# Singular Solutions with Asymptotic Expansion of Linear Partial Differential Equations in the Complex Domain

### Sunao Ōuchi

Sophia University, Tokyo, Japan

## Abstract

We consider a linear partial differential equation with holomorphic coefficients in a neighbourhood of *z*=0 in _ℂ_d+1, *P*(*z*, ∂) *u*(*z*) = *f*(*z*), where *u*(*z*) and *f*(*z*) admit singularities on the surface *K*={_z_0=0}. Our main result is the following: For the operator *P* we define an exponent γ* called the minimal irregularity of *K* and show that if *u*(*z*) grows at most exponentially with exponent γ* as _z_0 tends to 0 and if *f*(*z*) has a Gevrey type expansion of exponent γ* with respect to _z_0, then *u*(*z*) also has the same one.

## Cite this article

Sunao Ōuchi, Singular Solutions with Asymptotic Expansion of Linear Partial Differential Equations in the Complex Domain. Publ. Res. Inst. Math. Sci. 34 (1998), no. 4, pp. 291–311

DOI 10.2977/PRIMS/1195144627