非退化区域上的分歧模型

发布时间：2018-02-14 02:36

本文关键词： 奇点分歧 k-非退化分歧模型拓扑度 Banach空间 Laplace算子 Fredholm算子　出处：《东北师范大学》2016年博士论文　论文类型：学位论文

【摘要】：本文将奇点理论和非线性分析方法相结合,应用到无限维Banach空间中的分歧理论中去,主要研究单参数非线性分歧理论中分歧点的判定与识别问题,以及分歧点处的半解支数目问题.对无限维Banach空间中的一类偏微分方程的分歧现象,采用类似于光滑映射有限决定性的思想,建立描述此分歧现象的由有限个加权齐次多项式函数的零点集所构成的分歧模型,并运用此分歧模型讨论多重特征根是否为分歧点的判定与分歧类型的识别.文中的分歧模型是一类由非线性问题诱导出的映射芽的奇点集,这类单参数映射芽所含有的变量相互独立,于是可以讨论一般的映射芽在孤立奇点处的分支个数,通过得到的分支个数的拓扑度公式来表述出分歧模型的半解支个数,从而得出Banach空间中分歧问题在分歧点处的分支数目的拓扑度公式.本文是奇点理论在分歧理论上的应用,也是对非线性偏微分方程分歧问题的有益的探索与尝试.第一章是引言部分,简要介绍与本课题相关的奇点与分歧理论的历史研究概况,以及本课题的研究动机、目的和论文的结构.在第二章,定义了区域Ω的k-非退化条件,讨论了k-非退化条件的等价条件,建立了(m,k)-分歧模型,运用奇点理论证明了(m.k)-分歧模型与Lyapunov-Schmidt约化所得分歧方程的等价性.在第三章,对于k-非退化区域上的分歧模型,考虑分歧点处分支解的个数问题,得出了半解支个数的拓扑度计算公式,计算出几类特殊的二元分歧模型在平面上不同位置处的具体的半解支个数.在第四章,给出了n维矩体上的一类含有Laplace算子的偏微分方程的分歧模型的表达公式,对此表达公式进行退化检验,在2维矩形和3维矩体上更精确的给出了不同分歧点处的分歧模型,运用此模型讨论了这些分歧点的分歧类型和分歧点处的半解支个数.除了n维矩体之外,在第五章,简略的给出在圆盘、扇形、同心圆环、球体、同心球壳、2维球面、环面以及等边三角形等特殊区域上的分歧模型.非线性问题的可能分歧点是其线性化问题的奇点,在第六章,运用非线性分析算子广义逆方法,给出Banach流形中非线性算子的局部线性化定理.
[Abstract]:In this paper, the singular point theory is combined with the nonlinear analysis method and applied to the bifurcation theory in infinite dimensional Banach space. The problem of judging and identifying the bifurcation points in the single parameter nonlinear bifurcation theory is studied. The bifurcation phenomenon of a class of partial differential equations in infinite dimensional Banach spaces is similar to the finitely deterministic idea of smooth mapping. A bifurcation model consisting of 00:00 sets of finite weighted homogeneous polynomial functions is established to describe the bifurcation phenomenon. The bifurcation model is used to discuss whether multiple eigenvalues are bifurcation points and the recognition of bifurcation types. The bifurcation model in this paper is a kind of singular point set of mapping buds induced by nonlinear problems. The variables contained in this kind of one-parameter mapping germs are independent of each other, so we can discuss the number of branches of general mapping germs at isolated singularities. The number of semi-solution branches of the bifurcation model can be expressed by the topological degree formula of the number of branches obtained. The topological degree formula of the number of bifurcation problems at bifurcation points in Banach spaces is obtained. This paper is an application of singular point theory to bifurcation theory. It is also a useful exploration and attempt for the bifurcation problem of nonlinear partial differential equations. The first chapter is the introduction, which briefly introduces the historical research situation of singularity and bifurcation theory related to this topic, as well as the motivation of the research. In chapter 2, we define the k-nondegenerate condition of domain 惟, discuss the equivalent condition of k-nondegenerate condition, and establish a k-degenerate model. By using singularity theory, we prove the equivalence between the bifurcation model and the bifurcation equation obtained by Lyapunov-Schmidt reduction. In Chapter 3, we consider the number of bifurcation solutions for the bifurcation model on k-nondegenerate domain. The topological degree calculation formula of the number of semi-solution branches is obtained, and the specific number of half-solution branches at different positions of several special binary bifurcation models on the plane is calculated. In Chapter 4th, In this paper, the expression formulas of a class of partial differential equations with Laplace operator on n-dimensional moment are given, and the degeneracy test is carried out. The bifurcation models at different bifurcation points are given more accurately on 2-dimensional rectangular and 3-dimensional moment bodies. By using this model, we discuss the bifurcation types of these bifurcation points and the number of half-solution branches at the bifurcation points. In Chapter 5th, in addition to n-dimensional moment bodies, we briefly give 2-dimensional spherical surfaces in disk, sector, concentric ring, sphere and concentric spherical shell. Bifurcation models on special domains such as torus and equilateral triangles. The possible bifurcation points of nonlinear problems are singularities of their linearization problems. In Chapter 6th, the generalized inverse method of nonlinear analysis operator is used. The local linearization theorem of nonlinear operators in Banach manifolds is given.
【学位授予单位】：东北师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O177

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