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

Abstract

Consider the linear partial differential equation P(z, ∂_z_)u(z) = f(z) in ℂ_d_+1 , where f(z) is not holomorphic on K = {_z_0 = 0}, but it has an asymptotic expansion with respect to _z_0 as z_0 → 0 in some sectorial region. We show under some conditions on P(z, ∂_z) that there exists a solution u(z) which has an asymptotic expansion of the same type as that of f(z).