The overdetermined Cauchy problem

Abstract

We consider the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients. In order to avoid the problem of the formal coherence of the initial data, which leads to the notion of formally non-characteristic introduced by Andreotti-Nacinovich in the 80's, we allow formal solutions (in the sense of Whitney) of the given system on any closed subset as initial data. The problem is then to find classical smooth solutions of the system, whose restrictions in the sense of Whitney are the given initial data. This gives a unifying point of view encompassing several different problems, ranging from questions of smoothness of the solutions, to the classical Cauchy problem, to comparison of formal to actual solutions, to Hartog's type phenomena

The overdetermined Cauchy problem in some classes of ultradifferentiable functions

“…A further useful reduction is to substitute to the consideration of M that of the P -modules P /p for the prime ideals p belonging to Ass(M). In several interesting cases, and namely in investigations concerning the Cauchy problem, this can be achieved by employing the following propositions (see [32] and [11]): …”

“…As in [11], we shall say that the pair (K 1 , K 2 ) is of evolution (and of causality, hyperbolic) in the class (U(K 1 ), F(K 2 );Ũ(K 2 )) for A(D) if (C.P.) has at least one solution (and at most one solution, one and only one solution respectively) for each pair of admissible data.…”

“…As in [11], we formulate the Cauchy problem for general pairs (K 1 , K 2 ) of closed subsets of R N with K 1 ⊂ K 2 :…”

“…In this paper we continue the study of the Cauchy problem for overdetermined systems of linear partial differential operators, following [32], [34], [10], [11].…”

Local radial Phragmén-Lindelöf estimates for plurisubharmonic functions on analytic varieties

“…His characterization was given in terms of global and also of local conditions of Phragmén-Lindelöf type for plurisubharmonic functions on the zero variety of the symbol P . Since then, it was shown in a number of papers that similar Phragmén-Lindelöf conditions on algebraic varieties can be used to characterize other properties of (systems of) such operators (see, e.g., Andreotti and Nacinovich [1], Boiti and Nacinovich [3], Braun, Meise, and Vogt [9], Franken and Meise [11], Kaneko [13], Meise, Taylor, and Vogt [15], Momm [18], Palamodov [20], Zampieri [24]). …”