Functional and Real Analysis for Partial Differential Equations

在过去三年中，我们在助学金的帮助下取得的最大成就是，我们在东京科学大学神乐坂校区建立了一个非常活跃的研究小组。从1999年4月开始，我们每月举办一次公开研讨会“神乐坂分析研讨会”，其中有三场讲座。我们还在1999年11月举行了一次国际会议。这个研究项目的目的被描述为“准备泛函分析和实分析，以便为偏微分方程的研究提供必要的工具，特别是非线性方程”。沿着这条路线，不仅项目的注册研究者，而且与他们一起工作的研究生以及许多研讨会和国际会议的与会者都取得了大量成果。除了对其著作的英译本进行修订和更正外，小松还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于具体方程，如金兹堡-朗道方程。他还建立了线性算子的对数loga的一般理论。Hayashi和Kato主要研究演化型的非线性偏微分方程。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时的渐近行为。加藤建立了解的解析性。Nakamura考虑了弹性体的逆问题，并将其应用于层析成像。Miyachi在实解析方法定义的函数空间上得到了一些基本结果，这些结果在微分方程理论中越来越重要。

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.