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

Abstract

Let $P(z,\partial )$ be a linear partial differential operator with holomorphic coefficients in a neighborhood $\Omega$ of $z=0$ in $C^{d+1}$. Consider the equation $P(z,\partial )u(z)=f(z)$, 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 $z_0\to 0$. We study the possibility of asymptotic expansion of $u(z)$. We define the characteristic polygon of $P(z,\partial )$ 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].