The degenerate two well problem for piecewise affine maps

The two well problem consists in finding maps u which satisfy some boundary conditions and whose gradient Du assumes values in the two wells A,B . Here A (similarly B ) is the well generated by a 2 × 2 matrix A, i.e., A is the set of matrices of the form RA, where R is a rotation. We study specifically the case when at least one of the two matrices A, B is singular and we characterize piecewise affine maps u satisfying almost everywhere the differential inclusion Du(x)∈A∪B . In particular we describe the lamination and angle properties, which turn out to be different from those of the nonsingular case described in detail in [15]. We also show that the two well problem can be solved in some cases involving singular matrices, in strict contrast to the nonsingular (and not orthogonal) case.