The two well problem for piecewise affine maps

Abstract

In the \textit{two-well problem} we look for a map $u$ which satisfies Dirichlet boundary conditions and whose gradient $Du$ assumes values in $SO\left( 2\right) A\cup SO\left( 2\right) B=\mathbb{S}_{A}\cup \mathbb{S}_{B},$ for two given invertible matrices $A,B$ (an element of $SO\left( 2\right) A$ is of the form $RA$ where $R$ is a rotation). In the original approach by Ball and James [1], [2] $A$, $B$ are two matrices such that $\det B>\det A>0$ and $\operatorname*{rank}\left\{ A-B\right\} =1.$ It was proved in the 1990's (see [4], [5], [6], [7], [17]) that a map $u$ satisfying given boundary conditions and such that $Du\in\mathbb{S} _{A}\cup\mathbb{S}_{B}$ exists in the Sobolev class $W^{1,\infty} (\Omega;\mathbb{R}^{2})$ of Lipschitz continuous maps. However, for orthogonal matrices it was also proved (see [3], [8], [9], [10], [11], [12], [16]) that solutions exist in the class of piecewise-$C^{1}$ maps, in particular in the class of piecewise-affine maps. We prove here that this possibility does not exist for other nonsingular matrices $A$, $B$: precisely, the two-well problem can be solved by means of piecewise-affine maps only for orthogonal matrices.