Semi-convex viscosity solutions of the special Lagrangian equation

TL;DR

This paper establishes regularity, sharpness of conditions, and Liouville theorems for viscosity solutions to the special Lagrangian equation with almost negative phases, advancing understanding of their smoothness and growth properties.

Contribution

It provides new interior derivative estimates, demonstrates sharpness of phase and semi-convexity conditions, and introduces a Liouville theorem for solutions with subcritical phase.

Findings

Proved smoothness and interior derivative estimates for solutions.

Showed sharpness of phase range and semi-convexity conditions.

Established a new Liouville theorem for entire solutions with subcritical phase.

Abstract

We prove smoothness and interior derivative estimates for viscosity solutions to the special Lagrangian equation with almost negative phases and small enough semi-convexity. We show by example that the range of phases we consider and the semi-convexity condition are sharp. As an application, we find a new Liouville theorem for entire such solutions of the special Lagrangian equation with subcritical phase. We also find effective Hessian estimates with exponential dependence, which we show to be optimal.