Functional Analysis, Sobolev Spaces, and Calculus of Variations

• Record Nr.: UNINA9910831009703321
• Autore: Pedregal Pablo
• Pubbl/distr/stampa: Cham : , : Springer, , 2024; ©2024
• ISBN: 3-031-49246-3
• Edizione: [1st ed.]
• Descrizione fisica: 1 online resource (391 pages)
• Collana: Unitext Series ; ; v.157
• Soggetti: Anàlisi funcional; Espais de Sobolev; Equacions en derivades parcials; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: Intro -- Preface -- Contents -- 1 Motivation and Perspective -- 1.1 Some Finite-Dimensional Examples -- 1.2 Basic Examples -- 1.3 More Advanced Examples -- 1.3.1 Transit Problems -- 1.3.2 Geodesics -- 1.3.3 Dirichlet's Principle -- 1.3.4 Minimal Surfaces -- 1.3.5 Isoperimetric Problems -- 1.3.6 Hamiltonian Mechanics -- 1.4 The Model Problem, and Some Variants -- 1.5 The Fundamental Issues for a Variational Problem -- 1.6 Additional Reasons to Care About Classes of Functions -- 1.7 Finite Versus Infinite Dimension -- 1.8 Brief Historical Background -- 1.9 Exercises -- Part I Basic Functional Analysis and Calculus of Variations -- 2 A First Exposure to Functional Analysis -- 2.1 Overview -- 2.2 Metric, Normed and Banach Spaces -- 2.3 Completion of Normed Spaces -- 2.4 Lp-Spaces -- 2.5 Weak Derivatives -- 2.6 One-Dimensional Sobolev Spaces -- 2.6.1 Basic Properties -- 2.6.2 Weak Convergence -- 2.7 The Dual Space -- 2.8 Compactness and Weak Topologies -- 2.9 Approximation -- 2.10 Completion of Spaces of Smooth Functions with Respect to Integral Norms -- 2.11 Hilbert Spaces -- 2.11.1 Orthogonal Projection -- 2.11.2 Orthogonality -- 2.11.3 The Dual of a Hilbert Space -- 2.11.4 Basic Calculus in a Hilbert Space -- 2.12 Some Other Important Spaces of Functions -- 2.13 Exercises -- 3 Introduction to Convex Analysis: ; The Hahn-Banach and Lax-Milgram Theorems -- 3.1 Overview -- 3.2 The Lax-Milgram Lemma -- 3.3 The Hahn-Banach Theorem: Analytic Form -- 3.4 The Hahn-Banach Theorem: Geometric Form -- 3.5 Some Applications -- 3.6 Convex Functionals, and the Direct Method -- 3.7 Convex Functionals, and the Indirect Method -- 3.8 Stampacchia's Theorem: Variational Inequalities -- 3.9 Exercises -- 4 The Calculus of Variations for One-dimensional Problems -- 4.1 Overview -- 4.2 Convexity -- 4.3 Weak Lower Semicontinuity for Integral Functionals.; 4.4 An Existence Result -- 4.5 Some Examples -- 4.5.1 Existence Under Constraints -- 4.6 Optimality Conditions -- 4.7 Some Explicit Examples -- 4.8 Non-existence -- 4.9 Exercises -- Part II Basic Operator Theory -- 5 Continuous Operators -- 5.1 Preliminaries -- 5.2 The Banach-Steinhaus Principle -- 5.3 The Open Mapping and Closed Graph Theorems -- 5.4 Adjoint Operators -- 5.5 Spectral Concepts -- 5.6 Self-Adjoint Operators -- 5.7 The Fourier Transform -- 5.8 Exercises -- 6 Compact Operators -- 6.1 Preliminaries -- 6.2 The Fredholm Alternative -- 6.3 Spectral Analysis -- 6.4 Spectral Decomposition of Compact, Self-Adjoint Operators -- 6.5 Exercises -- Part III Multidimensional Sobolev Spaces and Scalar Variational Problems -- 7 Multidimensional Sobolev Spaces -- 7.1 Overview -- 7.2 Weak Derivatives and Sobolev Spaces -- 7.3 Completion of Spaces of Smooth Functions of Several Variables with Respect to Integral Norms -- 7.4 Some Important Examples -- 7.5 Domains for Sobolev Spaces -- 7.6 Traces of Sobolev Functions: The Space W1, p0(Ω) -- 7.7 Poincaré's Inequality -- 7.8 Weak and Strong Convergence -- 7.9 Higher-Order Sobolev Spaces -- 7.10 Exercises -- 8 Scalar, Multidimensional Variational Problems -- 8.1 Preliminaries -- 8.2 Abstract, Quadratic Variational Problems -- 8.3 Scalar, Multidimensional Variational Problems -- 8.4 A Main Existence Theorem -- 8.5 Optimality Conditions: Weak Solutions for PDEs -- 8.6 Variational Problems in Action -- 8.7 Some Examples -- 8.8 Higher-Order Variational Principles -- 8.9 Non-existence and Relaxation -- 8.10 Exercises -- 9 Finer Results in Sobolev Spaces and the Calculus of Variations -- 9.1 Overview -- 9.2 Variational Problems Under Integral Constraints -- 9.3 Sobolev Inequalities -- 9.3.1 The Case of Vanishing Boundary Data -- 9.3.1.1 The Subcritical Case -- 9.3.1.2 The Critical Case.; 9.3.1.3 The Supercritical Case -- 9.3.2 The General Case -- 9.3.3 Higher-Order Sobolev Spaces -- 9.4 Regularity of Domains, Extension, and Density -- 9.5 An Existence Theorem Under More General Coercivity Conditions -- 9.6 Critical Point Theory -- 9.7 Regularity. Strong Solutions for PDEs -- 9.8 Eigenvalues and Eigenfunctions -- 9.9 Duality for Sobolev Spaces -- 9.10 Exercises -- A Hints and Solutions to Exercises -- A.1 Chapter 1 -- A.2 Chapter 2 -- A.3 Chapter 3 -- A.4 Chapter 4 -- A.5 Chapter 5 -- A.6 Chapter 6 -- A.7 Chapter 7 -- A.8 Chapter 8 -- A.9 Chapter 9 -- B So Much to Learn -- B.1 Variational Methods and Related Fields -- B.1.1 Some Additional Sources for the Calculus of Variations -- B.1.2 Introductory Courses -- B.1.3 Indirect Methods -- B.1.4 Convex and Non-smooth Analysis -- B.1.5 Lagrangian and Hamiltonian Formalism -- B.1.6 Variational Inequalities -- B.1.7 Non-existence and Young Measures -- B.1.8 Optimal Control -- B.1.9 -Convergence -- B.1.10 Other Areas -- B.2 Partial Differential Equations -- B.2.1 Non-linear PDEs -- B.2.2 Regularity for PDEs: Regularity of Ω Is Necessary -- B.2.3 Numerical Approximation -- B.3 Sobolev Spaces -- B.3.1 Spaces of Bounded Variation, and More General Spaces of Derivatives -- B.4 Functional Analysis -- References -- Index.