Characterization for the solvability of nonlinear partial differential equations

Author:
Elemer E. Rosinger

Journal:
Trans. Amer. Math. Soc. **330** (1992), 203-225

MSC:
Primary 35D05; Secondary 35A05, 46F10

DOI:
https://doi.org/10.1090/S0002-9947-1992-1028764-7

MathSciNet review:
1028764

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Abstract | References | Similar Articles | Additional Information

Abstract: Within the nonlinear theory of generalized functions introduced earlier by the author a number of existence and regularity results have been obtained. One of them has been the first *global* version of the Cauchy-Kovalevskaia theorem, which proves the existence of generalized solutions on the *whole of the domain of analyticity* of arbitrary analytic nonlinear PDEs. These generalized solutions are *analytic everywhere, except for closed, nowhere dense subsets* which can be chosen to have *zero Lebesgue measure*.

This paper gives a certain extension of that result by establishing an *algebraic* necessary and sufficient condition for the existence of generalized solutions for arbitrary polynomial nonlinear PDEs with continuous coefficients. This algebraic characterization, given by the so-called *neutrix* or *off diagonal* condition, is proved to be equivalent to certain *densely vanishing* conditions, useful in the study of the solutions of general nonlinear PDEs.

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DOI:
https://doi.org/10.1090/S0002-9947-1992-1028764-7

Article copyright:
© Copyright 1992
American Mathematical Society