Monge-Ampère equation with Guillemin boundary condition in high dimension
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916395506204672 |
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| author | Huang, Genggeng Shen, Weiming |
| author_facet | Huang, Genggeng Shen, Weiming |
| contents | The Guillemin boundary condition naturally appears in the study of Kähler 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05471 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monge-Ampère equation with Guillemin boundary condition in high dimension Huang, Genggeng Shen, Weiming Analysis of PDEs 35J96, 35J75, 35J70, 58J60 The Guillemin boundary condition naturally appears in the study of Kähler 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. |
| title | Monge-Ampère equation with Guillemin boundary condition in high dimension |
| topic | Analysis of PDEs 35J96, 35J75, 35J70, 58J60 |
| url | https://arxiv.org/abs/2406.05471 |