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

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.