Boundary Value Problems and Index Theorem for D-Modules

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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.