## Overdetermined hyperbolic systems on l.e. convex sets.(English)Zbl 0736.35019

This article develops a general theory of systems of linear partial differential equations with constant coefficients related with relatively compact convex closed subsets of a half space. Relation between extension modules for the associated $${\mathcal P}$$-module $$M$$ of differential operators and local cohomology groups is studied with coefficient sheaves such as distributions or $$C^ \infty$$ functions, and interpreted from the viewpoint of the solvability of the Cauchy problem for systems. Results generalizing the unique continuation property of solutions for a single equation, or Hartogs type continuation theorem of homogeneous solutions are given. These are generalizations of the works of Ehrenpreis, Malgrange, Palamodov related with compact convex sets in the whole space. They generalize works of Malgrange for the same situation by the introduction of hyperbolicity for extension module. A similar study in hyperfunction solutions was made by A. Kaneko [J. Math. Soc. Japan 26, 92-123 (1974; Zbl 0265.35010)].
Reviewer: A.Kaneko (Komaba)

### MSC:

 35E20 General theory of PDEs and systems of PDEs with constant coefficients 35B60 Continuation and prolongation of solutions to PDEs 35N05 Overdetermined systems of PDEs with constant coefficients

Zbl 0265.35010
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### References:

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