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Synthetic research on differential equations

微分方程的综合研究

基本信息

批准号：
09304016
负责人：
IKAWA Mitsuru
金额：
$ 10.3万
依托单位：
Osaka University, Graduate School of Sciences
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

We studied differential equations, in the range from ordinary differential equations to partial differential equations and from linear ones to nonlinear ones. We cooperated with the Division of Functional Equations, Mathematical Scociety of Japan.As the result of a study including all the researchers of this field in Japan, we clarified the tendency of the forcecoming subjects. As one of the main results of our synthetic research, Atsushi Yagi published a monograph on evolution equations in Banach spaces. His theory made possible a systematic treatment of nonlinear problems in a form of evolution equations in Banach spaces. One point we want to emphasize is that his theory includes degenerate problems. As to another main subject of research, scattering theory, the Head of Investigaor is writing a monograph which represents the results of synthetic studies of partial differential equations related to scattering theory. Other subject, for instance, hypoellipticity of equations, by the studies in the space of analytic functions or of Gevrey class, the meanings of the conditions posed up to now has been made clear. In ordinary differential equations, by the studies of asymptotic behaviors of hypergeometric functions of congruent type made clear the structure of solutions. As to inverse problems of scattering problems, which are important even in the engineering, a theorem which guarantees the indentification of scatterers from scattering informations of a fixed enegy.We prepared the synthetic discussions on differential equations, but could not reach to results which sythesize the many subject of differential equations.

我们学习了微分方程，从常微分方程到偏微分方程，从线性方程到非线性方程。我们与日本数学会泛函方程部合作。作为一项包括日本该领域所有研究人员的研究结果，我们澄清了被迫成为受试者的趋势。作为我们综合研究的主要成果之一，Atsushi Yagi发表了关于Banach空间演化方程的专著。他的理论使得系统地处理巴拿赫空间中演化方程形式的非线性问题成为可能。我们要强调的一点是，他的理论包括退化问题。关于另一个主要的研究课题——散射理论，研究主任正在写一本专著，它代表了与散射理论有关的偏微分方程的综合研究结果。其他学科，例如方程的半椭圆性，通过对解析函数或格夫里类空间的研究，到目前为止所提出的条件的意义已经清楚了。在常微分方程中，通过对同余型超几何函数的渐近性质的研究，明确了解的结构。对于在工程中也很重要的散射逆问题，给出了从固定能量的散射信息中识别散射体的定理。我们准备了关于微分方程的综合讨论，但未能得出综合微分方程众多学科的结果。