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.References
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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.
- 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
Dedicated: Dedicated to V. P. Zaharjuta on the occasion of his 60th birthday