Holdings: Nonlinear functional analysis and its applications

Nonlinear functional analysis and its applications: 2,A Linear monotone operators

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Bibliographic Details
Main Author:	Zeidler, Eberhard 1940-2016 (Author)
Format:	Book
Language:	English
Published:	New York [u.a.] Springer 1990
Edition:	1. ed.
Subjects:	Anwendung Nichtlineare Funktionalanalysis
Online Access:	Inhaltsverzeichnis
Physical Description:	XVIII, 467 S. Ill.
ISBN:	3540968024 0387968024 9781461269717

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adam_text	Contents (Part II/A) Preface to Part И/А INTRODUCTION TO THE SUBJECT 1 CHAPTER 18 Variational Problems, the Ritz Method, and the Idea of Orthogonality 15 §18.1. The Space Cf (G) and the Variational Lemma 17 §18.2. Integration by Parts 19 §18.3. The First Boundary Value Problem and the Ritz Method 21 §18.4. The Second and Third Boundary Value Problems and the Ritz Method 28 §18.5. Eigenvalue Problems and the Ritz Method 32 §18.6. The Holder Inequality and its Applications 35 §18.7. The History of the Dirichlet Principle and Monotone Operators 40 §18.8. The Main Theorem on Quadratic Minimum Problems 56 §18.9. The Inequality of Poincaré-Friedrichs 59 §18.10. The Functional Analytic Justification of the Dirichlet Principle 60 § 18.11. The Perpendicular Principle, the Riesz Theorem, and the Main Theorem on Linear Monotone Operators 64 §18.12. The Extension Principle and the Completion Principle 70 §18.13. Proper Subregions 71 §18.14. The Smoothing Principle 72 §18.15. The Idea of the Regularity of Generalized Solutions and the Lemma of Weyl 78 §18.16. The Localization Principle 79 §18.17. Convex Variational Problems, Elliptic Differential Equations, and Monotonidty 81 xiii xiv Contents (Part II/A) §18.18. The General Euler-Lagrange Equations 85 §18.19. The Historical Development of the 19th and 20th Problems of Hubert and Monotone Operators 86 §18.20. Sufficient Conditions for Local and Global Minima and Locally Monotone Operators 93 CHAPTER 19 The Galerkin Method for Differential and Integral Equations, the Friedrichs Extension, and the Idea of Self-Adjointness 101 §19.1. Elliptic Differential Equations and the Galerkin Method 108 §19.2. Parabolic Differential Equations and the Galerkin Method 111 §19.3. Hyperbolic Differential Equations and the Galerkin Method 113 §19.4. Integral Equations and the Galerkin Method 115 §19.5. Complete Orthonormal Systems and Abstract Fourier Series 116 §19.6. Eigenvalues of Compact Symmetric Operators (Hilbert-Schmidt Theory) 119 §19.7. Proof of Theorem 19.B 121 §19.8. Self-Adjoint Operators 124 §19.9. The Friedrichs Extension of Symmetric Operators 126 §19.10. Proof of Theorem 19.C 129 §19.11. Application to the Poisson Equation 132 §19.12. Application to the Eigenvalue Problem for the Laplace Equation 134 § 19.13. The Inequality of Poincaré and the Compactness Theorem of Rellich 135 §19.14. Functions of Self-Adjoint Operators 138 §19.15. Application to the Heat Equation 141 §19.16. Application to the Wave Equation 143 §19.17. Semigroups and Propagators, and Their Physical Relevance 145 §19.18. Main Theorem on Abstract Linear Parabolic Equations 153 §19.19. Proof of Theorem 19.D 155 §19.20. Monotone Operators and the Main Theorem on Linear Nonexpansive Semigroups 159 §19.21. The Main Theorem on One-Parameter Unitary Groups 160 §19.22. Proof of Theorem 19.E 162 §19.23. Abstract Semilinear Hyperbolic Equations 164 §19.24. Application to Semilinear Wave Equations 166 §19.25. The Semilinear Schrödinger Equation 167 §19.26. Abstract Semilinear Parabolic Equations, Fractional Powers of Operators, and Abstract Sobolev Spaces 168 §19.27. Application to Semilinear Parabolic Equations 171 §19.28. Proof of Theorem 19.1 171 §19.29. Five General Uniqueness Principles and Monotone Operators 174 §19.30. A General Existence Principle and Linear Monotone Operators 175 CHAPTER 20 Difference Methods and Stability 192 §20.1. Consistency, Stability, and Convergence 195 §20.2. Approximation of Differential Quotients 199 Contents (Part Н/А) XV §20.3. Application to Boundary Value Problems for Ordinary Differential Equations 200 §20.4. Application to Parabolic