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Direct methods in the theory of elliptic equations. Transl. from the French. Editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader. (English) Zbl 1246.35005
Springer Monographs in Mathematics. Berlin: Springer (ISBN 978-3-642-10454-1/hbk; 978-3-642-10455-8/ebook). xvi, 372 p. (2012).
The book under review is the English translation of J. Nečas’ well known treatise [Les méthodes directes en théorie des équations elliptiques. Paris: Masson et Cie; Prague: Academia (1967; Zbl 1225.35003)] which was published by Academia, Prague 1967, and simultaneously by Masson, Paris. It was the first systematic discussion of the functional analytic approach to existence and regularity of solutions of elliptic boundary value problems within the Sobolev space setting and it represents the state of the art in the mid sixties.
The book includes many important results published as well as unpublished by several authors and results by J. Nečas himself. In addition, there are numerous bibliographical hints and many remarks, examples, exercises and problems. It should be noticed that the exercises amount to alternative proofs of results presented in the text, as well as to the proofs of related and even new results.
The material of the book is arranged in the following chapters. 1: Elementary description of principal results. 2: The spaces \(W^{k,p}\). 3: Existence, uniqueness and fundamental properties of solutions of boundary value problems. 4: Regularity of the solution. 5: Application of Rellich’s equalities and their generalizations to boundary value problems. 6: Boundary value problems in weighted Sobolev spaces. 7: Regularity of the solution for non-smooth coefficients and non-regular domains.
To the reviewer’s knowledge, Chapter 2 contains the basics of the spaces \(W^{k,p}(\Omega)\) (\(1\leq p<+\infty\)) for the first time in book form (see, e. g., the inverse trace theorems in 2.5.6, 2.5.7). The spaces which occur in Sobolev’s original papers are discussed in 2.7.3. Next, in Chapters 3, 4 and 5 the author provides a comprehensive study of the basic material of the \(L^2\)-theory of boundary value problems for elliptic boundary value problems including his own results (Section 3.7 and 5.3 give a brief introduction to elliptic systems, with an application to linear elasticity). Most of the topics presented in Chapter 6, have been part of J. Nečas’ research work. Finally, Chapter 7 contains results (some of them with simplified proofs) on weak solutions to elliptic equations with discontinuous coefficients.
Summarizing, the book continues to be one of the classics of the Sobolev space setting of linear elliptic boundary value problems. The reviewer is convinced that the now available English translation will be widely used by young researchers.

MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Jxx Elliptic equations and elliptic systems
01A60 History of mathematics in the 20th century
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