On complex singularity analysis for some linear partial differential equations in $\mathbb{C}^3$

TL;DR

This paper analyzes the singularity structure of solutions to certain linear PDEs in three complex variables, using series with infinitely many variables and fixed point methods to establish existence and bounds.

Contribution

It introduces a novel approach combining series of infinitely many variables and fixed point techniques to study singularities in complex PDE solutions.

Findings

Solutions develop exponential-type singularities along the variety.

Constructs solutions using series involving derivatives of all orders of a nonlinear PDE solution.

Establishes convergence and bounds via a majorant series and fixed point in Banach spaces.

Abstract

We investigate the existence of local holomorphic solutions $Y$ of linear partial differential equations in three complex variables whose coefficients are singular along an analytic variety $\Theta$ in $\mathbb{C}^{2}$. The coefficients are written as linear combinations of powers of a solution $X$ of some first order nonlinear partial differential equation following an idea we have initiated in a previous work \cite{mast}. The solutions $Y$ are shown to develop singularities along $\Theta$ with estimates of exponential type depending on the growth's rate of $X$ near the singular variety. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders of $X$ in one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series.