Liouville type theorems and Hessian estimates for special Lagrangian equations

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 (Comm Partial Differ Equ 33(4-6):922-932, 2008). Based on Warren-Yuan's work, our strategy is to show a global Hessian estimate of solutions via the Neumann- Poincare 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.

Automated summary

In this paper, a Liouville type theorem is obtained for the special Lagrangian equation with a certain 'convexity' condition. The strategy is to show global Hessian estimates of solutions and interior Hessian estimates on the gradient of the solutions by utilizing geometric measure theory and the Neumann-Poincare inequality on special Lagrangian graphs.