Asymptotic Expansion of Singular Solutions and the Characteristic Polygon of Linear Partial Differential Equations in the Complex Domain

Abstract

Let P(z, ∂) be a linear partial differential operator with holomorphic coefficients in a neighborhood Ω of z = 0 in ℂ\_d\_+1. Consider the equation P(z, ∂)u(z) = f(z), where u(z) admits singularities on the surface K = {\_z\_0 = 0} and f(z) has an asymptotic expansion of Gevrey type with respect to \_z\_0 as \_z\_0 → 0. We study the possibility of asymptotic expansion of u(z). We define the characteristic polygon of P(z, ∂) with respect to K and characteristic indices. We discuss the behavior of u(z) in a neighborhood of K, by using these notions. The main result is a generalization of that in \[6].