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

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.