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Variational integrals on Hessian spaces: partial regularity for critical points

Arunima Bhattacharya, Anna Skorobogatova

TL;DR

The paper addresses partial regularity for critical points of variational integrals on Hessian spaces, focusing on fourth-order double-divergence equations. It develops an ε-regularity framework: if a $W^{2,\infty}$-regular critical point has Hessian $D^{2}u$ in a convex Hessian region $U$ and $D^{2}u$ has small $\mathrm{BMO}$ modulus, then $u$ is smooth on a smaller domain, with a higher-integrability-based path to full regularity. A key outcome is a Hausdorff-dimension bound on the interior singular set, $\ ext{dim}_{\mathcal{H}}(\Sigma(u))\le n-p_0$ for some $p_0\in(2,3)$, derived via higher Sobolev estimates. The results are applied to Hamiltonian stationary equations and related Lagrangian-geometric variational problems, illustrating the necessity and impact of the a priori regularity assumption and the structural advantages of the double-divergence form. Overall, the work advances partial regularity theory for fourth-order Hessian-energy functionals, with implications for geometric analysis and elasticity-type problems.

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.

Variational integrals on Hessian spaces: partial regularity for critical points

TL;DR

The paper addresses partial regularity for critical points of variational integrals on Hessian spaces, focusing on fourth-order double-divergence equations. It develops an ε-regularity framework: if a -regular critical point has Hessian in a convex Hessian region and has small modulus, then is smooth on a smaller domain, with a higher-integrability-based path to full regularity. A key outcome is a Hausdorff-dimension bound on the interior singular set, for some , derived via higher Sobolev estimates. The results are applied to Hamiltonian stationary equations and related Lagrangian-geometric variational problems, illustrating the necessity and impact of the a priori regularity assumption and the structural advantages of the double-divergence form. Overall, the work advances partial regularity theory for fourth-order Hessian-energy functionals, with implications for geometric analysis and elasticity-type problems.

Abstract

We develop regularity theory for critical points of variational integrals defined on Hessian spaces of functions on open, bounded subdomains of , 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 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 , for some . 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.
Paper Structure (11 sections, 11 theorems, 88 equations)

This paper contains 11 sections, 11 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Applications.
  3. Structure of paper
  4. Preliminaries
  5. Regularity Theory for small BMO-Hessian
  6. Proofs of the main results
  7. Proof of Proposition \ref{['p:higher-int-D3-intro']} and Theorem \ref{['main1']}
  8. Proof of Corollary \ref{['c:dim-est']}
  9. Hamiltonian Stationary Equations
  10. Proof of Corollary \ref{['c:HS-reg']}
  11. A priori $W^{2,\infty}$ assumption

Key Result

Theorem 1.1

Suppose that $u\in W^{2,\infty}(B_{1})$ is a critical point of Ffunc where $F$ is smooth and uniformly convex or uniformly concave on $U$ and $D^2u(x) \in U$ for almost every $x\in B_1$. There exists $\omega(\Lambda,n,\|D^2u\|_{L^{\infty}(B_1)})>0$ such that if $D^2u\in\mathrm{BMO}(B_1)$ with modulu

Theorems & Definitions (31)

  • Definition 1.1: BMO
  • Remark 1.1
  • Theorem 1.1
  • Corollary 1.1
  • Proposition 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • ...and 21 more