Preface

This book is a product of the experience of the authors in teaching partial differential equations to students of mathematics, physics, and engineering over a period of 20 years. Our goal in writing it has been to introduce the subject with precise and rigorous analysis on the one hand, and interesting and significant applications on the other.

The starting level of the book is at the first-year graduate level in a U.S. university. Previous experience with partial differential equations is not required, but the use of classical analysis to find solutions of specific problems is not emphasized. From that perspective our treatment is decidedly theoretical. We have avoided abstraction and full generality in many situations, however. Our plan has been to introduce fundamental ideas in relatively simple situations and to show their impact on relevant applications. The student is then, we feel, well prepared to fight through more specialized treatises.

There are parts of the exposition that require Lebesgue integration, distributions and Fourier transforms, and Sobolev spaces. We have included a long appendix, Chapter 8, giving precise statements of all results used. This may be thought of as an introduction to these topics. The reader who is not familiar with these subjects may refer to parts of Chapter 8 as needed or become somewhat familiar with them as prerequisite and treat Chapter 8 as Chapter 0. On the other hand, this book is entirely self-contained with regard to partial differential equations. We have refrained from referring to more advanced treatises in order to complete an exposition. The exercises are an important part of the exposition: They serve both to illustrate and to extend the theory, as well as to train the student by having her or him fill in the details omitted in the text.

The book has six main chapters. Chapter 2 concerns the wave equation, Chapter 3 the heat equation, and Chapter 4 potential theory and the Laplace equation. Chapter 5 deals with second-order elliptic equations, Chapter 6 abstract evolution equations, and Chapter 7 hyperbolic conservation laws. Together with appropriate parts of Chapter 8, the book covers material for two one-semester courses in partial differential equations, an introductory course (Chapters l--4) and an advanced course (Chapters 5--7).

A book on this subject is certain to have a large overlap with existing expositions. There are, however, certain novel features, as follows.

Chapter 2 includes a section on linear hyperbolic systems with constant coefficients in two variables and an application to transmission-reflection problems in layered media. Chapter 3 contains a variety of problems from applications aimed at stimulating the interest of the reader. Chapter 4 presents Perron s method of subharmonic functions for the Dirichlet problem as well as a fairly complete discussion of the boundary integral equations method for the solution of the Dirichlet and Neumann problems. This includes certain interesting variants that have recently arisen especially in connection with applications to aerodynamics. In addition, a complete proof is given of Wieners ~ criterion for regularity of boundary points for the Laplace equation. The part of the theory of capacities required for this is developed in a manner bridging the gap between physics and modem potential theory. Furthermore, a theorem of Widman, which shows that first derivatives of harmonic functions are continuous up to the boundary for CIą~ boundary and data, is proven. This result has previously appeared only in the research literature. In Chapter 5 we present a proof of the celebrated De Giorgi--Nash--Moser theorem for linear elliptic divergence structure operators, and we give a self-contained exposition of the proof of existence for the nonparametric Plateau problem using Perrons method. Also, results on surfaces with constant mean curvature and capillary surfaces are proven. These results have hitherto been available only in research papers and monographs. Chapter 6 is mainly devoted to the Hilbert space theory of parabolic evolution equations and the study of the initial value problem for a quasilinear "viscous" conservation law together with some a priori estimates in the "vanishing viscosity limit." This enables us to give a proof of Kruzhkovs theorem concerning "entropy solutions" of the scalar conservation law in Chapter 7. In this chapter we have also given a new exposition of the proof of local existence of Lipschitz continuous solutions of quasilinear hyperbolic systems in one space variable along with a companion continuation theorem. (We have avoided systems of conservation laws in several space variables because more sophisticated techniques are required.) A proof of Glimms theorem on existence of weak solutions concludes Chapter 7. The exposition of this difficult theorem has been given in sufficient detail to put it at the level of this book.

Finally, each chapter includes applications that have not appeared outside of the research literature.

