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An application of the fundamental principle of Ehrenpreis to the existence of global Gevrey solutions of linear differential equations. (English) Zbl 0624.35011

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

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
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