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Remarks on the quadratic Hessian equation

Connor Mooney

TL;DR

This note advances the understanding of the quadratic Hessian equation $\boldsymbol{\sigma_2(D^2u)=1}$ by establishing a barrier-based strict $2$-convexity property: a viscosity solution cannot touch a harmonic function on a smooth minimal hypersurface from below, highlighting geometric rigidity near minimal surfaces. It also proves an interior $C^2$ estimate in terms of a $W^{2,p}$ norm for any $p>2$, strengthening regularity results and linking them to the Chou–Wang Pogorelov framework. Together, these results rule out simple barrier-based strategies for constructing singular viscosity solutions and constrain potential singular models to be more intricate than one-homogeneous or distance-function barriers. The work leverages a barrier construction, interior gradient estimates, and Krylov–Safonov/ABP techniques, and it interprets the linearized operator in a Riemannian-geometric framework to illuminate the role of strict $2$-convexity in the $\sigma_2$ problem. Overall, it narrows the landscape of possible singular behaviors for $\sigma_2(D^2u)=1$ and strengthens interior regularity theory for $p>2$.

Abstract

We prove that viscosity solutions to the quadratic Hessian equation $$σ_2(D^2u) = 1$$ cannot touch a harmonic function on a minimal surface from below. This can be viewed as a form of strict $2$-convexity. We also prove an a priori interior $C^2$ estimate in terms of the $W^{2,\,p}$ norm, for any $p > 2$. Finally, we discuss how these results rule out certain strategies for constructing counterexamples to regularity.

Remarks on the quadratic Hessian equation

TL;DR

This note advances the understanding of the quadratic Hessian equation by establishing a barrier-based strict -convexity property: a viscosity solution cannot touch a harmonic function on a smooth minimal hypersurface from below, highlighting geometric rigidity near minimal surfaces. It also proves an interior estimate in terms of a norm for any , strengthening regularity results and linking them to the Chou–Wang Pogorelov framework. Together, these results rule out simple barrier-based strategies for constructing singular viscosity solutions and constrain potential singular models to be more intricate than one-homogeneous or distance-function barriers. The work leverages a barrier construction, interior gradient estimates, and Krylov–Safonov/ABP techniques, and it interprets the linearized operator in a Riemannian-geometric framework to illuminate the role of strict -convexity in the problem. Overall, it narrows the landscape of possible singular behaviors for and strengthens interior regularity theory for .

Abstract

We prove that viscosity solutions to the quadratic Hessian equation cannot touch a harmonic function on a minimal surface from below. This can be viewed as a form of strict -convexity. We also prove an a priori interior estimate in terms of the norm, for any . Finally, we discuss how these results rule out certain strategies for constructing counterexamples to regularity.

Paper Structure

This paper contains 10 sections, 7 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structure
  4. Classical Solvability of the Dirichlet Problem
  5. Viscosity Solutions
  6. Interior Gradient Estimate
  7. Chou-Wang Estimate
  8. Proof of Theorem \ref{['Main1']}
  9. Proof of Theorem \ref{['Main2']}
  10. Discussion

Key Result

Theorem 1.1

Assume that $u$ solves (Sigma2) in the viscosity sense on $B_1 \subset \mathbb{R}^n$. Then for any smooth, open, embedded portion $\Sigma$ of a minimal hypersurface in $B_1$ and harmonic function $h$ on $\Sigma$, $u|_{\Sigma}$ cannot touch $h$ from below.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 15 more