We are indebted to Professor Victor Isakov and to Dr. Carlo Sinestrari for reading parts of the manuscript and suggesting valuable improvements. The figures were drawn by Megan Elcrat.

P. Bassanini Rome, Italy

A. R. Elcrat
Wichita, Kansas


Contents

1.

Introduction to Partial Differential Equations
1. Population Diffusion
2. Vibrating String Equation
3. Equations for Isentropic Flow of a Perfect Gas
4. Classification and Characteristics
Exercises
References
1
2
4
5
7
8
9


2.

Wave Equation
1. Initial Value Problem
2. Initial--Boundary Value Problems
3. Reflection Problem
4. Linear Hyperbolic Systems with Constant Coefficients in Two Variables
5. Wave Equation in Two and Three Dimensions
Exercises
References
11
11
16
24
29
32
43
51


3.

Heat Equation
1. Heat Kernel and Miscellaneous Solutions
2. Maximum Principle
3. Initial Value Problem
4. Inhomogeneous Initial Value Problem
5. Initial--Boundary Value Problems
Exercises
References
53
53
64
71
82
88
96
101


4.

Laplace Equation
1. Potential Theory: Basic Notions
2. General Properties of Harmonic Functions
     2.1. Greens Identities
     2.2. Maximum Principle
     2.3. Greens Function, Poissons Integral, and Mean Value Theorem
     2.4. Other Consequences of the Poisson Formula and the Mean Value Property
     2.5. Volume Potential and Poissons Equation
3. Dirichlet Problem
     3.1. Perrons Method
     3.2. C Regularity up to the Boundary
4. Integral Equation Formulations of Dirichlet and Neumann Problems
     4.1. Integral Operators with Weakly Singular Kernel
     4.2. Layer Potentials
     4.3. Layer Ansatz and Boundary Integral Equations
     4.4. Direct Method
     4.5. Poincare's Identity and Harmonic Vector Fields
5. Variational Theory
     5.1. Variational Solutions of Dirichlets Problem
     5.2. Variational Theory for Poissons Equation
     5.3. Laplace--Dirichlet Eigenvalue Problem
6. Capacity
7. Applications
     7.1. A Problem Concerning Asymptotic Efficiency of Cooling (Crushed Ice)
     7.2. A Free Boundary Problem Modeling Separated Flow of an Incompressible Fluid
Appendix to Section 1
Exercises
References
103
104
109
109
112
115
119
126
129
129
137
145
146
150
156
160
162
165
166
171
173
178
189
189
194
198
201
211


5.

Elliptic Partial Differential Equations of Second Order
1. Maximum Principle
2. Applications of the Maximum Principle
3. Equations with Discontinuous Coefficients
4. Nonlinear Elliptic Equations
     4.1. Monotone Operators
     4.2. Dirichlet Problem for the Minimal Surface Equation
Exercises
References
213
213
219
226
237
237
242
259
266


6.

Abstract Evolution Equations
1. Solution of the Heat Equation by Eigenfunction Expansions
2. Parabolic Evolution Equations
3. Nonlinear Initial Value Problem
Exercises
References
269
270
273
281
287
289


7.

Hyperbolic Systems of Conservation Laws in One Space Variable
1. Introduction
2. Local Existence Theorem for a.e. and Smooth Solutions of the Cauchy Problem
3. Scalar Equations
     3.1. Weak Solutions
     3.2. Kruzhkovs Theorem
     3.3. Riemann Problem
     3.4. Wave Interaction
4. Systems in One Space Variable
     4.1. Entropy Conditions
     4.2. Riemann Problem
5. Proof of Existence for Weak Solutions of Systems of Conservation Laws
Appendix to Section 5
Exercises
References
291
291
300
318
320
327
342
346
350
352
354
363
382
388
392


8.

Distributions and Sobolev Spaces
1. Banach and Hilbert Spaces
2. Theory of Distributions
3. Sobolev Spaces
4. Vector-Valued Functions
Exercises
References
Index
395
395
403
411
421
427
434
437

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