Optimal solvability of the exterior Dirichlet problem for the Monge–Ampère equation in dimension two

This paper is concerned with two optimal conditions for the existence of solutions to the two-dimensional Monge–Ampère equation. We prove that the exterior Dirichlet problem with prescribed asymptotic behavior at infinity is solvable if and only if the boundary value is “semi-convex”, which removes the \(C^2\) $C^2$ condition for the boundary value required in almost all related literatures. Additionally, we identify the sharp coefficient of the logarithmic term in the asymptotic behavior that ensures solvability, which completes the result on higher dimensional cases. A key challenge is handling the logarithmic term, which is unique to two-dimensional.