Characterization of generalized Young measures generated by $\mathcal A$-free measures

Abstract

We give two characterizations, one for the class of generalized Young measures generated by $\mathcal A$-free measures, and one for the class generated by $\mathcal B$-gradient measures $\mathcal Bu$. Here, $\mathcal A$ and $\mathcal B$ are linear homogeneous operators of arbitrary order, which we assume satisfy the constant rank property. The characterization places the class of generalized $\mathcal A$-free Young measures in duality with the class of $\mathcal A$-quasiconvex integrands by means of a well-known Hahn--Banach separation property. A similar statement holds for generalized $\mathcal B$-gradient Young measures. Concerning applications, we discuss several examples that showcase the rigidity or the failure of $\mathrm{L}^1$-compensated compactness when concentration of mass is allowed. These include the failure of $\mathrm{L}^1$-estimates for elliptic systems and the failure of $\mathrm{L}^1$-rigidity for the two-state problem. As a byproduct of our techniques we also show that, for any bounded open set $\Omega$, the inclusions \[ \mathrm{L}^1(\Omega) \cap \ker \mathcal A \hookrightarrow \mathcal M(\Omega) \cap \ker \mathcal A, \] \[ \{\mathcal B u\in \mathrm{C}^\infty(\Omega)\} \hookrightarrow \{\mathcal B u\in \mathcal M(\Omega)\}, \] are dense with respect to area-functional convergence of measures