Differential Equations 203 §20.5. Application to Elliptic Differential Equations 208 §20.6. The Equivalence Between Stability and Convergence 210 §20.7. The Equivalence Theorem of Lax for Evolution Equations 211 LINEAR MONOTONE PROBLEMS 225 CHAPTER 21 Auxiliary Tools and the Convergence of the Galerkin Method for Linear Operator Equations 229 §21.1. Generalized Derivatives 231 §21.2. Sobolev Spaces 235 §21.3. The Sobolev Embedding Theorems 237 §21.4, Proof of the Sobolev Embedding Theorems 241 §21.5. Duality in B-Spaces 251 §21.6. Duality in H-Spaces 253 §21.7. The Idea of Weak Convergence 255 §21.8. The Idea of Weak* Convergence 260 §21.9. Linear Operators 261 §21.10. Bilinear Forms 262 §21.11. Application to Embeddings 265 §21.12. Projection Operators 265 §21.13. Bases and Galerkin Schemes 271 §21.14. Application to Finite Elements 273 §21.15. Riesz-Schauder Theory and Abstract Fredholm Alternatives 275 §21.16. The Main Theorem on the Approximation-Solvability of Linear Operator Equations, and the Convergence of the Galerkin Method 279 §21.17. Interpolation Inequalities and a Convergence Trick 283 §21.18. Application to the Refined Banach Fixed-Point Theorem and the Convergence of Iteration Methods 285 §21.19. The Gagliardo-Nirenberg Inequalities 286 §21.20. The Strategy of the Fourier Transform for Sobolev Spaces 290 §21.21. Banach Algebras and Sobolev Spaces 292 §21.22. Moser-Type Calculus Inequalities 294 §21.23. Weakly Sequentially Continuous Nonlinear Operators on Sobolev Spaces 296 CHAPTER 22 Hubert Space Methods and Linear Elliptic Differential Equations 314 §22.1. Main Theorem on Quadratic Minimum Problems and the Ritz Method 320 §22.2. Application to Boundary Value Problems 325 §22.3. The Method of Orthogonal Projection, Duality, and a posteriori Error Estimates for the Ritz Method 335 §22.4. Application to Boundary Value Problems 337 xvi Contents (Part II/A) §22.5. Main Theorem on Linear Strongly Monotone Operators and the Galerkin Method 339 §22.6. Application to Boundary Value Problems 345 §22.7. Compact Perturbations of Strongly Monotone Operators, Fredholm Alternatives, and the Galerkin Method 347 §22.8. Application to Integral Equations 349 §22.9. Application to Bilinear Forms 350 §22.10. Application to Boundary Value Problems 351 §22.11. Eigenvalue Problems and the Ritz Method 352 §22.12. Application to Bilinear Forms 357 §22.13. Application to Boundary-Eigenvalue Problems 361 §22.14. Gårding Forms 364 §22.15. The Gårding Inequality for Elliptic Equations 366 §22.16. The Main Theorems on Gårding Forms 369 §22.17. Application to Strongly Elliptic Differential Equations of Order 2m 371 §22.18. Difference Approximations 374 §22.19. Interior Regularity of Generalized Solutions 376 §22.20. Proof of Theorem 22.H 378 §22.21. Regularity of Generalized Solutions up to the Boundary 383 §22.22. Proof of Theorem 22.1 384 CHAPTER 23 Hubert Space Methods and Linear Parabolic Differential Equations 402 §23.1. Particularities in the Treatment of Parabolic Equations 402 §23.2. The Lebesgue Space Lp(0, T; X) of Vector-Valued Functions 406 §23.3. The Dual Space to Lp(0,T;X) 410 §23.4. Evolution Triples 416 §23.5. Generalized Derivatives 417 §23.6. The Sobolev Space Wpl{0, T; V, H) 422 §23.7. Main Theorem on First-Order Linear Evolution Equations and the Galerkin Method 423 §23.8. Application to Parabolic Differential Equations 426 §23.9. Proof of the Main Theorem 430 CHAPTER 24 Hubert Space Methods and Linear Hyperbolic Differential Equations 452 §24.1. Main Theorem on Second-Order Linear Evolution Equations and the Galerkin Method 453 §24.2. Application to Hyperbolic Differential Equations 456 §24.3. Proof of the Main Theorem 459
