Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On the dependence of analytic solutions of partial differential equations on the right-hand side


Author: Siegfried Momm
Journal: Trans. Amer. Math. Soc. 345 (1994), 729-752
MSC: Primary 46E10; Secondary 32F05, 35B30, 35E10
MathSciNet review: 1254192
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Abstract: Given a nonzero polynomial $ P(z) = \sum\nolimits_{\vert\alpha \vert \leq m} {{a_\alpha }{z^\alpha }} $ on $ {\mathbb{C}^N}$, Martineau proved in the 1960s that for each convex domain G of $ {\mathbb{C}^N}$ the partial differential operator $ P(D)f = \sum\nolimits_{\vert\alpha \vert \leq m} {{a_\alpha }{f^{(\alpha )}}} $ acting on the Fréchet space $ A(G)$ of all analytic functions on G is surjective. In the present paper it is investigated whether solutions f of the equation $ P(D)f = g$ can be chosen as $ f = R(g)$ with a continuous linear operator $ R:A(G) \to A(G)$. For bounded G we give a necessary and sufficient condition for the existence of such an R.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1254192-7
Article copyright: © Copyright 1994 American Mathematical Society