Wen, Guo Chun
Begehr, Heinrich G.W.
|Title||Boundary Value Problems for Elliptic Equations and Systems|
|Publisher||Longman Scientific and Technical|
|Year of publication||1990|
|Reviewed by||Ioan A. Rus|
The theory of linear elliptic systems of two first order equations is very well developed and very well known (see Bers 1, Vekua 7, Gakhov 4, Gilberg 5, Bicadze 2, ...). The book under review deals with various boundary value problems for nonlinear elliptic systems of first and second order. The chapters of the book are the following: Boundary value problems for simple complex equations, Boundary value problems for elliptic complex equations of first order, Boundary value problems for elliptic equations of second order, Boundary value problems with piecewise continuous coefficients for elliptic equations and systems, Boundary value problems for elliptic systems of two second order equations, Boundary value problems for elliptic systems of several equations. The book contains an extensive list of references as well as detailed historical comments.
Most of the results of this book are given by the first named author and some of them are published here for the first time. Among the methods used in the book are:
(i) Transformation of the real equations into the complex forms.
(ii) Integral representation of the solutions.
(iii) Extremum principles (see Protter-Weinberger 6 for the real case).
(iv) Apriori extimates for solutions
(v) The method of continuity (see Dugundji-Granas 3).
(vi) Fixed point technique (see Dugundji-Granas 3).
This book is a research monograph written for people who already have some understanding of the field. According its originality and style I recommend this book to all interested in the theory of partial differential equations.
1. L. Bers, Theory of pseudoanalytic functions, Courant Institute, New York, 1953.
2. A.V. Bicadze, Some classes of partial differential equations, Gordon a. Breach, New York, 1988.
3. J. Dugundji, A. Granas, Fixed point theory, PWN, Warszawa, 1982.
4. F.D. Gakhov, Boundary value problems, Pergamon, Oxford, 1962.
5. R.P. Gilbert, Function theoretic methods in partial differential equations, Academic Press, New York, 1969.
6. M.H. Protter, H.F. Weinberger, Maximum principles in differential equations, Springer-Verlag, Berlin, 1984.
7. I.N. Vekua, Generalized analytic functions, Pergamon Press, Oxford, 1962.