The overdetermined Cauchy problem

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