## Abstract and Applied Analysis

### 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 394564, 30 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393512115

Digital Object Identifier
doi:10.1155/2013/394564

Mathematical Reviews number (MathSciNet)
MR3126755

Zentralblatt MATH identifier
1295.35013

#### Citation

Lastra, A.; Malek, S.; Stenger, C. On Complex Singularity Analysis for Some Linear Partial Differential Equations in ${ℂ}^{3}$. Abstr. Appl. Anal. 2013 (2013), Article ID 394564, 30 pages. doi:10.1155/2013/394564. https://projecteuclid.org/euclid.aaa/1393512115

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