Propagation of Analytic and Differentlable Singularities for Solutions of Partial Differential Equations

Abstract

In the first part of this paper, we study propagation of singularities for solutions of an analytic pseudo-differential equation, the characteristic set of which is a regular involutive manifold. There exists a natural foliation of this manifold, and (theorem 1.7) the analytic singular spectrum of a solution is a union of leaves—this is a joint work P. Schapira. The same result holds for differentiable singular spectrum (wave front) assuming Levi's condition. A similar result had been proved by J. Sjostrand for C°° pseudo-differential equations, but with additional assumptions on complex characteristics [14]. We prove also microlocal solvability for our operators (theorem 1.6). Complete proofs are given in [6] and [4]. In the second part of this paper (§ 4 to 7), we study operators the characteristic set of which contains a regular involutive manifold. The main result (theorem 5.6) is an analogue, for singular spectrum, to Holmgren theorem for support. Applications to propagation of analytic singularities and to uniqueness of Cauchy problem are given. Besides arguments used in the first part, we use results of Kashiwara [11], [12] and direct infinitesimal geometry. A more detailed exposition is given in [5].