Zampieri, Giuseppe An application of the fundamental principle of Ehrenpreis to the existence of global Gevrey solutions of linear differential equations. (English) Zbl 0624.35011 Boll. Unione Mat. Ital., VI. Ser., B 5, 361-392 (1986). This article applies Ehrenpreis’ fundamental principle, in the form of Hörmander, to the problem of surjectivity P(D)\({\mathcal E}^{\{d\}}(\Omega)={\mathcal E}^{\{d\}}(\Omega)\), of a linear partial differential operator with constant coefficients in the space of Gevrey class functions on an open set \(\Omega \subset {\mathbb{R}}^ n\). It gives a sufficient condition described by a Phragmén-Lindelöf type principle on the complex characteristic variety V of P. It also gives some stability of this problem with respect to perturbation of the index d. It follows especially that if the surjectivity holds for \(d=1\) (i.e. for real analytic case), then it holds also for \(1\leq d<c\) with some constant c depending only on the multiplicity of the characteristics of P. This implies, combined with Hörmander’s example of indeterminate quadratic operator, that the existence of a gap region of the form \(| \eta | =O(| \xi |^{1/d})\), which was introduced as a sufficient condition by Cattabriga, is not always necessary. (In the case of convolution operator on \({\mathbb{R}}^ 1\), this condition is shown to be necessary and sufficient by Meise.) Reviewer: A.Kaneko Cited in 7 Documents MSC: 35E10 Convexity properties of solutions to PDEs with constant coefficients 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B65 Smoothness and regularity of solutions to PDEs Keywords:Ehrenpreis’ fundamental principle; surjectivity; constant coefficients; Gevrey class; Phragmén-Lindelöf type principle; complex characteristic variety; stability; perturbation of the index; indeterminate quadratic operator; gap region PDFBibTeX XMLCite \textit{G. Zampieri}, Boll. Unione Mat. Ital., VI. Ser., B 5, 361--392 (1986; Zbl 0624.35011)