Sharp Small-Height Estimates for Subcritical Special Lagrangian Equations in Dimension Three

Abstract

We prove an interior Lipschitz estimate with a sharp height threshold for continuous viscosity solutions of the three-dimensional special Lagrangian equation with subcritical phase $|Θ|<π/2$. If \[ osc_{B_R}u\leq MR^2,\qquad 0\leq M<\secΘ, \] then $Lip_{B_{ρR}}u\leq CR$ for some $ρ>0$ and $C>0$ depending only on $|Θ|$ and $M$. The threshold is optimal: at $M=\secΘ$, there are entire real-analytic solutions whose Lipschitz seminorms are unbounded on every fixed interior ball. The proof uses a two-point maximum principle and algebraic identities specific to dimension three.