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

作　　者：冯美强[1] 张学梅[2] Meiqiang Feng;Xuemei Zhang(School of Applied Science,Beijing Information Science&Technology University,Beijing 100192;School of Mathematics and Physics,North China Electric Power University,Beijing 102206)

机构地区：[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

出　　处：《数学物理学报（A辑）》2023年第1期181-202,共22页Acta Mathematica Scientia

基　　金：北京市自然科学基金(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∈Ω,u(x)→+∞当dist(x,■Ω)→0.这里M[u]=det(ux_(i)x_(j))是Monge-Ampère算子,Ω是R^(N)(N≥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Ω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.This paper is dedicated to studying the optimal global estimates and nonexistence of strictly convex solutions to the boundary blow-up Monge-Ampere problem M[u](x)=K(x)f(u)for x∈Ω,u(x)+∞as dist(x,■Ω)→0.Here M[u]=det(ux_(i)x_(j)is the Monge-Ampere operator,andΩdenotes a smooth,bounded,strictly convex domain in R^(N)(N≥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,whenΩis a general region,we give some nonexistence results which is rarely discussed in previous literatures.

关 键 词：MONGE-AMPÈRE方程 边界爆破解 最优估计 严格凸解 不存在性 

分 类 号：O177.91[理学—数学]