# Striped periodic minimizers of a two-dimensional model for martensitic phase transitions - Mathematical Physics

Abstract: In this paper we consider a simplified two-dimensional scalar model for theformation of mesoscopic domain patterns in martensitic shape-memory alloys atthe interface between a region occupied by the parent austenite phase and aregion occupied by the product martensite phase, which can occur in twovariants twins. The model, first proposed by Kohn and Mueller, is defined bythe following functional: $${\cal E}u=\beta||u0,\cdot||^2 {H^{1-2}0,h}+\int {0}^{L} dx \int 0^h dy \big|u x|^2 + \epsilon |u {yy}| \big$$ where$u:0,L\times0,h\to R$ is periodic in $y$ and $u y=\pm 1$ almost everywhere.Conti proved that if $\beta\gtrsim\epsilon L-h^2$ then the minimal specificenergy scales like $\sim \min\{\epsilon\beta-L^{1-2}, \epsilon-L^{2-3}\}$,as $\epsilon-L\to 0$. In the regime $\epsilon\beta-L^{1-2}\ll\epsilon-L^{2-3}$, we improve Conti-s results, by computing exactly theminimal energy and by proving that minimizers are periodic one-dimensionalsawtooth functions.

Author: Alessandro Giuliani, Stefan Mueller

Source: https://arxiv.org/