Sergio Conti

Asymptotic self-similarity in a model of branching in microstructured materials

This talk addressed some properties of a scalar 2D model which has been proposed to describe microstructure in martensitic phase transformations, consisting in minimizing the bulk energy

$$E[u] = \int_0^l \int_0^h u_x^2 + \epsilon \vert u_{yy}\vert$$

where $\vert u_y\vert=1$ 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 the minimizers for $\epsilon>0$, 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

$$u^j(x,y) = \theta^{-2j/3} u(\theta^jx, \theta^{2j/3}y)$$

($0<\theta<1$) has a strongly converging subsequence in $W^{1,2}$.