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Book review

Author(s) Dzhuraev, A.
Title Methods of singular integral equations
Publisher Longman Scientific & Technical, John Wiley & Sons
Year of publication 1992
Reviewed by S.V. Rogozin

The book is rewritten and revised edition of the known monograph originally published in Russian. It is devoted to the investigation of solvability and solutions' properties of some classes of elliptic partial differential equations and systems of equations. Mainly it is connected with the problems stated for the bounded plane domains (in particular multiply connected domains). This determines the range of the using methods. First of all it is the method of singular integral equations. The latter is well-known tool for study of a great variety of problems (theoretical as well an applied). Classical theory of singular integral equations and their applications are presented in many books and surveys (the papers of F. Noether, M. Riesz, N.I. Muskhelishvili, A.P. Calderon & A. Zygmund, F.D. Gakhov, I.N. Vekua, A. Dzhuraev, E.M. Stein, S.G. Mikhlin & S. Prößdorf, E. Meister are the most known of them). The method of sin-gular integral equations (more precisely - modified version of it) is applied in the book to the investigation of partial differential equations (PDE) of nonclassical type, including the PDE of composite type. The most of presented results are obtained by the author in the last two-three decades. Besides are given some other methods for study a class of PDE and systems of PDE. These methods allowed the author to reach some results not only for two-dimensional case but also for multidimensional one and even for nonstationar systems of equations.

The book consists of five chapters and a short list of references (including only those papers which are directly using into the text).

The first chapter "Integral representation of functions" has an introductive character. It consists of all necessary definitions and results using in the next chapters. Are presented three type of them. First of all are given some pieces of analytic functions theory, especially those connected with Hilbert transform, singular integrals and singular integral equations and boundary value problems for analytic functions, at last with some Hilbert spaces of analytic functions. Besides are studied the properties of some analytic functions generalizations. Here on the discussion are solutions of inhomogeneous Cauchy-Riemann equations and of the generalized Cauchy-Riemann equations <formula>. The latter are also known as generalized analytic functions in sense of Bers-Vekua.

At last the great attention is paid in the chapter to the integral representations of the analytic or generalized analytic solutions of different boundary value problems.

Inspite that several results and definitions of the chapter 1 are well-known and prersented in many books and articles, they are obtained as detailed as possible and from the united position.

The central in the reviewed book are the chapters 2 and 3. In the first one (called "The theory of two-dimonsional singular integral equations") the theory of these equations is constructed for the bounded multiply connected domains on the complex plane. Are built some types of singular integral equations (SIE) and investigated certain of their properies. Then it is found the close connection between SIE and general conjugate boundary value problems for analytic functions.

The last part of the chapter is devoted to the investigation of the above mentioned type of the boundary value problems. The thing is that these problems studied for the first time by Boyarski at the end of fiftieth has no complete solution for the moment. Several results obtained by the author in this direction are presented in the book.

These investigations are then generalized in the case of boundary value problems for elliptic partial differential equations on the plane (chapter 3 "Boundary value problems for elliptic partial differential equations in the plane"). First of all are considered the integral representations of solutions the classical Dirichlet and Neumann problems in arbitrary multiply connected domains. Then are formulated and studied some new well-posed boundary value problems. The main attention is paid to the Fredholmity conditions of these problems. The method of investigation is based on the properties of the corresponding singular integral operators.

In the last two chapters on the consideration are some well-posed boundary value problems for the system of differential equations in nonstationar case. So it is not always possible to apply here some results of the previous chapters (in particular those of them in which are used the integral representations and singular integral equations method). Anyway some of results and technics can be generalized even for this situation.

In the chapter 4 "Initial and initial-boundary value probleas for nonclassical systems of partial differential equations" the mentioned problems are studied for such type of equations for complex valued functions which can be reduced to the above discussed problems. Thus the theory of singular integral-functional equations on the boundary of the considered domains still plays the main role in these investigations. The problems on the discussion are so galled boundary value problems for composite-type systems.

The author also has tried to generalize the results of the chapters 1-4 in the last one called "Multidimensional systems". Mainly this chapter is devoted to the formulation and study well-posed problems for some kind of systems of partial differential equations (first of all Moisil-Theodorescu systems). Here the only approach and small technical details are the same as in the previous chapters (in particular the singular integral equations method has much less applications). Anyway the ideas are very similar to the above mentioned because of the close connection (or more precisely - analogity) between generalized Cauchy-Riemann systems and Moisil-Theodorescu systems.

The book is written in quite classical manner, all the results have as many details as is needed, all the calculations are made very carefully. So the book will be extremely useful for the beginners in the problems of such a type. Besides it will be very interesting for the specialists in complex analysis and partial differential equations due to deep and original results presented in the monograph. At last all the investigations have mainly the theoretic character but all of them are prepared for the applications to different type of mechanical and physical problems. The main attractiveness of the book is in the potential possibility of these applications.