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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Parameter dependence of solutions of partial differential equations in spaces of real analytic functions


Authors: José Bonet and Pawel Domanski
Journal: Proc. Amer. Math. Soc. 129 (2001), 495-503
MSC (2000): Primary 35B30, 46E40, 46A63
Published electronically: August 28, 2000
MathSciNet review: 1800237
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Abstract:

Let $\Omega \subseteq \mathbb{R}^n$ be an open set and let $\mbox{A$(\Omega)$ }$ denote the class of real analytic functions on $\Omega$. It is proved that for every surjective linear partial differential operator $P(D,x):\mbox{A$(\Omega)$ }\to\mbox{A$(\Omega)$ }$ and every family $(f_ \lambda)\subseteq \mbox{A$(\Omega)$ }$depending holomorphically on $\lambda\in \mathbb{C}^m$ there is a solution family $(u_ \lambda)\subseteq\mbox{A$(\Omega)$ }$ depending on $\lambda$ in the same way such that \begin{equation*}P(D,x)u_ \lambda=f_ \lambda, \qquad \mbox{for } \lambda\in \mathbb{C}^m. \end{equation*} The result is a consequence of a characterization of Fréchet spaces $E$such that the class of ``weakly'' real analytic $E$-valued functions coincides with the analogous class defined via Taylor series. An example shows that the analogous assertions need not be valid if $\mathbb{C}^m$ is replaced by another set.


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Additional Information

José Bonet
Affiliation: Universidad Politécnica de Valencia, Departamento de Matemática Aplicada, E.T.S. Arquitectura, E-46071 Valencia, Spain
Email: jbonet@pleiades.upv.es

Pawel Domanski
Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University Poznań, Matejki 48/49, 60-769 Poznań, Poland
Email: domanski@amu.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05867-6
PII: S 0002-9939(00)05867-6
Keywords: Space of real analytic functions, linear partial differential operator, vector valued real analytic functions, Fr\'echet space, LB-space, surjectivity of convolution operators, parameter dependence of solutions
Received by editor(s): April 28, 1999
Published electronically: August 28, 2000
Additional Notes: The research of the first author was partially supported by DGICYT, grant no. PB 97-0333. The research of the second author was partially supported by the Committee of Scientific Research (KBN), Poland, grant 2 P03A 051 15.
Dedicated: Dedicated to V. P. Zaharjuta on the occasion of his 60th birthday
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society