A Maillet Type Theorem for First Order Singular Nonlinear Partial Differential Equations

Abstract

We shall study the first order singular nonlinear partial differential equation of the form f(x, u(x), ∂_x__u_(x)) = 0 with u(0) = 0, where x ∈ ℂ_n_ and f(x, u, ξ) is holomorphic in a neighborhood of the origin. This equation is said to be singular if f(0, 0, ξ) ≡ 0 for all ξ ∈ ℂ_n_. The purpose of this paper is to study the Maillet type theorem which means to determine the Gevrey order of a formal power series solution which may diverge.