On the multisummability of divergent solutions of linear partial differential equations with constant coefficients

Abstract

We consider the Cauchy problem for general linear partial differential equations in two complex variables with constant coefficients. We obtain necessary and sufficient conditions for the multisummability of formal solutions in terms of analytic continuation properties and growth estimates of the Cauchy data.

Analytic Solutions of Moment Partial Differential Equations with Constant Coefficients

“…We also introduce the concept of moment pseudodi¤erential operators, which generalise the pseudodi¤erential operators defined in [15,16]. …”

“…To avoid this inconvenience we shall choose some special solution which is called the normalised formal solution (see also Balser [3] and Michalik [15]). To this end we factorise the moment di¤erential operator Pðq m 1 ; t ; q m 2 ; z Þ as follows…”

“…; n À 1). We also give the necessary corrections of our previous papers [15,16,17] concerning summability of formal solutions of linear (fractional) PDEs.…”

“…The present paper is a generalisation of [15,17], where the characterisation of multisummable solutions of homogeneous and inhomogeneous linear PDEs with constant coe‰cients was given. The inspiration for our study was the paper of Balser and Yoshino [8], where the notion of moment di¤erentiation was introduced and the Gevrey order of formal solutions of general inhomogeneous linear moment PDEs with constant coe‰cients was determined.…”

“…The inspiration for our study was the paper of Balser and Yoshino [8], where the notion of moment di¤erentiation was introduced and the Gevrey order of formal solutions of general inhomogeneous linear moment PDEs with constant coe‰cients was determined. Finally, let us point out that the summability of formal solutions of homogeneous linear PDEs with constant coe‰cients was studied by Balser [1,3], Balser and Miyake [7], Ichinobe [9], Lutz, Miyake and Schäfke [10], Malek [12], Michalik [13,15,16] and Miyake [19]. The inhomogeneous case was investigated by Balser [4], Balser and Loday-Richaud [6], Balser, Duval and Malek [5] and Michalik [14,17].…”