Quasiconvex functions, $SO(n)$ and two elastic wells

Abstract

We use $W^{1,\infty }$ approximations of minimizing sequences to study the growth of some quasiconvex functions near their zero sets. We show that for $SO(n)$, the quasiconvexification of the distance function ${\mathrm{dist}}^2(\cdot ,SO(n))$ can be bounded below by the distance function itself. In certain cases of the incompatible two elastic well structure, we establish a similar result. We also prove that for small Lipschitz perturbations of $SO(n)$) and of the two well structure, the Young measure limits of gradients supported on these perturbed sets are Dirac masses.