权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

ALGEBRAIC-ANALYTICAL STUDY OF PSEUDO-DIFFERENTIAL AND CONVOLUTION ERUATIONS

伪微分和卷积方程的代数分析研究

基本信息

批准号：
11640153
负责人：
ISHIMURA Ryuichi
金额：
$ 2.18万
依托单位：
CHIBA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640153/
关键词：
Algebraic analysis Convolution equations Pseudo-differential equations Partial differential equations Differential equations of infinite ordre Micro-local study of sheaves Canchy problem 畳込み方程式代数解析

项目摘要

The aims of this research were as follows :[1] The algebraic-analytical study of convolution equations in the complex domains, using micro-local study of sheaves.[2] Study of the Fabry-Ehrenpreis-Kawai Theorem, applying the theory of analytical continuation of solutions to convolution equations.[3] The extension of the theory of the Cauchy problem for micro-differential equationts in the complex domains to the pseudo-differential case.For the problem [1] and [2], at first, we constructed good examples of convolution equations with elliptic codition in the several variables. And also, for the problem of the analytic continuation of the holomorphic solutions to the homogeneous convolution equation in the complex domains, we introduced its characteristic sets to be the natural extension of the case of usual constant coefficients linear partial differential equations and we could present the expicit form of the domains to which any solution is continued analytically, using the characteristic set. In particular, this resolves almost completely the problem of the analytic continuation to the infinite ordre differential-difference equations which are important examples of the functional-differential equaion. In the case of tube domains, one proved that, in a natural condition, the characteristic set coincides with the accumulating directions at infinity of the zeros of the symbol. However, for [3], we did not yet succeed to get an general theory. But we are now studying the problem using the sheaf theoritical study by means of inductive limits which is in the course of developpement.

本文的主要研究内容如下：[1]利用层的微观局部研究方法，对复域上卷积方程进行代数分析研究。[2]研究法布里-埃克塞特-卡瓦伊定理，将解的解析延拓理论应用于卷积方程。[3]将复域上微微分方程Cauchy问题的理论推广到伪微分情形，对于问题[1]和[2]，我们首先构造了几个具有椭圆条件的多元卷积方程的好例子。对于复域上齐次卷积方程全纯解的解析延拓问题，我们引入了其特征集，将其作为常系数线性偏微分方程情形的自然推广，并利用特征集给出了任意解在其上解析延拓的域的显式形式.特别地，这几乎完全解决了泛函微分方程的重要例子--无穷阶微分差分方程的解析延拓问题。在管域的情况下，证明了在自然条件下，特征集与符号的零点在无穷远处的积累方向一致。然而，对于[3]，我们还没有成功地得到一个普遍的理论。但我们目前正在利用处于发展过程中的归纳极限，用层理论研究这个问题。