Hessian and Gradient Estimates for Equations of Lagrangian Mean Curvature

We prove Hessian and gradient a priori estimates pertaining to solutions of equations involving Lagrangian Mean Curvature. We first consider the Lagrangian mean curvature type equations which model singularities of the Lagrangian mean curvature flow. We prove Hessian and gradient estimates for these singularities with hypercritical, supercritical, and critical Lagrangian phases. We then extend these results to a broader class of Lagrangian mean curvature type equations under reasonable structural conditions. Next, we consider the Lagrangian Mean Curvature equation in dimension two, and prove a priori Hessian and gradient estimates for solutions whose phases are allowed to pass through the critical phase.