Characterization of Generalized Young Measures Generated by A-free Measures

We give two characterizations, one for the class of generalized Young measures generated by A-free measures and one for the class generated by B-gradient measures Bu. Here, A and B are linear homogeneous operators of arbitrary order, which we assume satisfy the constant rank property. The first characterization places the class of generalized A-free Young measures in duality with the class of A-quasiconvex integrands by means of a well-known Hahn–Banach separation property. The second characterization establishes a similar statement for generalized B-gradient Young measures. Concerning applications, we discuss several examples that showcase the failure of L 1-compensated compactness when concentration of mass is allowed. These include the failure of L 1-estimates for elliptic systems and the lack of rigidity for a version of the two-state problem. As a byproduct of our techniques we also show that, for any bounded open set Ω , the inclusions L1(Ω)∩kerA↪M(Ω)∩kerA,{Bu∈C∞(Ω)}↪{Bu∈M(Ω)}are dense with respect to the area-functional convergence of measures.