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

### Sunao Ōuchi

Sophia University, Tokyo, Japan

## 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].

## Cite this article

Sunao Ōuchi, Asymptotic Expansion of Singular Solutions and the Characteristic Polygon of Linear Partial Differential Equations in the Complex Domain. Publ. Res. Inst. Math. Sci. 36 (2000), no. 4, pp. 457–482

DOI 10.2977/PRIMS/1195142869