The Two-Well Problem For Piecewise-Affine Maps

In the two-well problem we look for a map u which satisfies Dirichlet boundary conditions and whose gradient Du assumes values in SO (2) A boolean OR SO (2) B = S-A boolean OR S-B, 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 [1], [2] A, B are two matrices such that det B > det A > 0 and rank {A - B} = 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 is an element of S-A boolean OR S-B exists in the Sobolev class W-1,W-infinity (Omega; 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.