This book deals with several important topics in nonlinear analysis, presented both on their own and as a basic tool for solving a broad class of nonlinear problems. It includes problems arising in the theory of partial differential equations, in particular the theories of boundary value problems, control theory, and calculus of variations. The book is written as a self-contained textbook. The reader will be pleased to find, in a rather large appendix, all the basic facts on topology, measure theory and functional analysis. But even when reading the book from the beginning, the reader will find that the book can serve as a well-written textbook, providing the basic knowledge and containing material of a deeper level.
Successively, one learns about Hausdorff measures and capacity, covering theorems, Dini derivatives, area formulas, Lebesgue-Bochner and Sobolev spaces, vector valued integration, evolution triples needed for PDE theory, together with the standard inequalities and embedding theorems that are the core of the theory. The modern concepts of nonlinear operators and Young measures, also in the context of the Nemytskii operators, and the theory of superposed convergences are dealt with. The book will be valuable both for postgraduate students beginning their professional career in the field and for experts who are looking for a well-written and comprehensive handbook on the field.