The exterior Dirichlet problems of Monge–Ampère equations in dimension two

Abstract

Abstract In this paper, we study the Monge–Ampère equations det D 2 u = f $\det D^{2}u=f$ in dimension two with f being a perturbation of f 0 $f_{0}$ at infinity. First, we obtain the necessary and sufficient conditions for the existence of radial solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a unit ball. Then, using the Perron method, we get the existence of viscosity solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a bounded domain.

Keywords

• Monge–Ampère equations
• Exterior Dirichlet problem
• Asymptotic behavior