Gradient estimates for the Lagrangian mean curvature equation with Lipschitz critical phase

Abstract

Let $n\ge 3$. We prove a priori interior gradient estimate for the Lagrangian mean curvature equation \[ \sum_{i=1}^n\arctanλ_i(D^2u)=θ(x) \] under the critical and supercritical phase condition $\theta \ge (n-2)\pi /2$, assuming only that the Lipschitz norm of the prescribed phase is bounded.