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Generalized solutions of nonlinear partial differential equations. (English) Zbl 0635.46033

North-Holland Mathematics Studies, 146. Notas de Matemática (119). Amsterdam etc.: North-Holland. XVII, 409 p.; Dfl. 175.00 (1987).
The linear theory of distributions is an extremely powerfully tool in the theory of partial differential equations. Much less attention has been given to a parallel nonlinear theory, mainly due to the famous “impossibility result” of L. Schwartz [C. R. Acad. Sci. Paris 239, 847-848 (1954; Zbl 0056.106)] which states that, loosely speaking, multiplication operators cannot extend “reasonably” from functions to distributions. Nevertheless, in the last year several attempts have been made to develop a nonlinear theory of generalized solutions of nonlinear partial differential equations, for details see e.g. the monograph by E. E. Rosinger [Springer Lect. Notes Math. 684 (1978; Zbl 0469.35001)]. The same author presents now a fairly detailed account of this theory in book form which should be of interest to specialists in nonlinear analysis and partial differential equations.
The book consists of nine chapters with the following headings: classical versus distribution solution; impossiblity and degeneracy results in distributions: Limitations of the linear distribution theory; the differential algebra \({\mathcal G}\) as an extension of the \({\mathcal G}'\) distributions; generalized solutions for nonlinear partial differential equations; generalized solutions for linear partial differential equations; stability; generality, and exactness of generalized solutions; algebras of generalized functions; resolution of singularities of weak solutions for polynomial nonlinear partial differential equations.
Reviewer: J.Appell

MSC:

46F10 Operations with distributions and generalized functions
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
35F20 Nonlinear first-order PDEs
35G20 Nonlinear higher-order PDEs