Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains

Abstract

By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as \begin{document}$ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $\end{document} with zero boundary data, have unexpected degenerate nature.