotically self-similar, in the sense that the sequence $$u^j(x,y)={\theta }^{-2j/3}u({\theta }^{j}x,{\theta }^{2j/3}y)$$ ($0<\theta <1$) has a strongly converging subsequence in $W^{1,2}$.