Monge-Ampère方程边界爆破解的最优估计和不存在性

Monge-Ampère方程边界爆破解的最优估计和不存在性

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

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

冯美强^1,^*(),张学梅²()

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)

Feng Meiqiang^1,^*(),Zhang Xuemei²()

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, $\Omega$ 是 $\Bbb R^N (N\geq 2)$ 中的光滑有界严格凸区域. 文中不仅得到了 $K(x)$ 和 $f(u)$ 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

冯美强, 张学梅. Monge-Ampère方程边界爆破解的最优估计和不存在性[J]. 数学物理学报, 2023, 43(1): 181-202.

Feng Meiqiang, Zhang Xuemei. On the Optimal Global Estimates of Boundary Blow-up Solutions to the Monge-Ampère Equation[J]. Acta mathematica scientia,Series A, 2023, 43(1): 181-202.

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