Universitext. Series editors Sheldon Axler San Francisco State University. Carles Casacuberta Universitat de Barcelona

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1 Universitext

2 Universitext Series editors Sheldon Axler San Francisco State University Carles Casacuberta Universitat de Barcelona Angus MacIntyre Queen Mary, University of London Kenneth Ribet University of California, Berkeley Claude Sabbah École polytechnique, CNRS, Université Paris-Saclay, Palaiseau Endre Süli University of Oxford Wojbor A. Woyczyński Case Western Reserve University Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master s level and beyond. The books, often well class-tested by their author, may have an informal, personal even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, into very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may make their way into Universitext. More information about this series at

3 David Borthwick Introduction to Partial Differential Equations 123

4 David Borthwick Department of Mathematics and Computer Science Emory University Atlanta, GA USA ISSN ISSN (electronic) Universitext ISBN ISBN (ebook) DOI / Library of Congress Control Number: Springer International Publishing AG 2016, corrected publication 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

5 Dedicated to my parents

6 Preface Partial differential equations (PDE) first appeared over 300 years ago, and the vast scope of the theory and applications that have since developed makes it challenging to give a reasonable introduction in a single semester. The modern mathematical approach to the subject requires considerable background in analysis, including topics such as metric space topology, measure theory, and functional analysis. This book is intended for an introductory course for students who do not necessarily have this analysis background. Courses taught at this level traditionally focus on some of the more elementary topics, such as Fourier series and simple boundary value problems. This approach risks giving students a somewhat narrow and outdated view of the subject. My goal here is to give a balanced presentation that includes modern methods, without requiring prerequisites beyond vector calculus and linear algebra. To allow for some of the more advanced methods to be reached within a single semester, the treatment is necessarily streamlined in certain ways. Concepts and definitions from analysis are introduced only as they will be needed in the text, and the reader is asked to accept certain fundamental results without justification. The emphasis is not on the rigorous development of analysis in its own right, but rather on the role that tools from analysis play in PDE applications. The text generally focuses on the most important classical PDE, which are the wave, heat, and Laplace equations. Nonlinear equations are discussed to some extent, but this coverage is limited. (Even at a very introductory level, the nonlinear theory merits a full course to itself.) I have tried to stress the interplay between modeling and mathematical analysis wherever possible. These connections are vital to the subject, both as a source of problems and as an inspiration for the development of methods. vii

7 viii Preface I owe a great debt of gratitude to my colleague Alessandro Veneziani, with whom I collaborated on this project originally. The philosophy of the book and choice of topics were heavily influenced by our discussions, and I am grateful for his support throughout the process. I would also like to thank former student Dallas Albritton, for offering comments and suggestions on an early draft. Thanks also to the series editors at Springer, for comments that helped improve the writing and presentation. Atlanta, USA September 2016 David Borthwick

8 The original version of the book was revised: Belated corrections from author have been incorporated. The erratum to the book is available at _14 ix

9 Contents 1 Introduction Partial Differential Equations Example: d Alembert s Wave Equation Types of Equations Well Posed Problems Approaches Preliminaries Real Numbers Complex Numbers Domains in R n Differentiability Ordinary Differential Equations Vector Calculus Exercises Conservation Equations and Characteristics Model Problem: Oxygen in the Bloodstream Lagrangian Derivative and Characteristics Higher-Dimensional Equations Quasilinear Equations Exercises The Wave Equation Model Problem: Vibrating String Characteristics Boundary Problems Forcing Terms Model Problem: Acoustic Waves Integral Solution Formulas Energy and Uniqueness Exercises xi

10 xii Contents 5 Separation of Variables Model Problem: Overtones Helmholtz Equation Circular Symmetry Spherical Symmetry Exercises The Heat Equation Model Problem: Heat Flow in a Metal Rod Scale-Invariant Solution Integral Solution Formula Inhomogeneous Problem Exercises Function Spaces Inner Products and Norms Lebesgue Integration L p Spaces Convergence and Completeness Orthonormal Bases Self-adjointness Exercises Fourier Series Series Solution of the Heat Equation Periodic Fourier Series Pointwise Convergence Uniform Convergence Convergence in L Regularity and Fourier Coefficients Exercises Maximum Principles Model Problem: The Laplace Equation Mean Value Formula Strong Principle for Subharmonic Functions Weak Principle for Elliptic Equations Application to the Heat Equation Exercises Weak Solutions Test Functions and Weak Derivatives Weak Solutions of Continuity Equations Sobolev Spaces Sobolev Regularity

11 Contents xiii 10.5 Weak Formulation of Elliptic Equations Weak Formulation of Evolution Equations Exercises Variational Methods Model Problem: The Poisson Equation Dirichlet s Principle Coercivity and Existence of a Minimum Elliptic Regularity Eigenvalues by Minimization Sequential Compactness Estimation of Eigenvalues Euler-Lagrange Equations Exercises Distributions Model Problem: Coulomb s Law The Space of Distributions Distributional Derivatives Fundamental Solutions Green s Functions Time-Dependent Fundamental Solutions Exercises The Fourier Transform Fourier Transform Tempered Distributions The Wave Kernel The Heat Kernel Exercises Erratum to: Introduction to Partial Differential Equations... E1 Appendix A: Analysis Foundations References Index

12 Notations :¼ Equal by definition Equal except on a set of measure zero Asymptotic to (ratio approaches 1) Comparable to (ratio bounded above and below) kk Norm kk p L p norm h ; i Inner product (L 2 by default) ð ; Þ Distributional pairing A n Volume of the unit sphere S n 1 R n B n Open unit ball fjxj\1g R n Bðx 0 ; RÞ Open ball fjx x 0 j\rg R n c k ½ Fourier coefficient C Complex numbers C m ðωþ m-times continuously differentiable functions Ω! C C m ðω; RÞ Real valued C m functions C m ð ΩÞ C m functions that admit extension across oω Ccpt m ðωþ Cm functions with compact support in Ω χ A Characteristic (or indicator) function of a set A D Unit disk in R 2 D α Multivariable derivative δ x Dirac delta function oω Boundary of Ω D 0 ðωþ Distributions on Ω D f ½ Dirichlet energy ds Surface integral element E Energy of a solution F Fourier transform Γ Gamma function r Gradient operator H m ðωþ Sobolev space, functions with weak derivatives in L 2 to order m xv

13 xvi Notations Hloc m ðωþ H0 1ðΩÞ H t Local Sobolev functions Closure of C1 cpt ðωþ in H1 ðωþ Heat kernel Laplacian operator on R n p-th power integrable functions Ω! C Locally integrable functions Ω! C Δ L p ðωþ L 1 loc ðωþ p Discrete L p space N Natural numbers {1; 2; 3;...} N 0 Non-negative integers {0; 1; 2; 3;...} ν Outward unit normal Ω Domain (open, connected set) in R n Ω Closure of Ω (Ω [ oω) Φ Fundamental solution r jxj in R n R Real numbers R½ Rayleigh quotient S n 1 Unit sphere fjxj ¼ 1g R n SðR n Þ Schwartz functions (smooth functions with rapid decay) S 0 ðr n Þ Tempered distributions S n ½ Partial sum of Fourier series T R=2πZ W t Wave kernel Z Integers {...; 1; 0; 1;...}