Gradient estimates for the Lagrangian mean curvature equation with critical and supercritical phase

abstract: 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 by the first author, 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.