The two well problem for piecewise affine maps

created by paolini on 17 Feb 2012
modified on 14 Jun 2014

[BibTeX]

Published Paper

Inserted: 17 feb 2012
Last Updated: 14 jun 2014

In the 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 $A$, $B$ are two matrices such that $\det B>\det A>0$ and $\operatorname*{rank}\left\{ A-B\right\} =1.$ It was proved in the '90 that a map $u$ satisfying given boundary conditions and such that $Du\in\mathbb{S}% _{A}\cup\mathbb{S}_{B}$ exist in the Sobolev class $W^{1,\infty}% (\Omega;\mathbb{R}^{2})$ of Lipschitz continuous maps. However, for orthogonal matrices it was also proved 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.