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March 2011 Real analytic parameter dependence of solutions of differential equations over Roumieu classes
Paweł Domański
Funct. Approx. Comment. Math. 44(1): 79-109 (March 2011). DOI: 10.7169/facm/1301497748

Abstract

We consider the problem of real analytic parameter dependence of solutions of the linear partial differential equation $P(D)u=f$, i.e., the question if for every family $(f_\lambda)\subseteq\mathscr_{\{\omega\}}(\Omega)$ of ultradifferentiable functions of Roumieu type (in particular, of real analytic functions or of functions from Gevreyclasses) depending in a real analytic way on $\lambda\in U$, $U$ areal analytic manifold, there is a family of solutions$(u_\lambda)\subseteq \mathscr_{\{\omega\}}(\Omega)$ also depending analytically on$\lambda$ such that$$P(D)u_\lambda=f_\lambda \text{for every $\lambda\in U$},$$where $\Om\subseteq \mathbb{R}^d$ an open set. We solve the problem formany types of differential operators following a similar method asin the earlier paper of the same author for operators acting onspaces of distributions. We show for an operator $P(D)$ on thespace of real analytic functions $\mathscr{A}(\Omega)$, $\Omega \subseteq\mathbb{R}^d$ open convex, that it has real analytic parameter dependenceif and only if its principal part $P_p(D)$ has a continuous linearright inverse on the space $C^\infty(\Omega)$ (or, equivalently,on $\mathscr{D}'(\Omega)$). In particular, the property does not depend on the setof parameters $U$. Surprisingly, in all solved non-quasianalyticcases, it follows that the solution is positive if and only if$P(D)$ has a linear continuous right inverse.

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Paweł Domański. "Real analytic parameter dependence of solutions of differential equations over Roumieu classes." Funct. Approx. Comment. Math. 44 (1) 79 - 109, March 2011. https://doi.org/10.7169/facm/1301497748

Information

Published: March 2011
First available in Project Euclid: 30 March 2011

zbMATH: 1221.35050
MathSciNet: MR2807900
Digital Object Identifier: 10.7169/facm/1301497748

Subjects:
Primary: 35B30, 46E10
Secondary: 32U05, 35E20, 46A13, 46A63, 46F05, 46M18

Rights: Copyright © 2011 Adam Mickiewicz University

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Vol.44 • No. 1 • March 2011
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