On complex singularity analysis for some linear partial differential equations in C3

Abstract

We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variableswhose coefficients are holomorphic on some polydisc in C2 outside some singular set Theta. The coefficients are written as linearcombinations of powers of a solution X of some first-order nonlinear partial differential equation following an idea, we haveinitiated in a previous work (Malek and Stenger 2011). The solutions Y are shown to develop singularities along Theta with estimatesof exponential type depending on the growth's rate of X near the singular set.We construct these solutions with the help of seriesof functions with infinitely many variables which involve derivatives of all orders of X in one variable. Convergence and boundsestimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that containsdifferential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actuallyhave solutions in some Banach spaces of formal power series.