An introduction to variational inequalities and their applications. Reprint of the 1980 original. (English) Zbl 0988.49003

Kinderlehrer, David; Stampacchia, Guido

Classics in Applied Mathematics. 31. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). xx, 313 p. (2000).

Variational inequalities, which were introduced by Stampacchia in the early 1960s, have been proved to be an effective and powerful tool in studying very complicated and complex problems arising in various branches of pure and applied sciences. Variational inequalities provide us with a natural, novel, innovative and unified technique for developing various efficient numerical techniques for solving equilibrium problems. It is perhaps part of the fascination of this subject that so many branches of pure and applied sciences are involved in the development of the variational inequalities.
The original edition of the present book was published by Academic Press in 1980 (see the review in Zbl 0457.35001). In the development of this theory and its applications in equilibrium problems, this book has played a fundamental role. It has inspired many to apply the techniques and ideas of variational inequalities in different branches of science. There are now several books in this area to cover some special fields like transportation, elasticity and finance and optimizations. The reviewer of this book has the experience of teaching a one semester course based on the Chapters I, II, some parts of Chapters III and IV plus some topics based on Wiener-Hopf equations [see M. A. Noor, “Some Recent Advances in Variational Inequalities. Part I, and II”, N. Z. J. Math. 26, 53-80,229-255 (1997; Zbl 0886.49004 and Zbl 0889.49006)] at graduate level.
This book is written clearly and provides an introduction to the theory of variational inequalities. It emphasizes the applications in free boundary value problems associated with partial differential equations. This book has been used as a textbook at various universities and will inspire and motivate many in the future to discover new and innovative applications of variational inequalities in diverse fields.

Reviewer: Muhammad Aslam Noor (Halifax)

MSC:

49J40	Variational inequalities
35R35	Free boundary problems for PDEs
35J85	Unilateral problems; variational inequalities (elliptic type) (MSC2000)
47H05	Monotone operators and generalizations
47J20	Variational and other types of inequalities involving nonlinear operators (general)
90C33	Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)

Keywords:

regularity; existence; monotone operators; variational inequalities

Citations:

Zbl 0457.35001; Zbl 0086.49004; Zbl 0889.49006; Zbl 0886.49004