INTRO TO CALCUL VARIA (4TH ED)

by Bernard Dacorogna

Overview

The calculus of variations is one of the oldest subjects in mathematics, and it is very much alive and still evolving. Besides its mathematical importance and its links to other branches of mathematics, such as geometry or differential equations, it is widely used in physics, engineering, economics and biology.

This book serves both as a guide to the expansive existing literature and as an aid to the non-specialist — mathematicians, physicists, engineers, students or researchers — in discovering the subject's most important problems, results and techniques. Despite the aim of addressing non-specialists, mathematical rigor has not been sacrificed; most of the theorems are either fully proved or proved under more stringent conditions.

This new edition offers an entirely new chapter, as well as the addition of several new exercises. The book, containing a total of 147 exercises with detailed solutions, is well designed for a course at both undergraduate and graduate levels.

Contents:

• Preface to the Fourth English Edition
• About the Author
• Introduction
• Preliminaries
• Classical Methods
• Direct Methods: Existence
• Direct Methods: Regularity
• Minimal Surfaces
• Isoperimetric Inequality
• Geodesic
• Solutions to the Exercises
• Bibliography
• Index

Readership: This book is suitable for advanced undergraduate and graduate students, as well as researchers in the field of calculus of variations and differential equations. It would also be applicable to physicists, engineers economists or biologists more generally who are interested in mathematics.

Reviews of Previous Editions:A great feature is the [chapter] presenting complete solutions to all the exercises set earlier in the book … this is a well-thought-out selection, and Dacorogna's expert discussion is everywhere really clear and nicely motivated, with lots of detail put in. He obviously cares about actually teaching and not just covering material. - Professor L C EvansUniversity of California, BerkeleySIAM Review

This wonderful book is imbued with a marvelous historical perspective so that the reader is taught some very beautiful mathematics fitted in the proper historical perspective … it is full of terrific hard analysis focused on a general theme that is exemplified by the author's astute and elegant choice of topics … there are a lot of (outstanding) exercises and these are critical for a deeper understanding of the material. … it's a very beautiful treatment, and will reward the diligent reader with a solid introduction to a great and grand subject and to a lot of beautiful hard analysis. - Mathematical Association of America online book review

This book provides non-mathematics students with an easy way to grasp the basic idea of the calculus of variations, and its possible applications in their field of study. For mathematics students, the book leads them to the very directions which should be followed. - Professor Ji-Huan HeDonghua University, Shanghai

A lot of new material has been added, in particular, complements and exercises. The book is recommended not only for students but also for scientists from other disciplines that want to approach the fascinating field of variational problems. - Mathematical Reviews Clippings

Key Features:

• Serves as an excellent introduction to the calculus of variations
• Useful to researchers in different fields of mathematics who want to get a concise but broad introduction to the subject
• Includes nearly 150 exercises with solutions
• New edition offers a completely new chapter

Product Details

ISBN-13:	9781800615281
Publisher:	WSPC (EUROPE)
Publication date:	08/27/2024
Series:	ESSENTIAL TEXTBOOKS IN MATHEMATICS
Sold by:	Barnes & Noble
Format:	eBook
Pages:	368
File size:	50 MB
Note:	This product may take a few minutes to download.

Table of Contents

Prefaces to the English Edition ix

Preface to the French Edition xi

0 Introduction 1

0.1 Brief historical comments 1

0.2 Model problem and some examples 3

0.3 Presentation of the content of the monograph 7

1 Preliminaries 13

1.1 Introduction 13

1.2 Continuous and Höet;lder continuous functions 14

1.2.1 Exercises 18

1.3 L spaces 19

1.3.1 Exercises 26

1.4 Sobolev spaces 29

1.4.1 Exercises 42

1.5 Convex analysis 45

1.5.1 Exercises 48

2 Classical methods 51

2.1 Introduction 51

2.2 Euler-Lagrange equation 53

2.2.1 Exercises 64

2.3 Second form of the Euler-Lagrange equation 66

2.3.1 Exercises 68

2.4 Hamiltonian formulation 69

2.4.1 Exercises 76

2.5 Hamilton-Jacobi equation 77

2.5.1 Exercises 81

2.6 Fields theories 81

2.6.1 Exercises 86

3 Direct methods: existence 87

3.1 Introduction 87

3.2 The model case: Dirichlet integral 89

3.2.1 Exercise 92

3.3 A general existence theorem 92

3.3.1 Exercises 99

3.4 Euler-Lagrange equation 101

3.4.1 Exercises 107

3.5 The vectorial case 107

3.5.1 Exercises 115

3.6 Relaxation theory 118

3.6.1 Exercises 121

4 Direct methods: regularity 125

4.1 Introduction 125

4.2 The one dimensional case 126

4.2.1 Exercises 131

4.3 The difference quotient method: interior regularity 133

4.3.1 Exercises 139

4.4 The difference quotient method: boundary regularity 140

4.4.1 Exercises 143

4.5 Higher regularity for the Dirichlet integral 144

4.5.1 Exercises 146

4.6 Weyl lemma 147

4.6.1 Exercise 150

4.7 Some general results 150

4.7.1 Exercises 152

5 Minimal surfaces 155

5.1 Introduction 155

5.2 Generalities about surfaces 158

5.2.1 Exercises166

5.3 The Douglas-Courant-Tonelli method 167

5.3.1 Exercise 173

5.4 Regularity, uniqueness and non-uniqueness 173

5.5 Nonparametric minimal surfaces 175

5.5.1 Exercise 180

6 Isoperimetric inequality 181

6.1 Introduction 181

6.2 The case of dimension 2 182

6.2.1 Exercises 188

6.3 The case of dimension n 189

6.3.1 Exercises 196

7 Solutions to the Exercises 199

7.1 Chapter 1. Preliminaries 199

7.1.1 Continuous and Höet;lder continuous functions 203

7.1.2 L spaces 210

7.1.3 Sobolev spaces 217

7.1.4 Convex analysis 217

7.2 Chapter 2 Classical methods 224

7.2.1 Euler-Lagrange equation 224

7.2.2 Second form of the Euler-Lagrange equation 230

7.2.3 Hamiltonian formulation 231

7.2.4 Hamiltion-Jacobi equation 232

7.2.5 Fields theories 234

7.3 Chapter 3 Direct methods: existence 236

7.3.1 The model case: Dirichlet integral 236

7.3.2 A general existence theorem 236

7.3.3 Euler-Lagrange equation 239

7.3.4 The vectorial case 240

7.3.5 Relaxation theory 247

7.4 Chapter 4 Direct methods: regularity 251

7.4.1 The one dimensional case 251

7.4.2 The difference quotient method: interior regularity 254

7.4.3 The difference quotient method: boundary regularity 256

7.4.4 Higher regularity for the Dirichlet Integral 257

7.4.5 Weyl lemma 259

7.4.6 Some general results 260

7.5 Chapter 5 Minimal surfaces 263

7.5.1 Generalities about surfaces 263

7.5.2 The Douglas-Courant-Tonelli method 266

7.5.3 Nonparametric minimal surfaces 267

7.6 Chapter 6 Isoperimetric inequality 268

7.6.1 The case of dimension 2 268

7.6.2 The case of dimension n 271

Bibliography 275

Index 283