数学物理学报 ›› 2023, Vol. 43 ›› Issue (1): 181-202.

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Monge-Ampère方程边界爆破解的最优估计和不存在性

冯美强1,*(),张学梅2()   

  1. 1北京信息科技大学理学院 北京 100192
    2华北电力大学数理学院 北京 102206
  • 收稿日期:2021-11-24 修回日期:2022-04-24 出版日期:2023-02-26 发布日期:2023-03-07
  • 通讯作者: *冯美强, E-mail: meiqiangfeng@sina.com
  • 作者简介:张学梅, E-mail: zxm74@sina.com
  • 基金资助:
    北京市自然科学基金(1212003)

On the Optimal Global Estimates of Boundary Blow-up Solutions to the Monge-Ampère Equation

Feng Meiqiang1,*(),Zhang Xuemei2()   

  1. 1School of Applied Science, Beijing Information Science & Technology University, Beijing 100192
    2School of Mathematics and Physics, North China Electric Power University, Beijing 102206
  • Received:2021-11-24 Revised:2022-04-24 Online:2023-02-26 Published:2023-03-07
  • Supported by:
    Beijing Natural Science Foundation of China(1212003)

摘要:

该文致力于研究如下Monge-Ampère方程边界爆破解的最优估计和严格凸解的不存在性 $ M[u](x)=K(x)f(u),\ x \in \Omega,\; u(x)\rightarrow +\infty\ \mbox{ 当 }\ {\rm dist}(x,\partial \Omega)\rightarrow 0. $ 这里 $M[u]=\det\, (u_{x_{i}x_{j}})$ 是 Monge-Ampère 算子, $\Omega$$ \Bbb R^N (N\geq 2)$ 中的光滑有界严格凸区域. 文中不仅得到了$K(x)$$f(u)$ 的各种条件之间的关系, 还通过和已有文献中相关结果的比较明确了条件和估计之间的关系. 并且, 在 $\Omega$ 是一般区域的情况下给出了严格凸解不存在的结果, 而这在以往文献中尚未提及.

关键词: Monge-Ampère方程, 边界爆破解, 最优估计, 严格凸解, 不存在性

Abstract:

This paper is dedicated to studying the optimal global estimates and nonexistence of strictly convex solutions to the boundary blow-up Monge-Ampère problem $ M[u](x)=K(x)f(u) \mbox{ for } x \in \Omega,\; u(x)\rightarrow +\infty \mbox{ as } {\rm dist}(x,\partial \Omega)\rightarrow 0. $ Here $M[u]=\det\, (u_{x_{i}x_{j}})$ is the Monge-Ampère operator, and $\Omega$ denotes a smooth, bounded, strictly convex domain in $ \Bbb R^N (N\geq 2)$. The interesting features in our proof are that we not only obtain the relations among various conditions imposed on $K(x)$ and $f(u)$, but make comparison of some results of global estimates in previous literatures and make clear what conditions lead to what estimations. Moreover, when $\Omega$ is a general region, we give some nonexistence results which is rarely discussed in previous literatures.

Key words: Monge-Ampère equation, Boundary blow-up, Global estimates, Strictly convex solution, Nonexistence

中图分类号: 

  • O177.91