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 $f(x,u(x),{\partial }_{x}u(x))=0$ with $u(0)=0$, where $x\in C^n$ and $f(x,u,\xi )$ is holomorphic in a neighborhood of the origin. This equation is said to be singular if $f(0,0,\xi )\equiv 0$ for all $\xi \in C^n$. 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.