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Partial differential equations. Transl. from the German by C. B. and M. J. Thomas. (English) Zbl 0623.35006
Cambridge etc.: Cambridge University Press. XI, 518 p.; Cloth: £50.00; $ 79.50; Paper: £17.50; $ 29.95 (1987).
[For a review of the original German ed. (Teubner, Stuttgart 1982) see Zbl 0482.35001.]
This book is devoted to boundary value problems for elliptic differential operators and mixed problems for parabolic and hyperbolic equations for which the right-hand side is an elliptic differential operator. The main difference of this book from many others is an extensive use of Lopatinskij-Shapiro conditions (covering conditions) and modern methods (although the theory of pseudo-differential operators is not applied but the ideas of this theory are used in many constructions systematically) in the combination with the accessibility of the text. Classical boundary value problems (Dirichlet’s one and so on) are discussed in detail as examples.
The book contains six chapters. Chapter 1 ”Sobolev spaces” deals with an accurate description of the space \(W^ 1_ 2(\Omega)\) and different properties of this space that will be used in next chapters (in particular the Fourier transform). Chapter 2 ”Elliptic differential operators” deals with the Lopatinskij-Shapiro conditions and normal solvability for elliptic boundary value problems; Green’s formulae and the adjoint boundary value problems are considered, too. Chapter 3 ”Strongly elliptic differential operators and the method of variations” contains the Lax-Milgram theorem, Agmon’s theorem, theorems on the regularity of solutions and so on; the questions of connections between general methods of chapter 2 and variational methods of chapter 3 are considered in detail.
Chapter 4 ”Parabolic differential operators” deals with existence, uniqueness and regularity of solutions to such equations. Chapter 5 ”Hyperbolic differential operators” is devoted to analogous results for hyperbolic differential equations. The last chapter ”Difference processes for the calculation of the solution of the partial differential equation” is devoted to practical numerical methods for the solution of partial differential equations.
In general, the book is a very good introduction to modern theory of partial differential equations and its English edition will be useful to all mathematicians who wish to study modern methods in classical problems of mathematical physics.
Reviewer: P.P.Zabrejko

MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47F05 General theory of partial differential operators
47A53 (Semi-) Fredholm operators; index theories
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65N99 Numerical methods for partial differential equations, boundary value problems