On special Lagrangian equations

Abstract

In this paper we study the special Lagrangian equation and related equations. Special Lagrangian equation originates in the special Lagrangian geometry by Harvey-Lawson [HL1]. In subcritical phases, we construct singular solutions in dimension three and higher. We also convert our counterexamples to the ones for the minimal surface system equation. In critical and supercritical phases, we derive a priori Hessian estimates in general higher dimensions (n > 3). Our unified approach leads to sharper estimates for previously known three dimensional and convex solution cases.