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Boundary Value Problems and Index Theorem for D-Modules

D 模的边值问题和指数定理

基本信息

批准号：
12640172
负责人：
UCHIDA Motoo
金额：
$ 2.3万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

项目摘要

The purpose of this research project is to generalize the theory of elliptic pairs of Schapira and Schneiders (which originated in the work of Kashiwara on index theorem of constructible sheaves and the work of Kashiwara and Schapira on microlocal study of sheaves) in the boundary value problems for Dx-modules (i.e., for systems of differential equations). In this project, we found a natural formulation of boundary value problems for Dx-modules in which we can make clear the notion of "solution complex" of boonadry value problems. The next step of this research is to prove the finiteness of this solution complex on some ellipticity condition of boundary value problems and to describe its index in terms of its "characteristic cycle". These shall be left to the further research hereafter.Apart from the research above, we have made the following studies during the term of this research project.(1) Generalization of Bochner's extension theorem for solutions of differential equations from a microlocal point of view.(2) Boundary value problems for Dx-modules which are micro-hyperbolic at the boundary from the positive side.(3) Generalization of a theorem of Deslauriers and Dubuc on infinite convolution product of measures.(4) Regularity of weak solutions of semilinear elliptic differential equations.(5) Lacunas of fundamental solutions of hyperbolic differential operators with constant coefficients.(6) Holonomic character of the fundamental solutions of Schrodinger-type differential equations with constant coefficients.

本研究项目的目的是将Schapira和Schneiders的椭圆对理论（起源于Kashiwara关于可构造层的指标定理的工作以及Kashiwara和Schapira关于层的微局部研究的工作）推广到Dx-模的边值问题（即，对于微分方程系统）。在这个项目中，我们发现了Dx-模边值问题的一个自然形式，在这个形式中我们可以明确Boonadry值问题的“解复形”的概念。本文的下一步工作是证明该解复形在某些椭圆性条件下的有限性，并用其“特征圈”来描述其指数。除以上研究外，在本课题研究期间，我们还做了以下研究工作。(1)从微局部观点推广微分方程解的Bochner扩张定理。(2)在正边界上微双曲的Dx-模的边值问题。(3)Deslauriers和Dubuc关于测度的无限卷积积的一个定理的推广。(4)半线性椭圆型微分方程弱解的正则性。(5)常系数双曲型微分算子基本解的缺项。(6)常系数Schrodinger型微分方程基本解的完整性。