Symmetry of solutions of minimal gradient graph equations on punctured space

Abstract

In this paper, we study symmetry and existence of solutions of minimal gradient graph equations on punctured space ℝn \ {0}, which include the Monge—Ampère equation, inverse harmonic Hessian equation and the special Lagrangian equation. This extends the classification results on Monge—Ampère equations. Under some conditions, we also give a necessary and sufficient condition for the solvability of the exterior Dirichlet problem in terms of their asymptotic behaviors.