Liouville type theorems and Hessian estimates for special Lagrangian equations

TL;DR

This paper establishes Liouville theorems and Hessian estimates for special Lagrangian equations under convexity conditions, advancing understanding of solution behavior and regularity in geometric analysis.

Contribution

It introduces a global Hessian estimate for solutions to special Lagrangian equations using geometric measure theory, building on Warren-Yuan's convexity condition.

Findings

Proves a Liouville type theorem for solutions under convexity.

Derives interior Hessian estimates for solutions with supercritical phase.

Utilizes Neumann-Poincaré inequality and mean value inequality on Lagrangian graphs.

Abstract

In this paper, we get a Liouville type theorem for the special Lagrangian equation with a certain 'convexity' condition, where Warren-Yuan first studied the condition in [30]. Based on Warren-Yuan's work, our strategy is to show a global Hessian estimate of solutions via the Neumann-Poincar$\mathrm{\acute{e}}$ inequality on special Lagrangian graphs, and mean value inequality for superharmonic functions on these graphs, where we need geometric measure theory. Moreover, we derive interior Hessian estimates on the gradient of the solutions to the equation with this 'convexity' condition or with supercritical phase.