The exterior Dirichlet problem for special Lagrangian equations

We establish the existence and uniqueness theorem for the exterior Dirichlet problem for the special Lagrangian equation with prescribed asymptotic behavior at infinity, in both the viscosity setting for all the phases and classical setting for the critical and supercritical phases. These results generalize previous work by the second author by removing restrictive assumptions on the asymptotic matrix and improving the decay rate to the order $2-n$. We also solve the interior Dirichlet problem for the critical special Lagrangian equation and, as applications, all the above-mentioned corresponding problems for the three dimensional quadratic Hessian equation without any admissibility condition.