Gevrey regularity and summability of the formal solutions in time of some inhomogeneous linear systems of partial differential equations
[Régularité Gevrey et sommabilité des solutions formelles en temps de certains systèmes linéaires non homogènes d’équations aux dérivées partielles]
Remy, Pascal1
1Laboratoire de Mathématiques de Versailles, UVSQ (Paris-Saclay) & CNRS (UMR 8100), 45 avenue des Etats-Unis, 78035 Versailles cedex (France)
Annales de l'Institut Fourier, Online first, 65 p.

In this article, we investigate the Gevrey regularity and the summability of the formal solutions in time of some inhomogeneous linear systems of partial differential equations with analytic coefficients at the origin of $\mathbb{C}^2$. Given such a system, we first prove that the Gevrey regularity of its formal solutions follows a noteworthy dichotomy with respect to the $s$-Gevrey regularity of the inhomogeneity: for $s\ge s_0$ with $s_0$ a nonnegative rational number entirely determined by a convenient Newton polygon attached to the given system, the solutions inherit the Gevrey regularity of the inhomogeneity; for $s<s_0$, if any exists, the solutions keep the $s_0$-Gevrey regularity defined by the structure of the initial system. Then, assuming $s_0>0$ and some convenient additional assumptions on the system, we give a necessary and sufficient condition for the $1/s_0$-summability of the formal solutions in a given direction. Several examples illustrate these results.

Dans cet article, nous nous intéressons à la régularité Gevrey et à la sommabilité des solutions formelles en temps de certains systèmes linéaires non homogènes d’équations aux dérivées partielles à coefficients analytiques à l’origine de $\mathbb{C}^2$. Etant donné un tel système, nous montrons tout d’abord que la régularité Gevrey de ses solutions formelles suit une remarquable dichotomie liée à la régularité Gevrey-$s$ de son second membre : lorsque $s\ge s_0$, où $s_0$ est un nombre rationnel positif ou nul entièrement déterminé par le polygone de Newton associé au système, les solutions formelles héritent de cette régularité Gevrey-$s$ ; lorsque $s<s_0$, si une telle valeur existe, les solutions formelles conservent la régularité Gevrey-$s_0$ définie par la structure même du système. En supposant ensuite $s_0>0$ et en ajoutant quelques hypothèses supplémentaires sur le système, nous donnons une condition nécessaire et suffisante pour que ses solutions formelles soient $1/s_0$-sommables dans une direction donnée. Plusieurs exemples viennent illustrer ces différents résultats

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DOI : 10.5802/aif.3792
Classification : 35C10, 35C20, 35G35, 40B05
Keywords: Gevrey regularity, Summability, Inhomogeneous linear partial differential equation, Inhomogeneous linear system, Formal power series, Divergent power series, Newton polygon
Mots-clés : Régularité Gevrey, sommabilité, équation aux dérivées partielles linéaire non homogène, système linéaire non homogège, série entière formelle, série divergente, polygône de Newton

Remy, Pascal  1

1 Laboratoire de Mathématiques de Versailles, UVSQ (Paris-Saclay) & CNRS (UMR 8100), 45 avenue des Etats-Unis, 78035 Versailles cedex (France) 
Remy, Pascal. Gevrey regularity and summability of the formal solutions in time of some inhomogeneous linear systems of partial differential equations. Annales de l'Institut Fourier, Online first, 65 p.
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