Regularity for critical points of convex functionals on Hessian spaces

TL;DR

This paper proves that critical points of convex variational integrals on Hessian spaces are smooth when the Hessian's oscillation is small, advancing understanding of regularity in higher-order variational problems.

Contribution

It establishes a regularity result for critical points of convex functionals on Hessian spaces under small oscillation conditions, a novel insight in higher-order calculus of variations.

Findings

Critical points are smooth if the Hessian oscillation is small.

Convexity and smoothness of the functional lead to regularity of solutions.

The result applies to variational integrals involving second derivatives.

Abstract

We consider variational integrals of the form $\int F(D^2u)$ where $F$ is convex and smooth on the Hessian space. We show that a critical point $u\in W^{2,\infty}$ of such a functional under compactly supported variations is smooth if the Hessian of $u$ has a small oscillation.