  In this paper, we prove interior gradient estimates for the Lagrangian mean
curvature equation, if the Lagrangian phase is critical and supercritical and
$C^{2}$. Combined with the a priori interior Hessian estimates proved in
[Bha21, Bha22], this solves the Dirichlet boundary value problem for the
critical and supercritical Lagrangian mean curvature equation with $C^0$
boundary data. We also provide a uniform gradient estimate for lower regularity
phases that satisfy certain additional hypotheses.
