On Complex Singularity Analysis for Some Linear Partial Differential Equations in ℂ 3

Abstract

We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc in ${ℂ}^2$ outside some singular set $\mathrm{\Theta }$. 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 (Malek and Stenger 2011). The solutions Y are shown to develop singularities along $\mathrm{\Theta }$ with estimates of exponential type depending on the growth's rate of X near the singular set. 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.