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 $\mathcal{B}u$. 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 \[ \mathrm{L}^1(Ω) \cap \ker \mathcal A \hookrightarrow \mathcal M(Ω) \cap \ker \mathcal A, \] \[ \{\mathcal B u\in \mathrm{C}^\infty(Ω)\} \hookrightarrow \{\mathcal B u\in \mathcal M(Ω)\}, \] are dense with respect to area-functional convergence of measures

73 pages, 3 figures. Version 4 (accepted for publication in Arch. Ration. Mech. Anal.) incorporates the characterization of $\mathcal{B}$-gradient measures, several new examples and several new applications that discuss the failure of $L^1$ compensated compactness for elliptic systems and the $n$-state problem