JournalsprimsVol. 36, No. 4pp. 457–482

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

  • Sunao Ōuchi

    Sophia University, Tokyo, Japan
Asymptotic Expansion of Singular Solutions and the Characteristic Polygon of Linear Partial Differential Equations in the Complex Domain cover
Download PDF

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