Asymptotic behavior of solutions to the Monge--Ampère equations with slow convergence rate at infinity

Abstract

We consider the asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity and fulfill previous results under faster convergence rate by Bao-Li-Zhang [Calc. Var PDE. 52(2015). pp. 39-63]. Different from known results, we obtain the limit of Hessian and/or gradient of solution at infinity relying on the convergence rate. The basic idea is to use a revised level set method, the spherical harmonic expansion and the iteration method.