Nonsmooth Variational Problems and Their Inequalities - (Springer Monographs in Mathematics) by Siegfried Carl & Vy Khoi Le & Dumitru Motreanu

Book Synopsis

This is the first research monograph to focus on variational inequalities as part of nonsmooth variational systems research. The authors discuss partial differential equations, variational equations, variational and hemivariational inequalities, and related topics. With a wealth of problems and techniques from nonlinear and nonsmooth analysis, this versatile text is an important reference for mathematicians working in analysis, partial differential equations, elasticity, materials science and mechanics applications, as well as for physicists and engineers. It can serve as a main or supplemental text for a variety of specialized nonlinear analysis courses.

From the Back Cover

This monograph focuses primarily on nonsmooth variational problems that arise from boundary value problems with nonsmooth data and/or nonsmooth constraints, such as is multivalued elliptic problems, variational inequalities, hemivariational inequalities, and their corresponding evolution problems.

The main purpose of this book is to provide a systematic and unified exposition of comparison principles based on a suitably extended sub-supersolution method. This method is an effective and flexible technique to obtain existence and comparison results of solutions. Also, it can be employed for the investigation of various qualitative properties, such as location, multiplicity and extremality of solutions. In the treatment of the problems under consideration a wide range of methods and techniques from nonlinear and nonsmooth analysis is applied, a brief outline of which has been provided in a preliminary chapter in order to make the book self-contained.

This text is an invaluable reference for researchers and graduate students in mathematics (functional analysis, partial differential equations, elasticity, applications in materials science and mechanics) as well as physicists and engineers.

Review Quotes

From the reviews:

"This monograph presents in a systematic way the method of sub- and supersolutions for solving variational and hemivariational inequalities. ... Each chapter begins with a short overview presenting cases and ideas, and concludes with notes and remarks giving references to the literature. ... It is carefully written and suitable for advanced graduate students and researchers having a good background in functional analysis, basic partial differential equations, and critical point theory." (Thomas Bartsch, Mathematical Reviews, Issue, 2007 i)

"This monograph contains seven chapters, the bibliography of 223 entries, and an index. ... This well-written book contains large number of material. It can be useful for graduate students and researchers interested in variational methods. In the beginning of each chapter the authors give some motivation for the material of the chapter." (Alexander G. Ramm, Zentralblatt MATH, Vol. 1109 (11), 2007)

"Thepresent monograph develops in a careful and thorough way the theory of sub- and supersolutions for variational equalities and inequalities. ... The book is an important addition to the literature on nonlinear analysis, convex analysis, variational and hemivariational inequalities, nonlinear elliptic and parabolic partial differential equations, elasticity theory, fracture mechanics, and general obstacle and unilateral problems; it will be a welcome addition to the libraries of researchers and students of these areas." (Klaus Schmitt, SIAM Review, Vol. 49 (3), 2007)