  We establish a prior interior $C^{1,1}$ estimates for convex solutions and
supercritical phase solutions to the Lagrangian mean curvature equation with
sharp Lipschitz phase. Counter-examples exist when the phase is H\"{o}lder
continuous but not Lipschitz. As an application we obtain interior
$C^{2,\alpha}$ regularity for $C^0$ viscosity solutions on the first phase
interval $((n-2)\frac\pi2,n\frac\pi2)$.
