Partial Differential Equations 19

This is the second edition of the now definitive text on partial differential equations (PDE). It offers a comprehensive survey of modern techniques in the theoretical study of PDE with particular emphasis on nonlinear equations. Its wide scope and clear exposition make it a great text for a graduate course in PDE. For this edition, the author has made numerous changes, including * a new chapter on nonlinear wave equations, * more than 80 new exercises, * several new sections, * a significantly expanded bibliography. About the First Edition: I have used this book for both regular PDE and topics courses. It has a wonderful combination of insight and technical detail. ... Evans' book is evidence of his mastering of the field and the clarity of presentation. —Luis Caffarelli, University of Texas It is fun to teach from Evans' book. It explains many of the essential ideas and techniques of partial differential equations ... Every graduate student in analysis should read it. —David Jerison, MIT I use Partial Differential Equations to prepare my students for their Topic exam, which is a requirement before starting working on their dissertation. The book provides an excellent account of PDE's ... I am very happy with the preparation it provides my students. —Carlos Kenig, University of Chicago Evans' book has already attained the status of a classic. It is a clear choice for students just learning the subject, as well as for experts who wish to broaden their knowledge ... An outstanding reference for many aspects of the field. —Rafe Mazzeo, Stanford University Front Cover Contents Preface to the second edition Preface to the first edition Chapter 1. Introduction 1.1. Partial Differential Equations 1.2. Examples 1.3. Strategies for Studying PDE 1.4. Overview 1.5. Problems 1.6. References Part I. Representation Formulas for Solutions Chapter 2. Four Important Linear Partial Differential Equations 2.1. Transport Equation 2.2. Laplace’s Equation 2.3. Heat Equation 2.4. Wave Equation 2.5. Problems 2.6. References Chapter 3. Nonlinear First-Order PDE 3.1. Complete Integrals, Envelopes 3.2. Characteristics 3.3. Introduction to Hamilton–Jacobi Equations 3.4. Introduction to Conservation Laws 3.5. Problems 3.6. References Chapter 4. Other Ways to Represent Solutions 4.1. Separation of Variables 4.2. Similarity Solutions 4.3. Transform Methods 4.4. Converting Nonlinear into Linear PDE 4.5. Asymptotics 4.6. Power Series 4.7. Problems 4.8. References Part II. Theory for Linear Partial Differential Equations Chapter 5. Sobolev Spaces 5.1. Hölder Spaces 5.2. Sobolev Spaces 5.3. Approximation 5.4. Extensions 5.5. Traces 5.6. Sobolev Inequalities 5.7. Compactness 5.8. Additional Topics 5.9. Other Spaces of Functions 5.10. Problems 5.11. References Chapter 6. Second-Order Elliptic Equations 6.1. Definitions 6.2. Existence of Weak Solutions 6.3. Regularity 6.4. Maximum Principles 6.5. Eigenvalues and Eigenfunctions 6.6. Problems 6.7. References Chapter 7. Linear Evolution Equations 7.1. Second-Order Parabolic Equations 7.2. Second-Order Hyperbolic Equations 7.3. Systems of Hyperbolic First-Order Equations 7.4. Semigroup Theory 7.5. Problems 7.6. References Part III. Theory for Nonlinear Partial Differential Equations Chapter 8. The Calculus of Variations 8.1. Introduction 8.2. Existence of Minimizers 8.3. Regularity 8.4. Constraints 8.5. Critical Points 8.6. Invariance, Noether’s Theorem 8.7. Problems 8.8. References Chapter 9. Nonvariational Techniques 9.1. Monotonicity Methods 9.2. Fixed Point Methods 9.3. Method of Subsolutions and Supersolutions 9.4. Nonexistence of Solutions 9.5. Geometric Properties of Solutions 9.6. Gradient Flows 9.7. Problems 9.8. References Chapter 10. Hamilton–Jacobi Equations 10.1. Introduction, Viscosity Solutions 10.2. Uniqueness 10.3. Control Theory, Dynamic Programming 10.4. Problems 10.5. References Chapter 11. Systems of Conservation Laws 11.1. Introduction 11.2. Riemann’s Problem 11.3. Systems of Two Conservation Laws 11.4. Entropy Criteria 11.5. Problems 11.6. References Chapter 12. Nonlinear Wave Equations 12.1. Introduction 12.2. Existence of Solutions 12.3. Semilinear Wave Equations 12.4. Critical Power Nonlinearity 12.5. Nonexistence of Solutions 12.6. Problems 12.7. References Appendices Appendix A. Notation A.1. Notation for matrices A.2. Geometric notation A.3. Notation for functions A.4. Vector-valued functions A.5. Notation for estimates A.6. Some comments about notation Appendix B. Inequalities B.1. Convex functions B.2. Useful inequalities Appendix C. Calculus C.1. Boundaries C.2. Gauss–Green Theorem C.3. Polar coordinates, coarea formula C.4. Moving regions C.5. Convolution and smoothing C.6. Inverse Function Theorem C.7. Implicit Function Theorem C.8. Uniform convergence Appendix D. Functional Analysis D.1. Banach spaces D.2. Hilbert spaces D.3. Bounded linear operators D.4. Weak convergence D.5. Compact operators, Fredholm theory D.6. Symmetric operators Appendix E. Measure Theory E.1. Lebesgue measure E.2. Measurable functions and integration E.3. Convergence theorems for integrals E.4. Differentiation E.5. Banach space-valued functions Bibliography Index Back Cover "This is the second edition of the now definitive text on partial differential equations (PDE). It offers a comprehensive survey of modern techniques in the theoretical study of PDE with particular emphasis on nonlinear equations. Its wide scope and clear exposition make it a great text for a graduate course in PDE. For this edition, the author has made numerous changes, including: a new chapter on nonlinear wave equations, more than 80 new exercises, several new sections, and a significantly expanded bibliography."--Publisher's description. "This is the second edition of the now definitive text on partial differential equations (PDE). It offers a comprehensive survey of modern techniques in the theoretical study of PDE with particular emphasis on nonlinear equations. Its wide scope and clear exposition make it a great text for a graduate course in PDE. For this edition, the author has made numerous changes, including : a new chapter on nonlinear wave equations, more than 80 new exercises, several new sections, and a significantly expanded bibliography."--Résumé de l'éditeur This third printing of the second edition corrects all known mistakes, as of February, 2022.