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Doubling Argument of the Hessian Estimate for the Special Lagrangian Equation on General Phases with Constraints

Cheuk Yan Fung

TL;DR

This work develops a doubling-inequality framework to obtain Hessian estimates for the special Lagrangian equation under general phase constraints, avoiding the Michael-Simon mean value inequality. It first proves an Alexandrov-type theorem via compactness and a vector-valued Radon measure for the weak Hessian, then applies Savin's small perturbation theorem together with Jacobi inequalities to bootstrap regularity. The main result is a pointwise Hessian bound $|D^2u(0)|$ depending only on $n$, $\|u\|_{C^{0,1}}$, and the phase $\theta$, valid under the stated constraints, with a refined version in dimension three under a $\sigma_2$-type condition. Collectively, the paper extends Hessian-estimate techniques to general phases with constraints and strengthens interior regularity results without relying on the Michael-Simon inequality, building on and generalizing prior work by Zhou, Shankar, and Yuan.

Abstract

In this paper, we establish a doubling argument to obtain Hessian estimates for the special Lagrangian equation under general phase constraints. In particular, our approach does not rely on the Michael-Simon mean value inequality. As an intermediate step, we also establish Alexandrov-type theorems, which may be of independent interest.

Doubling Argument of the Hessian Estimate for the Special Lagrangian Equation on General Phases with Constraints

TL;DR

This work develops a doubling-inequality framework to obtain Hessian estimates for the special Lagrangian equation under general phase constraints, avoiding the Michael-Simon mean value inequality. It first proves an Alexandrov-type theorem via compactness and a vector-valued Radon measure for the weak Hessian, then applies Savin's small perturbation theorem together with Jacobi inequalities to bootstrap regularity. The main result is a pointwise Hessian bound depending only on , , and the phase , valid under the stated constraints, with a refined version in dimension three under a -type condition. Collectively, the paper extends Hessian-estimate techniques to general phases with constraints and strengthens interior regularity results without relying on the Michael-Simon inequality, building on and generalizing prior work by Zhou, Shankar, and Yuan.

Abstract

In this paper, we establish a doubling argument to obtain Hessian estimates for the special Lagrangian equation under general phase constraints. In particular, our approach does not rely on the Michael-Simon mean value inequality. As an intermediate step, we also establish Alexandrov-type theorems, which may be of independent interest.

Paper Structure

This paper contains 6 sections, 17 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Alexandrov-type theorem
  4. Doubling inequality
  5. Proof of Theorem \ref{['thm1.1']} and \ref{['thm1.2']}
  6. Appendix

Key Result

Theorem 1.1

Let $n\geq 3$ and let $u$ be a smooth solution to the special Lagrangian equation (SLE) in $B_2(0)$ with the phase constant $\theta \geq 0$. Suppose that $\lambda(D^2 u) \subset \overline{\Gamma^{\frac{1}{2}(n-2)}_{n-1}}$, then

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Lemma 2.6
  • ...and 19 more