Calculus of Variations

Description

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 G-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. Preface Contents Part I Basic Course 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 Microstructure in Crystals 1.9 Phase Transitions 1.10 Composite Elastic Materials 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 The General Theory of Convex Functions and Duality 3 Variations 3.1 The Euler–Lagrange Equation 3.2 Regularity of Minimizers 3.3 Lagrange Multipliers 3.4 Invariances and Noether's Theorem 3.5 Subdifferentials 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 5 Quasiconvexity 5.1 Quasiconvexity 5.2 Null-Lagrangians 5.3 A Jensen-Type Inequality for Gradient Young Measures 5.4 Rigidity for Gradients 5.5 Lower Semicontinuity 5.6 Integrands with u-Dependence 5.7 Regularity of Minimizers 6 Polyconvexity 6.1 Polyconvexity 6.2 Existence of Minimizers 6.3 Global Injectivity 7 Relaxation 7.1 Quasiconvex Envelopes 7.2 Relaxation of Integral Functionals 7.3 Generalized Convexity Notions and Envelopes 7.4 Young Measure Relaxation 7.5 Characterization of Gradient Young Measures Part II Advanced Topics 8 Rigidity 8.1 Two-Gradient Inclusions 8.2 Linear Inclusions 8.3 Relaxation and Quasiconvex Hulls of Sets 8.4 Multi-point Inclusions 8.5 The One-Well Inclusion 8.6 Multi-well Inclusions in 2D 8.7 Two-Well Inclusions in 3D 8.8 Compensated Compactness 9 Microstructure 9.1 Laminates and Hulls of Sets 9.2 Multi-well Inclusions 9.3 Convex Integration 9.4 Infinite-Order Laminates 9.5 Crystalline Microstructure in 3D 9.6 Stability of Gradient Distributions 9.7 Non-laminate Microstructures 9.8 Unbounded Microstructure 10 Singularities 10.1 Strict Convergence of Measures 10.2 Tangent Measures 10.3 Functions of Bounded Variation 10.4 Structure of Singularities 10.5 Convexity at Singularities 11 Linear-Growth Functionals 11.1 Extension of Functionals 11.2 Lower Semicontinuity 11.3 Relaxation 12 Generalized Young Measures 12.1 Functional Analysis Setup 12.2 Generation and Examples 12.3 Extended Representation 12.4 Strong Precompactness of Sequences 12.5 BV-Young Measures 12.6 Localization 12.7 Lower Semicontinuity 13 Γ-Convergence 13.1 Abstract Γ-Convergence 13.2 Sharp-Interface Limits 13.3 Higher-Order Sharp-Interface Limits 13.4 Periodic Homogenization 13.5 Convex Homogenization 13.6 Quadratic Homogenization A Prerequisites A.1 Linear Algebra A.2 Functional Analysis A.3 Measure Theory A.4 Vector Measures A.5 Sobolev and Other Function Spaces A.6 Harmonic Analysis Appendix References Index