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In this paper, we prove interior gradient estimates for the Lagrangian meancurvature 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 thecritical and supercritical Lagrangian mean curvature equation with $C^0$boundary data. We also provide a uniform gradient estimate for lower regularityphases that satisfy certain additional hypotheses.

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