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The boundary value problems of mathematical physics. Transl. from the Russian by Jack Lohwater. (English) Zbl 0588.35003
Applied Mathematical Sciences, 49. New York etc.: Springer-Verlag. XXX, 322 p. DM 198.00 (1985).
This English translation differs from the Russian original (1973; Zbl 0284.35001) by the inclusion of a ”Supplements and Problems” section located at the end of each chapter, which illustrate the possibilites of the methods contained in the book, and are to awake the student’s creativity, providing topics for independent work, as the author writes in the preface. These added sections contain about 40 pp. and make the book even more valuable.
The book contains 6 chapters, the first on functional analytic preliminaries and Sobolev spaces, three chapters on linear partial differential equations of second order of elliptic, parabolic and hyperbolic type resp., one on generalizations (higher order equations, systems) and the last on the finite difference method. Generalized solutions in the Sobolev sense are considered throughout, the leading pedagogical principle being ”to prove, as simply as possible,..., the solvability of basic boundary value (and initial-boundary value) problems... as consequence of the uniqueness theorems in a ”sufficiently large” function space.” A beautiful book, indeed.
Reviewer: J.Lorenz

MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J25 Boundary value problems for second-order elliptic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems