Calculus of Variations

Calculus of Variations

von Filip Rindler

This book offers advanced undergraduates, graduate students, and researchers a comprehensive introduction to both the classical and modern Calculus of Variations. It can serve as the main text for a lecture course, the foundation for a reading seminar, or as a companion for independent study. This thoroughly revised second edition features numerous improvements, including the addition of several new topics, an enhanced order of presentation, and expanded references to the literature.

Starting with a string of motivating examples, the first half of the book presents the central elements of the classical theory, including the Direct Method, the Euler–Lagrange equation, Lagrange multipliers, Noether’s theorem, and some regularity theory. Using the efficient framework of Young measures, the text then develops the vectorial theory of integral functionals, covering quasiconvexity, polyconvexity, relaxation, and Γ-convergence. The second half of the book introduces more recent developments, some of which have previously been accessible only in the research literature. Topics treated in detail include rigidity for differential inclusions, microstructure, convex integration, concentrations in measures, linear growth functionals on functions of bounded variation (BV), and generalized Young measures.

The reader is expected to be familiar with vector analysis, functional analysis, basic measure theory, and some Sobolev space theory; essential preliminaries are reviewed in an appendix.

This book offers advanced undergraduates, graduate students, and researchers a comprehensive introduction to both the classical and modern Calculus of Variations. It can serve as the main text for a lecture course, the foundation for a reading seminar, or as a companion for independent study. This thoroughly revised second edition features numerous improvements, including the addition of several new topics, an enhanced order of presentation, and expanded references to the literature.

Starting with a string of motivating examples, the first half of the book presents the central elements of the classical theory, including the Direct Method, the Euler–Lagrange equation, Lagrange multipliers, Noether’s theorem, and some regularity theory. Using the efficient framework of Young measures, the text then develops the vectorial theory of integral functionals, covering quasiconvexity, polyconvexity, relaxation, and Γ-convergence. The second half of the book introduces more recent developments, some of which have previously been accessible only in the research literature. Topics treated in detail include rigidity for differential inclusions, microstructure, convex integration, concentrations in measures, linear growth functionals on functions of bounded variation (BV), and generalized Young measures.

The reader is expected to be familiar with vector analysis, functional analysis, basic measure theory, and some Sobolev space theory; essential preliminaries are reviewed in an appendix.

Highlights

• A thorough introduction to both the classical and modern Calculus of Variations Leads the reader from basic principles to recent research topics Covers powerful modern techniques, such as Young measures and rigidity theory Includes 140 exercises to deepen the study and consolidate understanding

Über den Autor

Filip Rindler is Professor of Mathematics at the University of Warwick. His work focuses on the calculus of variations, PDEs, geometric measure theory, and mathematical material science and has been funded by two ERC grants as well as EPSRC and Turing Fellowships. For his research he has received the Whitehead Prize in 2018 and was named a 2025 Blavatnik Awards honoree.