INTRODUCTION TO THE CALCULUS OF VARIATIONS

By U. KATAY

CONTENTS INTRODUCTION BRIEF HISTORICAL COMMENTS MODEL PROBLEM AND SOME EXAMPLES PRESENTATION OF THE CONTENT OF THE MONOGRAPH PRELIMINARIES CONTINUOUS AND HÖLDER CONTINUOUS FUNCTIONS EXERCISES LP SPACES SOBOLEV SPACES CONVEX ANALYSIS CLASSICAL METHODS EULER-LAGRANGE EQUATION SECOND FORM OF THE EULER-LAGRANGE EQUATION HAMILTONIAN FORMULATION HAMILTON-JACOBI EQUATION FIELDS THEORIES DIRECT METHODS THE MODEL CASE: DIRICHLET INTEGRAL A GENERAL EXISTENCE THEOREM EULER-LAGRANGE EQUATIONS THE VECTORIAL CASE RELAXATION THEORY REGULARITY THE ONE DIMENSIONAL CASE SOME GENERAL RESULTS MINIMAL SURFACES GENERALITIES ABOUT SURFACES THE DOUGLAS-COURANT-TONELLI METHOD REGULARITY, UNIQUENESS AND NON UNIQUENESS NONPARAMETRIC MINIMAL SURFACES ISOPERIMETRIC INEQUALITY THE CASE OF DIMENSION 2 THE CASE OF DIMENSION N SOLUTIONS TO THE EXERCISES CHAPTER 1: PRELIMINARIES CHAPTER 2: CLASSICAL METHODS CHAPTER 3: DIRECT METHODS CHAPTER 4: REGULARITY CHAPTER 5: MINIMAL SURFACES CHAPTER 6: ISOPERIMETRIC INEQUALITY