Doubling and the two-dimensional critical valued Lagrangian phase

TL;DR

This paper develops new interior Hessian and gradient estimates for the two-dimensional Lagrangian mean curvature equation at critical phases, overcoming degeneracy issues with a novel doubling technique.

Contribution

It introduces a modified doubling method to handle degenerate Jacobi inequalities at critical phases in two dimensions, enabling interior estimates.

Findings

Established interior Hessian estimates at critical phase

Developed a new doubling technique for degenerate inequalities

Addressed degeneracy in Jacobi inequalities for 2D Lagrangian equations

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.