Topics in analytic theory of partial differential equations

Abstract: The thesis consists of three papers I, II III devoted to some topics in analytic theory of partial differential equations. The main paper is II. In this paper we study a version of the classical problem on the convergence of formal solutions of systems of partial differential equations. A necessary and sufficient condition for the convergence of a given formal solution (found by any method) is proved. This convergence criterion applies to systems of partial differential equations (possibly, nonlinear) solved for the highest-order derivatives or, which is most important,"almost solved for the highest-order derivatives".In the paper I, systems of linear partial differential equations with constant coefficients are considered. The space of formal and analytic solutions of such systems are described by elementary algebraic methods. The Hilbert and Hilbert-Samuel polynomials for systems of linear partial differential equations with constant coefficients are defined.In the paper III, systems of linear partial differential equations with analytic coefficients are considered. Suppose the coefficients of a system are defined in a domain U in n-dimensional complex space. We study the space of germs of formal and analytic solutions of the system at a point u of the domain U. We discuss the following questions. 1) How to pose "proper" initial conditions for formal and analytic solutions for the system. 2) How depend dimensions of the space of k-jets of germs of formal and analytic solutions of the system at a point u in the domain U on the positive integer k and the point u.