Hessian estimates for Lagrangian mean curvature equation with Lipschitz critical and supercritical phases

In this paper, we develop a new strategy to study Lagrangain mean curvature equation on open sets of ${\mathbb{R}}^n(n\ge 2)$. By establishing an Allard-type regularity theorem, we obtain an interior Hessian estimate of solutions to this equation with prescribed Lipschitz critical and supercritical phases. Here, our condition on the phases is sharp. The proof heavily relies on geometric measure theory, geometry of Lagrangian graphs, and De Giorgi-Nash-Moser iteration. We expect that the techniques and ideas developed here can be used in some other equations.