On the structure of the quasiconvex hull in planar elasticity

Let K 1 and K 2 be compact sets of real 2 × 2 matrices with positive determinant. Suppose that both sets are frame invariant, meaning invariant under the left action of the special orthogonal group. Then we give an algebraic characterization for K 1 and K 2 to be incompatible for homogeneous gradient Young measures. This result can be used to determine the structure of the quasiconvex hull for sets of energy wells in planar elasticity.