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.

DOI:
https://doi.org/10.1090/S0002-9939-00-05227-8

Published electronically:
February 23, 2000

MathSciNet review:
1653401

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Abstract | References | Similar Articles | Additional Information

Given a certain kind of linear operator (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 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.

**1.**H. Akiyama,*Nonstandard representations of generalized sections of vector bundles*, Osaka J. Math.,**32**(1995), 817-822. MR**97a:46113****2.**V. Guillemin and S. Sternberg,*Geometric Asymptotics*, rev. ed., Mathematical surveys and monographs No. 14, Amer. Math. Soc., Providence, R.I., 1990. MR**58:24404****3.**A. E. Hurd and P. A. Loeb,*An Introduction to Nonstandard Real Analysis*, Academic Press, New York, 1985. MR**87d:03184****4.**T. Lindstrm,*An invitation to nonstandard analysis*, in ``Nonstandard Analysis and its Applications'' (N. Cutland, ed.), London Math. Soc. Student Texts 10, Cambridge Univ. Press, 1988, pp. 1-105. MR**89m:03060****5.**S. Sternberg,*Lectures on Differential Geometry*, 2nd ed., Chelsea, New York, 1983. MR**33:1797**; MR**88f:58001****6.**T. Todorov,*An existence result for a class of partial differential equations with smooth coefficients*, in ``Advances in Analysis, Probability and Mathematical Physics: Contributions of Nonstandard Analysis'' (S. Albeverio, W. A. J. Luxemburg, M. P. H. Wolff, eds.), Math. Appl. Vol. 314, Kluwer Acad. Publ., Dordrecht, 1995, pp. 107-121. MR**96m:35005****7.**T. Todorov,*An existence result for linear partial differenial equations with coefficients in an algebra of generalized functions*, Trans. Amer. Math. Soc.,**348**(1996), 673-689. MR**96m:35004**

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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:
https://doi.org/10.1090/S0002-9939-00-05227-8

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