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

 

Nonstandard solvability for linear operators between sections of vector bundles


Author: Hiroshi Akiyama
Journal: Proc. Amer. Math. Soc. 128 (2000), 2129-2135
MSC (1991): Primary 46S20, 03H05, 35D05, 47B38; Secondary 46F10, 47A50, 58G99.
Published electronically: February 23, 2000
MathSciNet review: 1653401
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Abstract | References | Similar Articles | Additional Information

Abstract:

Given a certain kind of linear operator $A$ (possibly a differential operator or a properly supported pseudodifferential operator) between sections of Hermitian vector bundles over a Riemannian manifold, a necessary and sufficient condition is obtained for the operator $A$ to be solvable in a class of nonstandard sections in a generalized sense of weak solutions. The existence of a fundamental-solution-like internal section is established in the solvable case.


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Additional Information

Hiroshi Akiyama
Affiliation: Department of Applied Mathematics, Faculty of Engineering, Shizuoka University, Hamamatsu 432-8561, Japan
Email: tshakiy@eng.shizuoka.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05227-8
PII: S 0002-9939(00)05227-8
Keywords: Nonstandard analysis, nonstandard extension, transfer principle, saturation principle, hyperfinite-dimensional internal vector space, Hermitian vector bundle, generalized section
Received by editor(s): March 30, 1998
Received by editor(s) in revised form: September 4, 1998
Published electronically: February 23, 2000
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2000 American Mathematical Society