In recent years the need to extend the notion of degree to nonsmooth functions has been triggered by developments in nonlinear analysis and some of its applications. This new study relates several approaches to degree theory for continuous functions and incorporates newly obtained results for Sobolev functions. These results are put to use in the study of variational principles in nonlinear elasticity. Several applications of the degree are illustrated in the theories of ordinary and partial differential equations. Other topics include multiplication theorem, Hopf's theorem, Brower's fixed...

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...

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...