Boundary estimates of solutions for a class of Monge–Ampère equations

Abstract

On a smooth strictly convex bounded domain in R2, we establish some boundary estimates of the solution to the complete hyperbolic affine sphere equation. On the unit ball in Rn, we also give some sharp estimates of the solutions of a class of Monge–Ampère equations. As an application, we prove the relative hyperbolic hypersurface constructed by solving two linked Monge–Ampère equations, has constant Gauss–Kronecker curvature and has bounded principal curvatures.