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.

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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