Its main topic is the analysis of necessary and sufficient conditions for lower semicontinuity on L p-spaces, as well as of relaxation techniques. “This is the first of a two-volume introduction into direct methods in the calculus of variations. … 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. The main objective of this book is to introduce necessary and sufficient conditions for sequential lower semicontinuity of functionals on L p-spaces. "This book is the first of two volumes in the calculus of variations and measure theory. Balder, Mathematical Reviews, Issue 2008 m) Several open problems are indicated as well. … interesting examples and exercises help to keep the reader on track. "This book is intended as a graduate textbook and reference for those who work in the modern calculus of variations. 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. Giovanni Leoni is also a professor 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. 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. Therefore,it may be used both as a graduate textbook as well as a reference text for researchers in the field. 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 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 addresses fundamental questions related to lower semicontinuity and relaxation of functionals within the unconstrained setting, mainly in L^p spaces. 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. Substantially revised and corrected by the translator, this inexpensive ne edition will be welcomed by advanced undergraduate and graduate students of mathematics and physics.This is the first of two books on methods and techniques in the calculus of variations. Two appendices and suggestions for supplementary reading round out the text. The problems following each chapter were made specially for this English-language edition, and many of them comment further on corresponding parts of the text. Chapter seven considers the application of variational methods to the study of systems with infinite degrees of freedom, and Chapter eight deals with direct methods in the calculus of variations. Students wishing a more extensive treatment, however, will find the first six chapters comprise a complete university-level course in the subject, including the theory of fields and sufficient conditions for weak and strong extrema. The reader who merely wishes to become familiar with the most basic concepts and methods of the calculus of variations need only study the first chapter. Considerable attention is devoted to physical applications of variational methods, e.g., canonical equations, variational principles of mechanics, and conservation laws. The aim is to give treatment of the elements of the calculus of variations in a form both easily understandable and sufficiently modern. Based on a series of lectures given by I.M.Gelfand at Moscow State University, this book actually goes considerably beyond the material presented in the lectures.
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