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An existence result for linear partial differential equations with $C^\infty$ coefficients in an algebra of generalized functions


Author: Todor Todorov
Journal: Trans. Amer. Math. Soc. 348 (1996), 673-689
MSC (1991): Primary 35A05, 35D05, 35E20, 46S10, 46S20
DOI: https://doi.org/10.1090/S0002-9947-96-01450-X
MathSciNet review: 1316863
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Abstract: We prove the existence of solutions for essentially all linear partial differential equations with $C^\infty$-coefficients in an algebra of generalized functions, defined in the paper. In particular, we show that H. Lewy’s equation has solutions whenever its right-hand side is a classical $C^\infty$-function.


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Additional Information

Todor Todorov
Affiliation: Department of Mathematics, California Polytechnic State University, San Luis Obispo, California 93407
Email: ttodorov@oboe.calpoly.edu

Keywords: Existence of generalized solutions, Schwartz distribution, nonstandard function, nonstandard functional analysis, nonstandard extension, transfer principle, saturation principle
Received by editor(s): March 7, 1994
Received by editor(s) in revised form: November 28, 1994
Article copyright: © Copyright 1996 American Mathematical Society