On Exceptional Cases of Cauchy Problems for Fuchslan Partial Differential Operators

Cauchy problems for Fuchsian partial differential operators have been investigated by several authors, in the space of holomorphic functions ([2], [6], [9]) or in the space of C°° functions ([8], [10]). To guarantee the unique solvability of the problem, they impose a certain condition, the condition (C) in Proposition 1.1, on the indicial polynomial. This condition is equivalent to that the Taylor expansion of the solution is uniquely determined by the Cauchy data. In the space of C°° functions, the procedure to solve the Cauchy problem is divided into two steps: First, to find the Taylor expansion of the solution and then to solve the fiat Cauchy problem. As for the flat Cauchy problem, the unique solvability is independent of the condition (C). Thus, under the assumption that the flat Cauchy problem is uniquely solvable, we want to consider the case when the condition (C) is not satisfied, by extending the space of admissible solutions. In this article, for simplicity, we shall consider only Fuchsian equations in the space of C°° functions. Our arguments are, however, essentially how to get formal solutions, hence similar arguments go well in various other situations: Fuchsian equations in the space of holomorphic functions ([2], [6], [9]), parabolic equations of "Fuchs type" ([1], [7]), etc. Our program is as follows. In Section 1, we state the main result. After reviewing some basic results on Fuchsian equations in Section 2, we prove the main result in Section 3. Finally, in Section 4, we treat Fuchsian systems.