Calculus of variations Filip Rindler

Format:

Book

Author/Creator:

Rindler, Filip, author.

Series:

Graduate texts in mathematics ; 2197-5612 310

Graduate texts in mathematics 2197-5612 volume 310

Language:

English

Subjects (All):

Calculus of variations.

Physical Description:

1 online resource

Edition:

Second edition

Place of Publication:

Cham Springer 2026

Summary:

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 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

Contents:

Intro

Preface

Contents

Part I Basic Course

Chapter 1 Introduction

1.1 The Brachistochrone Problem

1.2 The Isoperimetric Problem

1.3 Electrostatics

1.4 Stationary States in Quantum Mechanics

1.5 Optimal Saving and Consumption

1.6 Sailing Against the Wind

1.7 Hyperelasticity

1.8 Linearized Hyperelasticity

1.9 Microstructure in Crystals

1.10 Composite Elastic Materials

1.11 Phase Transitions

Chapter 2 Convexity

2.1 The Direct Method

2.2 Functionals with Convex Integrands

2.3 Integrands with u-Dependence

2.4 The Lavrentiev Gap Phenomenon

2.5 Integral Side Constraints

2.6 Convex Functions and Duality

2.7 Subdifferentials

Notes and Historical Remarks

Problems

Chapter 3 Variations

3.1 The Euler-Lagrange Equation

3.2 Regularity of Minimizers

3.3 Lagrange Multipliers

3.4 Invariances and the Noether Theorem

Chapter 4 Young Measures

4.1 The Fundamental Theorem

4.2 Examples

4.3 Young Measures and Notions of Convergence

4.4 Gradient Young Measures

4.5 Homogeneous Gradient Young Measures

Problems

Chapter 5 Quasiconvexity

5.1 Quasiconvexity

5.2 Null-Lagrangians

5.3 Rank-One Convexity

5.4 A Jensen-Type Inequality for Gradient Young Measures

5.5 Lower Semicontinuity

5.6 Integrands with u-Dependence

5.7 Regularity of Minimizers

Chapter 6 Polyconvexity

6.1 Polyconvexity

6.2 Existence of Minimizers

6.3 Global Injectivity

Chapter 7 Relaxation

7.1 Quasiconvex Envelopes

7.2 Rigidity for Gradients

7.3 Relaxation of Integral Functionals

7.4 Relaxation via Young Measures

7.5 Mean Coercivity

Chapter 8 Quasiconvexity vs. Rank-One Convexity

8.1 Comparison Between Generalized Convexity Notions

8.2 Compensated Compactness

8.3 Characterization of Gradient Young Measures

Chapter 9 -Convergence

9.1 Abstract -Convergence

9.2 Periodic Homogenization

9.3 Convex Homogenization

9.4 Quadratic Homogenization

Part II Rigidity & Microstructure

Chapter 10 Rigidity

10.1 Rigidity for Gradient Inclusions

10.2 Linear Inclusions

10.3 The Young Measure Approach

10.4 Three-Gradient Inclusions

10.5 Four- and Five-Gradient Inclusions

10.6 The One-Well Inclusion

10.7 Multi-Well Inclusions in 2D

10.8 Two-Well Inclusions in 3D

Chapter 11 Microstructure

11.1 The T4-Configuration

11.2 Hulls of Sets

11.3 Multi-Well Inclusions

11.4 Non-Laminate Microstructures

11.5 Unbounded Microstructure

Chapter 12 Convex Integration

Part I: Basic Course

1 Introduction

2 Convexity

3 Variations

4 Young Measures

5 Quasiconvexity

6 Polyconvexity

7 Relaxation

8 Quasiconvexity vs. Rank-One Convexity

9 Γ-Convergence

Part II: Rigidity & Microstructure

10 Rigidity

11 Microstructure

12 Convex Integration

Part III: Linear-Growth Theory

13 Concentrations

14 Linear-Growth Functionals

15 Generalized Young Measures

Appendix A: Prerequisites

Notes:

Includes bibliographical references and index

Online resource; title from PDF title page (SpringerLink, viewed June 5, 2026)

ISBN:

9783032244833

3032244838

OCLC:

1593911534

Access Restriction:

Restricted for use by site license