adam_txt	Contents (Part II/A) Preface to Part И/А INTRODUCTION TO THE SUBJECT 1 CHAPTER 18 Variational Problems, the Ritz Method, and the Idea of Orthogonality 15 §18.1. The Space Cf (G) and the Variational Lemma 17 §18.2. Integration by Parts 19 §18.3. The First Boundary Value Problem and the Ritz Method 21 §18.4. The Second and Third Boundary Value Problems and the Ritz Method 28 §18.5. Eigenvalue Problems and the Ritz Method 32 §18.6. The Holder Inequality and its Applications 35 §18.7. The History of the Dirichlet Principle and Monotone Operators 40 §18.8. The Main Theorem on Quadratic Minimum Problems 56 §18.9. The Inequality of Poincaré-Friedrichs 59 §18.10. The Functional Analytic Justification of the Dirichlet Principle 60 § 18.11. The Perpendicular Principle, the Riesz Theorem, and the Main Theorem on Linear Monotone Operators 64 §18.12. The Extension Principle and the Completion Principle 70 §18.13. Proper Subregions 71 §18.14. The Smoothing Principle 72 §18.15. The Idea of the Regularity of Generalized Solutions and the Lemma of Weyl 78 §18.16. The Localization Principle 79 §18.17. Convex Variational Problems, Elliptic Differential Equations, and Monotonidty 81 xiii xiv Contents (Part II/A) §18.18. The General Euler-Lagrange Equations 85 §18.19. The Historical Development of the 19th and 20th Problems of Hubert and Monotone Operators 86 §18.20. Sufficient Conditions for Local and Global Minima and Locally Monotone Operators 93 CHAPTER 19 The Galerkin Method for Differential and Integral Equations, the Friedrichs Extension, and the Idea of Self-Adjointness 101 §19.1. Elliptic Differential Equations and the Galerkin Method 108 §19.2. Parabolic Differential Equations and the Galerkin Method 111 §19.3. Hyperbolic Differential Equations and the Galerkin Method 113 §19.4. Integral Equations and the Galerkin Method 115 §19.5. Complete Orthonormal Systems and Abstract Fourier Series 116 §19.6. Eigenvalues of Compact Symmetric Operators (Hilbert-Schmidt Theory) 119 §19.7. Proof of Theorem 19.B 121 §19.8. Self-Adjoint Operators 124 §19.9. The Friedrichs Extension of Symmetric Operators 126 §19.10. Proof of Theorem 19.C 129 §19.11. Application to the Poisson Equation 132 §19.12. Application to the Eigenvalue Problem for the Laplace Equation 134 § 19.13. The Inequality of Poincaré and the Compactness Theorem of Rellich 135 §19.14. Functions of Self-Adjoint Operators 138 §19.15. Application to the Heat Equation 141 §19.16. Application to the Wave Equation 143 §19.17. Semigroups and Propagators, and Their Physical Relevance 145 §19.18. Main Theorem on Abstract Linear Parabolic Equations 153 §19.19. Proof of Theorem 19.D 155 §19.20. Monotone Operators and the Main Theorem on Linear Nonexpansive Semigroups 159 §19.21. The Main Theorem on One-Parameter Unitary Groups 160 §19.22. Proof of Theorem 19.E 162 §19.23. Abstract Semilinear Hyperbolic Equations 164 §19.24. Application to Semilinear Wave Equations 166 §19.25. The Semilinear Schrödinger Equation 167 §19.26. Abstract Semilinear Parabolic Equations, Fractional Powers of Operators, and Abstract Sobolev Spaces 168 §19.27. Application to Semilinear Parabolic Equations 171 §19.28. Proof of Theorem 19.1 171 §19.29. Five General Uniqueness Principles and Monotone Operators 174 §19.30. A General Existence Principle and Linear Monotone Operators 175 CHAPTER 20 Difference Methods and Stability 192 §20.1. Consistency, Stability, and Convergence 195 §20.2. Approximation of Differential Quotients 199 Contents (Part Н/А) XV §20.3. Application to Boundary Value Problems for Ordinary Differential Equations 200 §20.4. Application to Parabolic Differential Equations 203 §20.5. Application to Elliptic Differential Equations 208 §20.6. The Equivalence Between Stability and Convergence 210 §20.7. The Equivalence Theorem of Lax for Evolution Equations 211 LINEAR MONOTONE PROBLEMS 225 CHAPTER 21 Auxiliary Tools and the Convergence of the Galerkin Method for Linear Operator Equations 229 §21.1. Generalized Derivatives 231 §21.2. Sobolev Spaces 235 §21.3. The Sobolev Embedding Theorems 237 §21.4, Proof of the Sobolev Embedding Theorems 241 §21.5. Duality in B-Spaces 251 §21.6. Duality in H-Spaces 253 §21.7. The Idea of Weak Convergence 255 §21.8. The Idea of Weak* Convergence 260 §21.9. Linear Operators 261 §21.10. Bilinear Forms 262 §21.11. Application to Embeddings 265 §21.12. Projection Operators 265 §21.13. Bases and Galerkin Schemes 271 §21.14. Application to Finite Elements 273 §21.15. Riesz-Schauder Theory and Abstract Fredholm Alternatives 275 §21.16. The Main Theorem on the Approximation-Solvability of Linear Operator Equations, and the Convergence of the Galerkin Method 279 §21.17. Interpolation Inequalities and