Functional and Real Analysis for Partial Differential Equations

在助学金的帮助下，我们在过去三年中取得的最大成就是，我们创建了一个以东京科学大学鹿儿町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--《鹿儿崎分析研讨会》，共举办三场讲座。我们还在1999年11月举行了一次国际会议。这项研究项目的目的被描述为“准备泛函分析和实数分析，以便为研究偏微分方程组，特别是研究非线性方程提供必要的工具”。在这方面，不仅该项目的注册研究人员，而且与他们一起工作的研究生以及研讨会和国际会议的许多与会者都取得了大量的结果。除了对他的书的英译本进行修改和更正外，小松还发现了一些关于超函数卷积等的新结果。冈泽将线性发展方程和非线性半群的抽象理论应用于具体的方程，如Ginzburg-Landau方程。他还建立了线性算子的对数对数A的一般理论。Hayashi和Kato主要研究发展型的非线性偏微分方程组。对于许多重要的方程，Hayashi证明了解的存在性，并分析了当时间趋于无穷大时解的渐近行为。Kato建立了解的解析性。Nakamura考虑了弹性体反问题并将其应用于层析成像。Miyachi在函数空间上获得了基本结果，这些函数空间由真实的解析方法定义，在微分方程组理论中变得越来越重要。

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.