Modern Methods in the Calculus of Variations: L^p Spaces

ISBN-13: 9781441922601 / Angielski / Miękka / 2010 / 600 str.

Irene Fonseca; Giovanni Leoni

Opis

In recent years there has been renewed interest in the calculus of variations, motivated in part by ongoing research in materials science and other dis- plines. Often, the study of certain material instabilities such as phase tran- tions, formation of defects, the onset of microstructures in ordered materials, fractureanddamage, leadstothesearchforequilibriathroughaminimization problem of the type min{I(v): v?V}, wheretheclassV ofadmissiblefunctionsv isasubsetofsomeBanachspaceV. This is the essence of the calculus of variations: the identi?cation of nec- sary and su?cient conditions on the functionalI that guarantee the existence of minimizers. These rest on certain growth, coercivity, and convexity con- tions, which often fail to be satis?ed in the context of interesting applications, thusrequiringtherelaxationoftheenergy.Newideaswereneeded, andthe- troduction of innovative techniques has resulted in remarkable developments in the subject over the past twenty years, somewhat scattered in articles, preprints, books, or available only through oral communication, thus making it di?cult to educate young researchers in this area. This is the ?rst of two books in the calculus of variations and measure theory in which many results, some now classical and others at the forefront of research in the subject, are gathered in a uni?ed, consistent way. A main concern has been to use contemporary arguments throughout the text to - visit and streamline well-known aspects of the theory, while providing novel contributio

Recenzje

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 L^p-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 L^p-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)

Spis treści

Measure Theory and Lp Spaces.- Measures.- Lp Spaces.- The Direct Method and Lower Semicontinuity.- The Direct Method and Lower Semicontinuity.- ConvexAnalysis.- Functionals Defined on Lp.- Integrands f = f (z).- Integrands f = f (x, z).- Integrands f = f (x, u, z).- Young Measures.

Blurb

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.