Monge-Amp\`ere equation with Guillemin boundary condition in high dimension

Abstract

The Guillemin boundary condition naturally appears in the study of K\"ahler geometry of toric manifolds. In the present paper, the following Guillemin boundary value problem is investigated \begin{align} \label{eq1} &\det D^2 u=\frac{h(x)}{\prod_{i=1}^N l_i(x)},\quad\text{in}\quad\quad P\subset\mathbb R^n, \quad\quad \quad \quad\quad \quad \quad \quad\quad (1)\\ \label{bdy1} &u(x)-\sum_{i=1}^N l_i(x)\ln l_i(x)\in C^\infty(\overline{P}). \quad\quad\quad\quad \quad \quad\quad \quad \quad \quad\quad\quad (2) \end{align} Here \begin{equation*} 0<h(x)\in C^\infty(\overline{P}),\quad P=\cap_{i=1}^N \{l_i(x)>0\} \end{equation*} is a simple convex polytope in $\mathbb R^n$. The solvability of (1)-(2) is given under the necessary and sufficient condition. The key issue in the proof is to obtain the boundary regularity of $u(x)-\displaystyle \sum_{i=1}^N l_i(x)\ln l_i(x)$. Due to the difficulty caused by the structure of the equation itself and the singularity of $\partial P$, special attention is required to understand the influence of different singularity types at various positions on $\partial P$ and how these impact the behavior of $u$ in its vicinity.