A note on the two dimensional Lagrangian mean curvature equation

TL;DR

This paper derives a simplified Hessian bound for solutions to the two-dimensional Lagrangian mean curvature equation using Warren-Yuan's super isoperimetric inequality, avoiding complex inequalities used previously.

Contribution

It introduces a new, simplified method to bound the Hessian of solutions in 2D, leveraging a super isoperimetric inequality specific to two dimensions.

Findings

Derived a modified Hessian bound for 2D solutions

The approach avoids Michael-Simon inequalities

Applicable to supercritical Lagrangian phase with bounded second derivatives

Abstract

In this note, we use Warren-Yuan's super isoperimetric inequality on the level sets of subharmonic functions, which is available only in two dimensions, to derive a modified Hessian bound for solutions of the two dimensional Lagrangian mean curvature equation. We assume the Lagrangian phase to be supercritical with bounded second derivatives. Unlike the previous approach, the simplified approach in this proof does not require the Michael-Simon mean value and Sobolev inequalities on generalized submanifolds of $\mathbb{R}^n$.