The two well problem for piecewise affine maps

In the two well problem we look for a map uu which satisfies Dirichlet boundary conditions and whose gradient Du assumes values in SO(2)A∪SO(2)B=SA∪SB, for two given invertible matrices A,B (an element of SO(2)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 detB>detA>0 and rank⁡{A−B}=1. It was proved in the '90 that a map uu satisfying given boundary conditions and such that Du∈A∪B exist in the Sobolev class W1,∞(Ω;R2) of Lipschitz continuous maps. However, for orthogonal matrices it was also proved that solutions exist in the class of piecewise C1 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.