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Semi-convex viscosity solutions of the special Lagrangian equation

Connor Mooney, Ravi Shankar

TL;DR

The paper analyzes the regularity of viscosity solutions to the special Lagrangian equation $F(D^2u)=\Theta$ in the subcritical, almost negative phase regime under small semi-convexity. It develops a framework using Legendre transform touching reversal, CSY rotations, and a constant rank approach to propagate regularity from rotated equations back to the original problem, yielding analytic solutions with Hessian bounds that grow exponentially in $\|Du\|_{}\infty$. The authors prove a Liouville-type theorem for entire subcritical solutions and establish sharpness: both the phase/semi-convexity thresholds and the exponential rate in the Hessian bounds are essentially optimal. These results deepen understanding of gradient graphs as minimal Lagrangian objects under subcritical phases and introduce robust techniques (rotation, Legendre transform, and quantified Harnack arguments) that may extend to related fully nonlinear PDEs.

Abstract

We prove smoothness and interior derivative estimates for viscosity solutions to the special Lagrangian equation with almost negative phases and small enough semi-convexity. We show by example that the range of phases we consider and the semi-convexity condition are sharp. As an application, we find a new Liouville theorem for entire such solutions of the special Lagrangian equation with subcritical phase. We also find effective Hessian estimates with exponential dependence, which we show to be optimal.

Semi-convex viscosity solutions of the special Lagrangian equation

TL;DR

The paper analyzes the regularity of viscosity solutions to the special Lagrangian equation in the subcritical, almost negative phase regime under small semi-convexity. It develops a framework using Legendre transform touching reversal, CSY rotations, and a constant rank approach to propagate regularity from rotated equations back to the original problem, yielding analytic solutions with Hessian bounds that grow exponentially in . The authors prove a Liouville-type theorem for entire subcritical solutions and establish sharpness: both the phase/semi-convexity thresholds and the exponential rate in the Hessian bounds are essentially optimal. These results deepen understanding of gradient graphs as minimal Lagrangian objects under subcritical phases and introduce robust techniques (rotation, Legendre transform, and quantified Harnack arguments) that may extend to related fully nonlinear PDEs.

Abstract

We prove smoothness and interior derivative estimates for viscosity solutions to the special Lagrangian equation with almost negative phases and small enough semi-convexity. We show by example that the range of phases we consider and the semi-convexity condition are sharp. As an application, we find a new Liouville theorem for entire such solutions of the special Lagrangian equation with subcritical phase. We also find effective Hessian estimates with exponential dependence, which we show to be optimal.
Paper Structure (15 sections, 13 theorems, 109 equations)

This paper contains 15 sections, 13 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Viscosity solutions
  4. Lipschitz semi-convex functions
  5. Touching reversal under Legendre transform
  6. CSY rotation and touching preservation
  7. A bound on the non-maximal Hessian eigenvalues
  8. Proof of Theorem \ref{['thm:reg']} (Regularity) and Theorem \ref{['thm:liou']} (Liouville)
  9. Preservation of semi-convex solutions under rotation
  10. Higher regularity by convexity of rotated equation
  11. Regularity of viscosity solution by constant rank theorem
  12. Liouville theorem
  13. Effective estimate
  14. Proof of Theorem \ref{['thm:Ex']} (Sharpness of assumptions)
  15. Proof of Theorem \ref{['thm:Ex2']} (Sharpness of effective bound)

Key Result

Theorem 1.1

Let $\Theta \in (-(n-2)\pi/2,\,\pi/2)$ and define Assume that $u$ is a viscosity solution in $B_1 \subset \mathbb{R}^n$ to and satisfies in addition that Then $u$ is analytic, and moreover we have for $k \geq 2$ that

Theorems & Definitions (28)

  • Theorem 1.1: Regularity
  • Theorem 1.2: Sharpness of assumptions
  • Theorem 1.3: Sharpness of effective bound
  • Theorem 1.4: Liouville
  • Lemma 2.1: Touching reversal of transforms
  • proof
  • Proposition 2.2: Touching preservation of rotation
  • Lemma 2.3
  • proof
  • Proposition 3.1: Supersolution preservation, CSY
  • ...and 18 more