On Complex Singularity Analysis for Some Linear Partial Differential Equations in ℂ3

Abstract EN

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 ℂ2 outside some singular set Θ.

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 Θ 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.