Synopses & Reviews
This is the first of two books on methods and techniques in the calculus of variations. Contemporary arguments are used throughout the text to streamline and present in a unified way classical results, and to provide novel contributions at the forefront of the theory. This book addresses fundamental questions related to lower semicontinuity and relaxation of functionals within the unconstrained setting, mainly in L^p spaces. It prepares the ground for the second volume where the variational treatment of functionals involving fields and their derivatives will be undertaken within the framework of Sobolev spaces. This book is self-contained. All the statements are fully justified and proved, with the exception of basic results in measure theory, which may be found in any good textbook on the subject. It also contains several exercises. Therefore,it may be used both as a graduate textbook as well as a reference text for researchers in the field. Irene Fonseca is the Mellon College of Science Professor of Mathematics and is currently the Director of the Center for Nonlinear Analysis in the Department of Mathematical Sciences at Carnegie Mellon University. Her research interests lie in the areas of continuum mechanics, calculus of variations, geometric measure theory and partial differential equations. Giovanni Leoni is also a professor in the Department of Mathematical Sciences at Carnegie Mellon University. He focuses his research on calculus of variations, partial differential equations and geometric measure theory with special emphasis on applications to problems in continuum mechanics and in materials science.
Review
From the reviews: "This book is intended as a graduate textbook and reference for those who work in the modern calculus of variations. ... interesting examples and exercises help to keep the reader on track. Several open problems are indicated as well. ... excellent presentation." (Erik J. Balder, Mathematical Reviews, Issue 2008 m) "This book is the first of two volumes in the calculus of variations and measure theory. The main objective of this book is to introduce necessary and sufficient conditions for sequential lower semicontinuity of functionals on Lp-spaces. ... This book is very nicely written, self-contained and it is an excellent and modern introduction to the calculus of variations." (Jean-Pierre Raymond, Zentrablatt MATH, Vol. 1153, 2009) "This is the first of a two-volume introduction into direct methods in the calculus of variations. Its main topic is the analysis of necessary and sufficient conditions for lower semicontinuity on Lp-spaces, as well as of relaxation techniques. ... The book provides a well-written and self-contained introduction to an active area of research and will be valuable both to graduate students as an introduction and to researchers in the field as a reference work." (M. Kunzinger, Monatshefte für Mathematik, Vol. 156 (4), April, 2009)
Synopsis
This book presents in a unified way both classical and contemporary results in the Calculus of Variations. In recent years there has been a remarkable and renewed interest in this area, motivated in part by applications of allied disciplines. The need to bring together in one place the contemporary developments in the calculus of variations led to the writing of this book, based on a series of lectures by Fonseca at carnegie Mellon University. It is divided into two parts: Part I is devoted to Calculus of Variations in a Sobolev setting, and Part 2 addresses variational methods in function spaces allowing for discontinuities of the underlying potentials, e.g. the space of functions of bounded variation. Because it is largely self-contained, it will appeal to non- specialists and new students in this area. It is intended to be a graduate text and a reference for more experienced researchers working in the area.
Synopsis
This book is a unified presentation of both classical and contemporary results in the calculus of variations. It offers a comprehensive analysis of necessary and sufficient conditions for sequential lower semicontinuity of functionals on Lp spaces, followed by relaxation techniques. In recent years there has been a remarkable and renewed interest in this area, motivated in part by applications of allied disciplines. Based on a series of lectures by Irene Fonseca at Carnegie Mellon University, this book was written in response to the need to bring together in one volume contemporary developments in the calculus of variations. Because it is largely self-contained, the book will appeal to non-specialists and new students in this discipline. It is intended for use as a graduate text and as a reference for more experienced researchers working in the area.
Synopsis
This is the first of two books in the Calculus of Variations and Measure Theory where many results, some now classical and others at the forefront of research in the subject, are gathered in a unified, consistent way. Contemporary arguments are used throughout the text, streamlining well-known aspects of the theory, while providing novel contributions. The first book prepares the ground for the second in that it introduces and develops the basic tools in the Calculus of Variations and in Measure Theory needed to address fundamental questions in the treatment of functionals involving derivatives.
This book offers a comprehensive and unified treatment of lower semicontinuity and relaxation of functionals without derivatives. It is self-contained in the sense that, with the exception of fundamentally basic results in Measure Theory which may be found in any textbook in the subject (e.g. Lebesgue Dominated Convergence Theorem), all the statements are fully justified and proved. This book is written in a multilayered way in that graduate students and researchers with different backgrounds may access different parts of the text.
Table of Contents
Preface.- Measure Theory and Lp Spaces.- Measures.- Lp spaces.- The Direct method and lower semicontinuity.- Convex analysis.- Functionals defined on Lp.- Integrands f = f (z).- Integrands f = f (x; z).- Integrands f = f (x; u; z).- Young measures.- A Appendix.- B Notes and open problems.- C Notation and List of symbols.- Index.