Doubling method for Monge-Ampere and special Lagrangian type equations

We present a recent new, doubling method for establishing a priori estimates, then classical solvability and
regularity for solutions to fully nonlinear PDEs.  The method produces the missing estimates for the quadratic Hessian
and prescribed hypersurface scalar curvature PDEs in dimension four.  It also gives new proofs of the estimates and regularity for Monge–Ampère and special Lagrangian equations, and provides prospects for classical solvability of alternative Dirichlet problems.