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 Omega, the inclusions L-1 (Omega) boolean AND ker A -> M(Omega) boolean AND ker A, {Bu epsilon C-infinity (Omega)} -> {Bu epsilon M(Omega)} are dense with respect to the area-functional convergence of measures.