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Nonstandard solvability for linear operators between sections of vector bundles
Author(s):
Hiroshi
Akiyama
Journal:
Proc. Amer. Math. Soc.
128
(2000),
2129-2135.
MSC (1991):
Primary 46S20, 03H05, 35D05, 47B38;
Secondary 46F10, 47A50, 58G99.
Posted:
February 23, 2000
MathSciNet review:
1653401
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Abstract:
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.
References:
- 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. Lindstr
 m, 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
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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:
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
Posted:
February 23, 2000
Communicated by:
Carl G. Jockusch,~Jr.
Copyright of article:
Copyright
2000,
American Mathematical Society
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