This textbook provides a comprehensive introduction to the classical and modern calculus of variations, serving as a useful reference to advanced undergraduate and graduate students as well as researchers in the field. Starting from ten motivational examples, the book begins with the most important aspects of the classical theory, including the Direct Method, the Euler-Lagrange equation, Lagrange multipliers, Noether's Theorem and some regularity theory. Based on the efficient Young measure approach, the author then discusses the vectorial theory of integral functionals, including quasiconvexity, polyconvexity, and relaxation. In the second part, more recent material such as rigidity in differential inclusions, microstructure, convex integration, singularities in measures, functionals defined on functions of bounded variation (BV), and -convergence for phase transitions and homogenization are explored.

While predominantly designed as a textbook for lecture courses on the calculus of variations, this book can also serve as the basis for a reading seminar or as a companion for self-study. The reader is assumed to be familiar with basic vector analysis, functional analysis, Sobolev spaces, and measure theory, though most of the preliminaries are also recalled in the appendix.

Presents several strands of the most recent research on the calculus of variations Builds on powerful analytical techniques such as Young measures to provide the reader with an effective toolkit for the analysis of variational problems in the vectorial setting Includes 120 exercises to consolidate understanding Includes supplementary material: sn.pub/extras

Autorentext
Filip Rindler is Associate Professor of Mathematics at the University of Warwick, UK. His research focuses on the calculus of variations and the theory of PDEs. For 2018-2023, he leads the ERC-funded SINGULARITY project.

Inhalt
Part I Basic Course.- 1 Introduction.- 2 Convexity.- 3 Variations.- 4 Young Measures.- 5 Quasiconvexity.- 6 Polyconvexity.- 7 Relaxation.- Part II Advanced Topics.- 8 Rigidity.- 9 Microstructure.- 10 Singularities.- 11 Linear-Growth Functionals.- 12 Generalized Young Measures.- 13 G-Convergence.- A Prerequisites.- References.- Index.