UNITEXT La Matematica per il 3+2 Volume 87
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Sandro Salsa Gianmaria Verzini Partial Differential Equations in Action Complements and Exercises
Sandro Salsa Dipartimento di Matematica Politecnico di Milano Milano, Italy Gianmaria Verzini Dipartimento di Matematica Politecnico di Milano Milano, Italy Translated by Simon G. Chiossi, UFBA Universidade Federal da Bahia, Salvador (Brazil). Translation from the Italian language edition: Equazioni a derivate parziali. Complementi ed esercizi, Sandro Salsa e Gianmaria Verzini, Springer-Verlag Italia, Milano 2005. All rights reserved. UNITEXT La Matematica per il 3+2 ISBN 978-3-319-15415-2 ISBN 978-3-319-15416-9 (ebook) DOI 10.1007/978-3-319-15416-9 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2015930285 Springer International Publishing Switzerland 2015 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. Cover Design: Simona Colombo, Giochi di Grafica, Milano, Italy Typesetting with L A TEX: PTP-Berlin, Protago TEX-Production GmbH, Germany (www.ptp-berlin.eu) Springer is a part of Springer Science+Business Media (www.springer.com)
Preface This book is designed for advanced undergraduate students from various disciplines, including applied mathematics, physics, and engineering. It evolved during the PDE courses that both authors have taught during recent decades at the Politecnico di Milano, and consists of problems of various types and difficulties. In the first part of the book, while much emphasis is placed on the most common methods of resolution, such as separation of variables or the method of characteristics, we also invite the student to handle the basic theoretical tools and properties of the solutions to the fundamental equations of mathematical physics. The second part is slightly more advanced and requires basic tools from functional analysis. A small number of exercises aims to familiarize the student with the first elements of the theory of distributions and of the Hilbertian Sobolev spaces. The focus then switches to the variational formulation of the most common boundary value problems for uniformly elliptic equations. A substantial number of problems is devoted to the use of the Riesz representation and the Lax-Milgram theorems together with Fredholm alternative to analyse well posedness or solvability of those problems. Next, a number of problems addresses the analysis of weak solutions to initial-boundary value problems for the heat or the wave equation. The text is completed by two short appendixes, the first dealing with Sturm-Liouville problems and Bessel functions and the second listing frequently used formulas. Each chapter begins with a brief review of the main theoretical concepts and tools that constitute necessary prerequisites for a proper understanding. The text Partial Differential Equation in Action [18], by S. Salsa, is the natural theoretical reference. Within each chapter, the problems are divided into two sections. In the first one we present detailed solutions and comments to provide the student with a reasonably complete guide. In the second section, we propose a set of problems that each student should try to solve by him- or herself. In each case, a solution can be found at the end of the chapter. Some problems are proposed as theoretical complements and may prove particularly challenging; this is especially true of those marked with one or two asterisks. Milano, January 2015 Sandro Salsa Gianmaria Verzini
Contents 1 Diffusion... 1 1.1 Backgrounds... 1 1.2 SolvedProblems... 3 1.2.1 The method of separation of variables... 3 1.2.2 Use of the maximum principle... 20 1.2.3 Applying the notion of fundamental solution... 25 1.2.4 Use of Fourier and Laplace transforms... 37 1.2.5 Problems in dimension higher than one... 43 1.3 FurtherExercises... 50 1.3.1 Solutions..... 56 2 The Laplace Equation... 81 2.1 Backgrounds... 81 2.2 SolvedProblems... 84 2.2.1 General properties of harmonic functions..... 84 2.2.2 Boundary-value problems. Solution methods... 95 2.2.3 Potentials and Green functions...117 2.3 FurtherExercises...124 2.3.1 Solutions.....130 3 First Order Equations...149 3.1 Backgrounds...149 3.2 SolvedProblems...152 3.2.1 Conservation laws and applications.....152 3.2.2 Characteristics for linear and quasilinear equations...181 3.3 FurtherExercises...194 3.3.1 Solutions.....197 4 Waves...215 4.1 Backgrounds...215 4.2 SolvedProblems...217
viii Contents 4.2.1 One-dimensional waves and vibrations...217 4.2.2 Canonical forms. Cauchy and Goursat problems.....238 4.2.3 Higher-dimensional problems...247 4.3 FurtherExercises...255 4.3.1 Solutions.....259 5 Functional Analysis...273 5.1 Backgrounds...273 5.2 SolvedProblems...278 5.2.1 Hilbert spaces......278 5.2.2 Distributions...291 5.2.3 Sobolev spaces.....298 5.3 FurtherExercises...310 5.3.1 Solutions.....314 6 Variational Formulations...333 6.1 Backgrounds...333 6.2 SolvedProblems...336 6.2.1 One-dimensional problems......336 6.2.2 Elliptic problems...346 6.2.3 Evolution problems...366 6.3 FurtherExercises...381 6.3.1 Solutions.....385 Appendix A. Sturm-Liouville, Legendre and Bessel Equations...405 A.1 Sturm-Liouville Equations....405 A.1.1 Regular equations....405 A.1.2 Legendre s equation.......406 A.2 Bessel sequationandfunctions...407 A.2.1 Bessel functions....407 A.2.2 Bessel s equation...410 Appendix B. Identities...413 B.1 Gradient,Divergence,Curl,Laplacian...413 B.2 Formulas...415 B.3 FourierTransforms...416 B.4 LaplaceTransforms...417 References...419