a Convergence Trick 283 §21.18. Application to the Refined Banach Fixed-Point Theorem and the Convergence of Iteration Methods 285 §21.19. The Gagliardo-Nirenberg Inequalities 286 §21.20. The Strategy of the Fourier Transform for Sobolev Spaces 290 §21.21. Banach Algebras and Sobolev Spaces 292 §21.22. Moser-Type Calculus Inequalities 294 §21.23. Weakly Sequentially Continuous Nonlinear Operators on Sobolev Spaces 296 CHAPTER 22 Hubert Space Methods and Linear Elliptic Differential Equations 314 §22.1. Main Theorem on Quadratic Minimum Problems and the Ritz Method 320 §22.2. Application to Boundary Value Problems 325 §22.3. The Method of Orthogonal Projection, Duality, and a posteriori Error Estimates for the Ritz Method 335 §22.4. Application to Boundary Value Problems 337 xvi Contents (Part II/A) §22.5. Main Theorem on Linear Strongly Monotone Operators and the Galerkin Method 339 §22.6. Application to Boundary Value Problems 345 §22.7. Compact Perturbations of Strongly Monotone Operators, Fredholm Alternatives, and the Galerkin Method 347 §22.8. Application to Integral Equations 349 §22.9. Application to Bilinear Forms 350 §22.10. Application to Boundary Value Problems 351 §22.11. Eigenvalue Problems and the Ritz Method 352 §22.12. Application to Bilinear Forms 357 §22.13. Application to Boundary-Eigenvalue Problems 361 §22.14. Gårding Forms 364 §22.15. The Gårding Inequality for Elliptic Equations 366 §22.16. The Main Theorems on Gårding Forms 369 §22.17. Application to Strongly Elliptic Differential Equations of Order 2m 371 §22.18. Difference Approximations 374 §22.19. Interior Regularity of Generalized Solutions 376 §22.20. Proof of Theorem 22.H 378 §22.21. Regularity of Generalized Solutions up to the Boundary 383 §22.22. Proof of Theorem 22.1 384 CHAPTER 23 Hubert Space Methods and Linear Parabolic Differential Equations 402 §23.1. Particularities in the Treatment of Parabolic Equations 402 §23.2. The Lebesgue Space Lp(0, T; X) of Vector-Valued Functions 406 §23.3. The Dual Space to Lp(0,T;X) 410 §23.4. Evolution Triples 416 §23.5. Generalized Derivatives 417 §23.6. The Sobolev Space Wpl{0, T; V, H) 422 §23.7. Main Theorem on First-Order Linear Evolution Equations and the Galerkin Method 423 §23.8. Application to Parabolic Differential Equations 426 §23.9. Proof of the Main Theorem 430 CHAPTER 24 Hubert Space Methods and Linear Hyperbolic Differential Equations 452 §24.1. Main Theorem on Second-Order Linear Evolution Equations and the Galerkin Method 453 §24.2. Application to Hyperbolic Differential Equations 456 §24.3. Proof of the Main Theorem 459
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spellingShingle	Zeidler, Eberhard 1940-2016 Nonlinear functional analysis and its applications Anwendung (DE-588)4196864-5 gnd Nichtlineare Funktionalanalysis (DE-588)4042093-0 gnd
subject_GND	(DE-588)4196864-5 (DE-588)4042093-0
title	Nonlinear functional analysis and its applications
title_alt	Vorlesungen über nichtlineare Funktionalanalysis
title_auth	Nonlinear functional analysis and its applications
title_exact_search	Nonlinear functional analysis and its applications
title_exact_search_txtP	Nonlinear functional analysis and its applications
title_full	Nonlinear functional analysis and its applications 2,A Linear monotone operators Eberhard Zeidler
title_fullStr	Nonlinear functional analysis and its applications 2,A Linear monotone operators Eberhard Zeidler
title_full_unstemmed	Nonlinear functional analysis and its applications 2,A Linear monotone operators Eberhard Zeidler
title_short	Nonlinear functional analysis and its applications
title_sort	nonlinear functional analysis and its applications linear monotone operators
topic	Anwendung (DE-588)4196864-5 gnd Nichtlineare Funktionalanalysis (DE-588)4042093-0 gnd
topic_facet	Anwendung Nichtlineare Funktionalanalysis
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015310992&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
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work_keys_str_mv	AT zeidlereberhard vorlesungenubernichtlinearefunktionalanalysis AT zeidlereberhard nonlinearfunctionalanalysisanditsapplications2a

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