Quantified rigidity of $\mathcal{A}$-free Young measure concentrations

Abstract

We establish quantitative stability of the De Philippis--Rindler rigidity on conical convex regions. By constructing $\mathcal{A}$-quasiconvex functions with a tailored concavity---which additionally yields a new, elementary proof of the original theorem for constant-rank operators---we prove a spreading inequality that strictly limits the angular concentration of $\mathcal{A}$-free measures. This geometric bound establishes that singular concentrations cannot cluster arbitrarily close to a direction outside the Tartar wave cone. As a consequence, we obtain a new $L^1$ compensated compactness result: $\mathcal{A}$-free sequences with uniformly bounded mass are forced to be equi-integrable, thereby preventing the formation of mass concentrations, provided their targets are asymptotically restricted to cones whose aperture is controlled by a power of the distance to the wave cone of $\mathcal{A}$.

34 pages (changed the title, added Theorem A, corrected minor errors, updated with relevant recent bibliography)