A Bernstein problem for special Lagrangian equations in exterior domains

Abstract

We establish quadratic asymptotics for solutions to special Lagrangian equations with supercritical phases or with semiconvexity on solutions in exterior domains. The method for “convex” case is based on an “exterior” Evans-Krylov or exterior Liouville type result for general fully nonlinear elliptic equations toward constant asymptotics of bounded Hessian, and also certain rotation arguments toward Hessian bound. Our unified approach also leads to quadratic asymptotics for convex solutions to Monge-Ampère equations (previously known), quadratic Hessian equations, and inverse harmonic Hessian equations over exterior domains. The semiconvex case is based on Allard-Almgren's uniqueness of tangent cones.