本文關(guān)鍵詞： 奇點 分歧 k-非退化 分歧模型 拓撲度 Banach空間 Laplace算子 Fredholm算子　出處：《東北師范大學》2016年博士論文　論文類型：學位論文

【摘要】：本文將奇點理論和非線性分析方法相結(jié)合,應用到無限維Banach空間中的分歧理論中去,主要研究單參數(shù)非線性分歧理論中分歧點的判定與識別問題,以及分歧點處的半解支數(shù)目問題.對無限維Banach空間中的一類偏微分方程的分歧現(xiàn)象,采用類似于光滑映射有限決定性的思想,建立描述此分歧現(xiàn)象的由有限個加權(quán)齊次多項式函數(shù)的零點集所構(gòu)成的分歧模型,并運用此分歧模型討論多重特征根是否為分歧點的判定與分歧類型的識別.文中的分歧模型是一類由非線性問題誘導出的映射芽的奇點集,這類單參數(shù)映射芽所含有的變量相互獨立,于是可以討論一般的映射芽在孤立奇點處的分支個數(shù),通過得到的分支個數(shù)的拓撲度公式來表述出分歧模型的半解支個數(shù),從而得出Banach空間中分歧問題在分歧點處的分支數(shù)目的拓撲度公式.本文是奇點理論在分歧理論上的應用,也是對非線性偏微分方程分歧問題的有益的探索與嘗試.第一章是引言部分,簡要介紹與本課題相關(guān)的奇點與分歧理論的歷史研究概況,以及本課題的研究動機、目的和論文的結(jié)構(gòu).在第二章,定義了區(qū)域Ω的k-非退化條件,討論了k-非退化條件的等價條件,建立了(m,k)-分歧模型,運用奇點理論證明了(m.k)-分歧模型與Lyapunov-Schmidt約化所得分歧方程的等價性.在第三章,對于k-非退化區(qū)域上的分歧模型,考慮分歧點處分支解的個數(shù)問題,得出了半解支個數(shù)的拓撲度計算公式,計算出幾類特殊的二元分歧模型在平面上不同位置處的具體的半解支個數(shù).在第四章,給出了n維矩體上的一類含有Laplace算子的偏微分方程的分歧模型的表達公式,對此表達公式進行退化檢驗,在2維矩形和3維矩體上更精確的給出了不同分歧點處的分歧模型,運用此模型討論了這些分歧點的分歧類型和分歧點處的半解支個數(shù).除了n維矩體之外,在第五章,簡略的給出在圓盤、扇形、同心圓環(huán)、球體、同心球殼、2維球面、環(huán)面以及等邊三角形等特殊區(qū)域上的分歧模型.非線性問題的可能分歧點是其線性化問題的奇點,在第六章,運用非線性分析算子廣義逆方法,給出Banach流形中非線性算子的局部線性化定理.
[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.
【學位授予單位】：東北師范大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O177

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