Higher-dimensional rigidity for the area functional

Determine whether the rigidity property established for the area functional in two-dimensional domains—specifically, the absence of nontrivial $T_N$ configurations in the differential inclusion associated with critical points of the area functional—continues to hold in higher dimensions $m\geq 3$.

Background

The paper proves a rigidity theorem for approximate solutions of the prescribed vertical mean curvature system when the domain dimension is two. In particular, equibounded Lipschitz approximate solutions whose residuals vanish strongly in $L^1$ have gradients converging strongly in $L^1$. As a consequence, the differential inclusion associated with the area functional in two dimensions contains no $T_N$ configurations, which are structures used in known constructions of irregular critical points for other polyconvex energies.

The authors identify the corresponding rigidity question as unresolved when the domain dimension satisfies $m\geq 3$. Establishing or refuting this higher-dimensional rigidity would clarify whether analogous oscillatory constructions and irregular Lipschitz critical points can occur for the area functional in higher-dimensional domains.