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 et al. [Monge-Ampère equation on exterior domains, Calc. Var PDE. 52 (2015), 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.

Keywords

• monge-ampère equation
• asymptotic behavior
• slow convergence rate
• 35j96
• 35b40
• 35j25