Analysis of non-linear singularities by microlocal methods

微局部方法的非线性奇点分析

• 批准号: 10440037
• 负责人: KATAOKA Kiyoomi
• 金额: $ 1.15万
• 依托单位: THE UNIVERSITY OF TOKYO
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10440037/
• 关键词: microlocal analysis; second microlocal analysis; Lagrangian; microfunctions; Mellin transformations; pseudo-differential operators; holomorphic functions; boundary values; 第二超局所解析; 偏微分方程式; 佐藤超関数; コロンボ超関数; FBI変換

Our researches concerning non-linear analysis in our project had almost no progress because we encountered a unexpected difficulties. However our researches concerning linear analysis had a remarkable progress due to KATAOKA's group ; in particular, the second microlocal analysis and an elementary construction of the microlocal theory of systems of equations.1. In the microlocal analysis, it is indispensable to employ expressions of singular solutions or kernel functions by boundary values of pseudodifferential operators. Concerning these expressions, KATAOKA's group obtained a necessary and sufficient condition on growth orders of lower order terms of formal symbols of pseudo-differential operators for taking boundary values. This result was applied to the analysis of microlocal solutions of some Fuchsian type equations. Further this result is deeply connected to the theory of small second hyperfunctions in the second microlocal analysis.2. KATAOKA's group succeeded in obtaining an integral formula of Mellin's type for holomorphic functions with any growth order at the corner in one-dimensional angled domains ; the main idea is to tilte the integral paths. Such a formula was indispensable for extending FUNAKOSHI's resullt on second microlocal analysis along Lagrangian submanifolds.

我们在项目中关于非线性分析的研究几乎没有任何进展，因为我们遇到了意想不到的困难。然而，由于KATAOKA的研究，我们在线性分析方面的研究取得了显著的进展；特别是二阶微局部分析和方程组微局部理论的初步构造。在微局部分析中，用伪微分算子的边值表示奇异解或核函数是不可缺少的。关于这些表达式，KATAOKA小组得到了伪微分算子形式符号的低阶项取边值的生长阶的一个充分必要条件。该结果应用于某些Fuchsian型方程微局部解的分析。这一结果与二次微局部分析中的小二次超功能理论密切相关。KATAOKA小组成功地获得了一维角域上任意生长阶全纯函数的Mellin型积分公式；主要思想是倾斜整条路径。这个公式对于沿着拉格朗日子流形推广FUNAKOSHI关于二阶微局部分析的结果是必不可少的。

项目成果

H.MATANO, T.OGIWARA: "Stability analysis in order-preserving systems in the presence of symmetry"Proceedings of Royal Society Edinburrgh. 129. 397-442 (1999)

H.MATANO、T.OGIWARA：“存在对称性的保序系统中的稳定性分析”爱丁堡皇家学会论文集。

H.MATANO, T.OGIWARA: "Monotonicity and convergenceresults in order-preserving systems in the presence of symmetry"Discrete and Continuous Dynamical Systems. 5. 1-34 (1999)

H.MATANO，T.OGIWARA：“在对称性存在的情况下保序系统的单调性和收敛结果”离散和连续动力系统。

片岡清臣: "Some micro differential oquations for microfunctions with a holomorphic parameter their formal symbol type solutions"京都大学数理解析研究所講究録「超局所解析と複素領域における微分方程式」に載る予定.

Kiyoomi Kataoka：“具有全纯参数的微函数的一些微微分方程及其形式符号类型解决方案”将发表在京都大学数学科学研究所讲座记录“复杂域中的超局部分析和微分方程”中。

T.AOKI, K.KATAOKA, S.YAMAZAKI: "Construction of kernel functions of pseudodifferential operators of infinite order, Actual problems in Mathematical Analysis"Proceedings of the conference dedicated to the seventieth birthday of Professor Yu.F.Korobeinik, R

T.AOKI、K.KATAOKA、S.YAMAZAKI：“无限阶伪微分算子的核函数的构造，数学分析中的实际问题”纪念 Yu.F.Korobeinik, R 教授七十岁生日的会议记录

K.YAJIMA: "Boundedness and continuity of the fundamental solution of time dependent Schrodinger equation with singular potentials"Tohoku Mathematical journal. 50. 577-595 (1998)

K.YAJIMA：“具有奇异势的时变薛定谔方程基本解的有界性和连续性”东北数学杂志。