Variational integrals on Hessian spaces: Partial regularity for critical points

Abstract

We develop regularity theory for critical points of variational integrals defined on Hessian spaces of functions on open, bounded subdomains of $R^n$, under compactly supported variations. We show that for smooth convex functionals, a $W^{2,\infty }$ critical point with bounded Hessian is smooth provided that its Hessian has a small bounded mean oscillation (BMO). We deduce that the interior singular set of a critical point has Hausdorff dimension at most $n-p_0$, for some $p_0\in (2,3)$. We state some applications of our results to variational problems in Lagrangian geometry. Finally, we use the Hamiltonian stationary equation to demonstrate the importance of our assumption on the a priori regularity of the critical point.