Numerous line drawings clarify the mathematics.Įach chapter ends with recommended readings that introduce the student to the relevant scientific literature and with exercises that consolidate understanding. The exposition starts with simple integrals containing a single independent variable, a single dependent variable, and a single derivative, subject to weak variations, but steadily moves on to more advanced topics, including multivariate problems, constrained extrema, homogeneous problems, problems with variable endpoints, broken extremals, strong variations, and sufficiency conditions. The text lays out important necessary and sufficient conditions for extrema in historical order, and it illustrates these conditions with numerous worked-out examples from mechanics, optics, geometry, and other fields. The reader will learn methods for finding functions that maximize or minimize integrals. Kunzinger, Monatshefte für Mathematik, Vol.This book is intended for a first course in the calculus of variations, at the senior or beginning graduate level. … 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. 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. One of the aims is to serve as a guide in the extensive existing. Furthermore several aspects of the calculus of variations are not discussed here.
Every individual chapter can be, on its own, the subject of a book, For example, I have written one that, essentially, covers the subject of Chapter 3. … 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. phasized that it is not a reference book. 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. This is the first of two books on methods and techniques in the calculus of variations.
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