Divergence Property of Formal Solutions for Singular First Order Linear Partial Differential Equations

Abstract

This paper is concerned with the study of the convergence and the divergence of formal power series solutions of the following first order singular linear partial differential equation with holomorphic coefficients at the origin: P(x,D)u(x) = \sum^d_{\iota=1}a_\iota(x)D_\iota u(x)+b(x)u(x) = f(x), with f(x) holomorphic at the origin. Here the equation is said to be singular if a_\iota(0)=0 ( \iota = 1, ...,d ). In this case, it is known that under the so-called Poincare condition, if \{a_\iota(x)\}^d_{\iota=1 } generates a simple ideal, every formal solution is convergent. However if we remove these conditions, we shall see that the formal solution, if it exists, may be divergent. More precisely, we will characterize the rate of divergence of formal solutions via Gevrey order of formal solutions determined by a Newton Polyhedron, a generalization of Newton Polygon which is familiar in the study of ordinary differential equations with an irregular singular point.