Examples of microstructures related to the theory of martensitic phase transformations

Abstract

We consider the problem I^{\infty}=\underset{u\in\mathcal{W}}{inf}\underset{\Omega}{\int}\varphi(\nabla u(x,y))dxdy in the class \mathcal{W}=\{u\in W^{1,\infty}(\Omega):u/\Gamma_{0}=0,\left|u_{y}\right|=1\, a.e.\}, where \Omega is either the rectangle (0,1)^{2} or the parallelogram \{(x,y)\in\mathbb{R}^{2}:0<y<1,y<x<y+1\} and \Gamma_{0} denotes the boundary part {0}x[0,1] in the first case, for the parallelogram we let \Gamma_{0}=\{(x,x):0\leq x\leq1\}. The function \varphi:\mathbb{R}^{2}\rightarrow[0,\infty) is an elastic potential with wells in (0,\pm1). We prove that I^{\infty}=0 by considering minimizing sequences which differ substantially for both cases.