Boundary Value Problems and Index Theorem for D-Modules

本研究的目的是推广Dx-模边值问题(即微分方程组的边值问题)中的Schapira和Schneiders椭圆对理论(源于Kashiwara关于可构造层指数定理的工作和Kashiwara和Schapira关于层的微局部研究)。在这个项目中，我们发现了Dx-模的边值问题的一个自然的公式，在这个公式中，我们可以明确边值问题的“解复”的概念。本研究的下一步是证明该解的复形在某些椭圆性条件下是有限的，并用它的“特征圈”来刻画它的指标。这些将留待以后的进一步研究。除了以上的研究，在本研究项目期间，我们做了以下工作：(1)从微局部的角度推广Bochner微分方程解的延拓定理；(2)Dx-模在边界上从正侧微双曲的边值问题；(3)Deslaurier和Dubuc关于测度的无限卷积积的一个定理的推广；(4)半线性椭圆型方程弱解的正则性；(5)常系数双曲型微分方程解的缺陷性。(6)常系数薛定谔方程基本解的完整性质。

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.