Functional and Real Analysis for Partial Differential Equations

偏微分方程的泛函分析和实分析

基本信息

批准号：
09440069
负责人：
KOMATSU Hikosaburo
金额：
$ 6.78万
依托单位：
Science 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：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440069/
关键词：
Hyperfunctions Logarithms of operators Nonlinear partial differential equations Existence of solutions Analyticity of solutions Asymptotic behavior of solutions Inverse problems for elasticity Function spaces defined by real analysis 分数冪対数

项目摘要

The biggest achievement we have made for the last three years with the help of the Grant-in-Aid would be that we have created a very active group of researchers centered at the Kagurazaka Campus of the Science University of Tokyo. Since April, 1999, we are conducting an open seminar "Kagurazaka Seminar on Analysis" every month, in which three lectures are presented. We also had an international conference in November, 1999. The purpose of this research project was described as "to prepare Functional Analysis and Real Analysis so that necessary tools would be furnished for the research of partial differential equations and, in particular, to nonlinear equations". A big amount of results have been obtained along this line by not only the registered investigators of the project but also graduate students who worked with them and many of the attendants of the seminar and the international conference.Besides revision and correction of an English translation of his books, Komatsu discovered a few new results on convolutions of hyperfunctions etc.Okazawa applied abstract theories of linear evolution equations and nonlinear semigroups to concrete equations such as the Ginzburg-Landau equations. He also established a general theory of the logarithms log A of linear operators.Hayashi and Kato studied nonlinear partial differential equations mainly of evolution type. Hayashi proved the existence of solutions and analyzed their asymptotic behavior as the time tends to infinity for many important equations. Kato established the analyticity of solutions.Nakamura considered the inverse problem of elastic bodies and applied it to the tomography.Miyachi has obtained basic results on function spaces which are defined by real analytic methods and are getting more and more important in the theory of differential equations.

过去三年，我们在援助金的帮助下取得的最大成就是，我们创建了一个以东京科学大学神乐坂校区为中心的非常活跃的研究小组。自1999年4月起，我们每月举办一次公开研讨会“神乐坂分析研讨会”，共进行三场讲座。我们还在1999年11月召开了一次国际会议。这个研究项目的目的被描述为“准备泛函分析和实分析，以便为偏微分方程，特别是非线性方程的研究提供必要的工具”。沿着这条线，不仅该项目的注册研究员、与他们一起工作的研究生以及研讨会和国际会议的许多参加者都取得了大量成果。在他的书中，小松发现了一些关于超函数卷积等的新结果。冈泽将线性演化方程和非线性半群的抽象理论应用到具体的方程中，例如Ginzburg-Landau方程。他还建立了线性算子的对数log A的一般理论。Hayashi和Kato研究的非线性偏微分方程主要是演化型的。 Hayashi 证明了许多重要方程解的存在性，并分析了它们在时间趋于无穷大时的渐近行为。加藤建立了解的解析性。中村考虑了弹性体的反问题并将其应用到层析成像中。宫内得到了用实解析方法定义的函数空间的基本结果，在微分方程理论中变得越来越重要。。