Variational integrals on Hessian spaces: partial regularity for critical points

TL;DR

This paper develops a regularity theory for critical points of variational integrals on Hessian spaces, showing smoothness under small BMO Hessian conditions and analyzing the size of singular sets.

Contribution

It introduces new regularity results for critical points of fourth order nonlinear equations in Hessian spaces, especially regarding smoothness and singular set dimension.

Findings

Critical points with bounded Hessian and small BMO are smooth.

The interior singular set has Hausdorff dimension at most n - p_0.

Applications to variational problems in Lagrangian geometry and Hamiltonian stationary equations.

Abstract

We develop regularity theory for critical points of variational integrals defined on Hessian spaces of functions on open, bounded subdomains of $\mathbb{R}^n$, under compactly supported variations. The critical point solves a fourth order nonlinear equation in double divergence form. 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.