Geometric rigidity principles for $\mathcal{A}$-free Young measures

We study generalized Young measures generated by $\mathcal{A}$-free measures, $\mathcal{A}$ a constant-rank operator, in a compactification of the value space adapted to an energy. Given an $\mathcal{A}$-quasiconvex integrand $G$ of linear growth with coercive recession function $G^{\mathrm{\infty }}$, the unit sphere at infinity is replaced by the indicatrix ${\mathbb{S}}_G=\{G^{\mathrm{\infty }}=1\}$, which need be neither convex nor symmetric. The two compactifications are equivalent, but in the adapted one the equality cases of the Jensen inequalities characterizing $\mathcal{A}$-free Young measures become statements of convex geometry about ${\mathbb{S}}_G$. We prove the characterization in this setting, with a new streamlined proof of the inhomogenization step based on a dyadic martingale argument. We then show that if $\mathcal{A}$-free measures ${\mu }_j\mu$ and their $G$-energies converge, then at $\mathrm{\mid }{\mu }^s\mathrm{\mid }$-almost every point the concentration-angle measure is supported on the smallest exposed face of $\overline{\mathrm{c}\mathrm{o}}({\mathbb{S}}_G)$ containing the $G$-polar of ${\mu }^s$; in particular, it never charges the set where $G^{\mathrm{\infty }}$ differs from its convex envelope, which, by the automatic convexity of Kirchheim and Kristensen, does not meet the wave cone. The concentration on the singular part of $\mu$ is elementary for every such sequence if and only if each point of the wave cone on ${\mathbb{S}}_G$ is an extreme point of $\overline{\mathrm{c}\mathrm{o}}({\mathbb{S}}_G)$, as for the Euclidean norm underlying Reshetnyak’s continuity theorem; on flat faces of ${\mathbb{S}}_G$, rigidity degrades from a point to a face. Finally, we prove a transfer principle for the convergence of energies of $\mathrm{B}\mathrm{V}$ maps under Dirichlet boundary conditions.