© 2018

Calculus of Variations

  • 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


Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Basic Course

    1. Front Matter
      Pages 1-1
    2. Filip Rindler
      Pages 3-22
    3. Filip Rindler
      Pages 23-45
    4. Filip Rindler
      Pages 47-79
    5. Filip Rindler
      Pages 81-103
    6. Filip Rindler
      Pages 105-133
    7. Filip Rindler
      Pages 135-151
    8. Filip Rindler
      Pages 153-182
  3. Advanced Topics

    1. Front Matter
      Pages 183-183
    2. Filip Rindler
      Pages 185-225
    3. Filip Rindler
      Pages 227-268
    4. Filip Rindler
      Pages 269-299
    5. Filip Rindler
      Pages 301-329
    6. Filip Rindler
      Pages 331-368
    7. Filip Rindler
      Pages 369-408
  4. Back Matter
    Pages 409-444

About this book


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 Γ-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.


calculus of variations PDE partial differential equations variational problem minimization problem Euler-Lagrange equation Young measure rigidity differential inclusion microstructure convex integration Gamma-convergence homogenization MSC (2010): 49-01, 49-02, 49J45, 35J50, 28B05, 49Q20

Authors and affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUnited Kingdom

About the authors

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.

Bibliographic information


“I have never seen a text which successfully treats Young measures, relaxation and the rather modern concept of polyconvexity at this level. … Each chapter starts with an introductory section where the problem(s) are illustrated using simple settings, and at the end we find a short section on the history of the material and further developments, as well as a problem section. Summing up, this is a well-written and most welcome addition to the (textbook) literature.” (René L. Schilling, The Mathematical Gazette, Vol. 104 (559), March, 2020)

“The purpose of this textbook is to give a comprehensive introduction to the classical and modern calculus of variations; it serves as useful reference to advanced undergraduate and graduate students as well as researchers in the field. … This book is interesting to a reading seminar or a companion for self-study.” (Hengyou Lan, zbMATH 1402.49001, 2019)