奇异非线性椭圆型方程Dirichlet问题解的精确边界行为 Alternative Title The exact boundary behavior of solutions to Dirichlet problems for singular nonlinear elliptic equations 李波 Thesis Advisor 张志军 2015-05-28 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 博士 Keyword 半线性椭圆方程 Dirichlet 问题 对流项 精确边界行为 局部比较原理 Karamata正规变化理论 上下解 比较函数 Abstract 本文主要研究三类奇异非线性椭圆型方程 Dirichlet 问题的古典解在边界附件的精确渐近行为.首先,对问题(P1)我们在边界附近建立了 解的比较原理, 随后应用Karamata 正规变化理论仔细研究了一个积分方程的唯一解 0 处附近的精确渐近行为. 再应用摄动方法, 当权重项b满足适当的结构条件时, 构造了恰当的比较函数, 得到了问题(P1)(的所有解在边界附近具有相同的精确渐近行为.对于问题(P2),通过建立新的局部比较原理, 结合Poisson方程Dirichlet 问题的唯一解的整体估计, 在a满足适当条件下,我们揭示了非线性项a(x)f(u)对问题 (P2)任一解在边界附近的精确渐近行为不产生影响.对于问题(P3). 首先, 当q=2 时, 我们通过一个非线性变换 将问题(P3)转换成等价的问题(P2);当 q 介于 0 和 1 之间时, 我们建立了类似于问题(P1)的局部比较原理; 而当 q 介于 1 和 2 之间时,应用一个不等式将问题(P3)转化为新的问题,建立了类似于问题(P2) 的局部比较原理. 揭示出了对流项对问题（P3）任一解在边界附近的精确渐近行为不构成影响. Other Abstract In this paper we mainly study the exact asymptotic behavior near the boundary of classical solutions to a singular Dirichlet problem forthree types of nonlinear elliptic equation. First,for the problem (P1),we first establish the local comparison principle of solutions, then by Karamata regular variation theory, then we study carefully the exact asymptotic behavior of the unique solution to a integral equation. Moreover, by a perturbation method, when b satisfies a proper structure condition ,we construct proper comparison function, and reveal that all the solutions of problem(P1) have the same asymptotic behavior near the boundary. For the problem(P2),by constructing new local comparison principle, combining with the global estimates of the unique solution to the Dirichlet problem of Poisson equation and when a satisfies a proper condition, we reveal that the nonlinear term a(x)f(u) does not affect the exact asymptotic behavior of any solution near the boundary to problem (P2). For the problem（P3）, First, when q=2, nonlinear transformation transforms problem(P3) into the equivalent problem(P2); when q belongs to (0, 1) , we construct a local comparison principle similar to(P1); and when q belongs to [1, 2) ,by using an inequility, the problem(P3)can be changed into a new problem. we construct a local comparison principle similar to(P2), and reveal that the convection term does not affect the exact asymptotic behavior of any solution near the boundary to problem(P3). URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/224816 Collection 数学与统计学院 Recommended CitationGB/T 7714 李波. 奇异非线性椭圆型方程Dirichlet问题解的精确边界行为[D]. 兰州. 兰州大学,2015.
 Files in This Item: There are no files associated with this item.
 Related Services Recommend this item Bookmark Usage statistics Export to Endnote Altmetrics Score Google Scholar Similar articles in Google Scholar [李波]'s Articles Baidu academic Similar articles in Baidu academic [李波]'s Articles Bing Scholar Similar articles in Bing Scholar [李波]'s Articles Terms of Use No data! Social Bookmark/Share
No comment.