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Doubling and the two-dimensional critical valued Lagrangian phase

Arunima Bhattacharya, Ravi Shankar, Jeremy Wall

TL;DR

This work analyzes the two-dimensional Lagrangian mean curvature equation $\arctan\lambda_1+\arctan\lambda_2=\Theta(x)$ in the critical regime where the phase changes sign. A newly developed modified doubling method addresses the degeneracy of the Jacobi inequality at $\Theta=0$, enabling interior Hessian and gradient bounds. The authors establish a scale-invariant gradient estimate and a volume bound, derive an Alexandrov-type regularity theorem for viscosity solutions, and complete the argument with eigenvalue bounds via one-dimensional auxiliary constructions. These results yield interior Hessian control and, via perturbation and regularity theory, contribute to Dirichlet solvability in subcritical settings and a broader understanding of fully nonlinear geometric PDEs with variable right-hand sides. The methodology combines a degenerate Jacobi framework, doubling techniques, Alexandrov-type results, and modern perturbation theory in a cohesive, dimension-two setting.

Abstract

In this paper, we establish interior Hessian and gradient estimates for the two-dimensional Lagrangian mean curvature equation when the phase changes signs, provided the gradient of the phase vanishes along its zero set. At the critical phase in two dimensions, the Jacobi inequality degenerates, preventing the use of higher-dimensional methods to obtain Hessian estimates. To address this difficulty, we introduce a modified doubling technique that applies to degenerate Jacobi inequalities and yields interior estimates.

Doubling and the two-dimensional critical valued Lagrangian phase

TL;DR

This work analyzes the two-dimensional Lagrangian mean curvature equation in the critical regime where the phase changes sign. A newly developed modified doubling method addresses the degeneracy of the Jacobi inequality at , enabling interior Hessian and gradient bounds. The authors establish a scale-invariant gradient estimate and a volume bound, derive an Alexandrov-type regularity theorem for viscosity solutions, and complete the argument with eigenvalue bounds via one-dimensional auxiliary constructions. These results yield interior Hessian control and, via perturbation and regularity theory, contribute to Dirichlet solvability in subcritical settings and a broader understanding of fully nonlinear geometric PDEs with variable right-hand sides. The methodology combines a degenerate Jacobi framework, doubling techniques, Alexandrov-type results, and modern perturbation theory in a cohesive, dimension-two setting.

Abstract

In this paper, we establish interior Hessian and gradient estimates for the two-dimensional Lagrangian mean curvature equation when the phase changes signs, provided the gradient of the phase vanishes along its zero set. At the critical phase in two dimensions, the Jacobi inequality degenerates, preventing the use of higher-dimensional methods to obtain Hessian estimates. To address this difficulty, we introduce a modified doubling technique that applies to degenerate Jacobi inequalities and yields interior estimates.
Paper Structure (10 sections, 8 theorems, 94 equations)

This paper contains 10 sections, 8 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A degenerate Jacobi inequality
  4. Doubling property of the Hessian
  5. Alexandrov-type Theorem
  6. Scale Invariant Gradient Estimate
  7. Volume bound by the gradient
  8. Proof of the Main Theorem
  9. Proof of Equation \ref{['lambda2']}
  10. Proof of Equation \ref{['lambda1']}

Key Result

Theorem 1.1

Let $u$ be a smooth solution of s in $B_R(0)\subset\mathbb{R}^2$, where $-\pi<\Theta < \pi$, $D\Theta=0$ on the level set $\{\Theta = 0\}$ and $\Theta\in C^2(B_R(0))$. Then the Hessian of $u$ admits the bound

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 3.1
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • proof
  • Theorem 5.1
  • ...and 7 more