(In)finiteness of slopes for Monge-Ampère equations at flat boundary pieces

Abstract

On any bounded convex domain $Ω\subset\mathbb{R}^n$ with a $d$-dimensional flat boundary piece $F\subset\partialΩ$, we consider convex solutions with zero boundary values to the Monge-Ampère equations $$ \det\mathsf{D}^2 u= \mathrm{dist}(\,\cdot\,,\mathrm{Aff}(F))^α,\quad\det\mathsf{D}^2 u=\mathrm{dist}(\,\cdot\,,\partialΩ)^α,\quad \det\mathsf{D}^2 u=(-u)^α, $$ where $\mathrm{dist}$ stands for Euclidean distance and $\mathrm{Aff}(F)$ is the affine hull of $F$. We show that the threshold of the exponent $α\in\mathbb{R}$ for $u$ to have infinite slope at $F$ is $α\leq 2d-n$. For the first two equations, we also show that no solution exists when $α\leq 2d-2n$. Moreover, when $Ω$ satisfies the ``exterior cone condition with a $d$-dimensional ridge'' at $F$, both thresholds are sharp for the first equation, and the infinite-slope threshold is sharp for the last equation.