Asymptotic Expansion of Singular Solutions and the Characteristic Polygon of Linear Partial Differential Equations in the Complex Domain
Sunao Ōuchi · Publications of the Research Institute for Mathematical Sciences · 2000
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].