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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Parameter dependence of solutions of partial differential equations in spaces of real analytic functions
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by José Bonet and Paweł Domański PDF
Proc. Amer. Math. Soc. 129 (2001), 495-503 Request permission

Abstract:

Let $\Omega \subseteq \mathbb {R}^n$ be an open set and let $\mathrm {A}(\Omega )$ denote the class of real analytic functions on $\Omega$. It is proved that for every surjective linear partial differential operator $P(D,x):\mathrm {A}(\Omega )\to \mathrm {A}(\Omega )$ and every family $(f_ \lambda )\subseteq \mathrm {A}(\Omega )$ depending holomorphically on $\lambda \in \mathbb {C}^m$ there is a solution family $(u_ \lambda )\subseteq \mathrm {A}(\Omega )$ depending on $\lambda$ in the same way such that \begin{equation*} P(D,x)u_ \lambda =f_\lambda , \qquad \mathrm {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
  • ORCID: 0000-0002-9096-6380
  • Email: jbonet@pleiades.upv.es
  • Paweł Domański
  • Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University Poznań, Matejki 48/49, 60-769 Poznań, Poland
  • Email: domanski@amu.edu.pl
  • 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
  • © Copyright 2000 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 129 (2001), 495-503
  • MSC (2000): Primary 35B30, 46E40, 46A63
  • DOI: https://doi.org/10.1090/S0002-9939-00-05867-6
  • MathSciNet review: 1800237