Compensated compactness of $\mathcal A$-free measures on cones with quantified aperture

Abstract

We establish the stability of the De Philippis--Rindler rigidity on conical convex regions of quantified aperture. By constructing specialized $\mathcal A$-free quasiconvex functions with certain concavity properties -- which additionally yields a new proof of the original theorem by lower semicontinuity arguments -- we prove a spreading inequality for $\mathcal A$-free Young measure concentrations. This shows such microstructures cannot have arbitrarily small variance if their barycenter lies outside the wave cone. More precisely, we obtain the following quantitative $L^1$ compensated compactness result: $\mathcal A$-free sequences with uniformly bounded mass are forced to be equi-integrable, preventing singular mass concentrations, provided their targets are asymptotically restricted to cones around an elliptic subspace with an aperture bounded by a power of the subspace's distance to the wave cone.