Solutions holomorphes locale et globale pour un opérateur différentiel linéaire à plusieurs variables Fuchsiennes

Abstract

We consider linear partial differential equations with several Fuchsian variables in the sense of M.S. Baouendi and C. Goulaouic [1]. For a holomorphic Fuchsian operator with holomorphic Fuchsian principal part, we prove existence and uniqueness of a holomorphic local solution. Our theorem generalizes the results of ([3, 1, 11]), precises the one of [4] and reduces the proof of their theorems to the proof of the fixed-point theorem. For a holomorphic Fuchsian operator with constant Fuchsian principal part, we establish the existence and uniqueness of a holomorphic global solution. Our aim is to simplify its proof. The methods of proof are based on the application of the fixed-point theorem in some Banach spaces defined by majorant functions that are suitable to this kind of equations.