Maillet-type theorem for some inhomogeneous linear systems of regular Gevrey-type partial differential equations

Abstract

In this article, we are interested in the Gevrey properties of the formal power series solutions in time of some inhomogeneous linear systems of regular Gevrey-type partial differential equations with analytic coefficients at the origin of C N + 1 . We systematically examine the cases where the inhomogeneity is s 0 -Gevrey for any s 0 ≥ 0 , in order to carefully distinguish the influence of the data (and their degree of regularity) from that of the system (and its structure). We thus prove that we have a noteworthy dichotomy with respect to a nonnegative real number s c fully determined by the Newton polygon associated with the given system: for any s 0 ≥ s c , the formal solutions inherit the s 0 -Gevrey regularity of the inhomogeneity; for any s 0 < s c , the formal solutions are generically s c -Gevrey. In the latter case, we give an explicit example in which the solution is s 0 ′ -Gevrey for no s 0 ′ < s c . As an illustration, we give several examples of concrete applications to classical physics problems, as well as to functional analysis problems such as the study of certain generating functions.