**Graduate Studies in Mathematics**

Volume: 205;
2020;
319 pp;
Hardcover

MSC: Primary 35;

**Print ISBN: 978-0-8218-3640-8
Product Code: GSM/205**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

**Electronic ISBN: 978-1-4704-5697-9
Product Code: GSM/205.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

#### Supplemental Materials

# Invitation to Partial Differential Equations

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*Mikhail Shubin*

Edited by Maxim Braverman, Robert McOwen, and Peter Topalov.

This book is based on notes from a beginning graduate course on partial differential equations. Prerequisites for using the book are a solid undergraduate course in real analysis. There are more than 100 exercises in the book. Some of them are just exercises, whereas others, even though they do require new ideas to solve them, provide additional important information about the subject.

#### Readership

Graduate students interested in partial differential equations.

#### Reviews & Endorsements

It is a great pleasure to see this book—written by a great master of the subject—finally in print. This treatment of a core part of mathematics and its applications offers the student both a solid foundation in basic calculations techniques in the subject, as well as a basic introduction to the more general machinery, e.g., distributions, Sobolev spaces, etc., which are such a key part of any modern treatment. As such this book is ideal for more advanced undergraduates as well as mathematically inclined students from engineering or the natural sciences. Shubin has a lovely intuitive writing style which provides a gentle introduction to this beautiful subject. Many good exercises (and solutions) are provided!

-- Rafe Mazzeo, Stanford University

This text provides an excellent semester's introduction to classical and modern topics in linear PDE, suitable for students with a background in advanced calculus and Lebesgue integration. The author intersperses treatments of the Laplace, heat, and wave equations with developments of various functional analytic tools, particularly distribution theory and spectral theory, introducing key concepts while deftly avoiding heavy technicalities.

-- Michael Taylor, University of North Carolina, Chapel Hill

#### Table of Contents

# Table of Contents

## Invitation to Partial Differential Equations

- Cover Cover11
- Title page iii4
- Foreword xi12
- Preface xiii14
- Selected notational conventions xvii18
- Chapter 1. Linear differential operators 120
- 1.1. Definition and examples 120
- 1.2. The total and the principal symbols 221
- 1.3. Change of variables 423
- 1.4. The canonical form of second-order operators with constant coefficients 524
- 1.5. Characteristics. Ellipticity and hyperbolicity 726
- 1.6. Characteristics and the canonical form of second-order operators and second-order equations for 𝑛=2 827
- 1.7. The general solution of a homogeneous hyperbolic equation with constant coefficients for 𝑛=2 1130
- 1.8. Appendix. Tangent and cotangent vectors 1231
- 1.9. Problems 1433

- Chapter 2. One-dimensional wave equation 1534
- 2.1. Vibrating string equation 1534
- 2.2. Unbounded string. The Cauchy problem. D’Alembert’s formula 2039
- 2.3. A semibounded string. Reflection of waves from the end of the string 2443
- 2.4. A bounded string. Standing waves. The Fourier method (separation of variables method) 2645
- 2.5. Appendix. The calculus of variations and classical mechanics 3352
- 2.6. Problems 3958

- Chapter 3. The Sturm-Liouville problem 4362
- Chapter 4. Distributions 5776
- 4.1. Motivation of the definition. Spaces of test functions 5776
- 4.2. Spaces of distributions 6382
- 4.3. Topology and convergence in the spaces of distributions 6786
- 4.4. The support of a distribution 7089
- 4.5. Differentiation of distributions and multiplication by a smooth function 7493
- 4.6. A general notion of the transposed (adjoint) operator. Change of variables. Homogeneous distributions 87106
- 4.7. Appendix. Minkowski inequality 91110
- 4.8. Appendix. Completeness of distribution spaces 94113
- 4.9. Problems 96115

- Chapter 5. Convolution and Fourier transform 99118
- 5.1. Convolution and direct product of regular functions 99118
- 5.2. Direct product of distributions 101120
- 5.3. Convolution of distributions 104123
- 5.4. Other properties of convolution. Support and singular support of a convolution 107126
- 5.5. Relation between smoothness of a fundamental solution and that of solutions of the homogeneous equation 109128
- 5.6. Solutions with isolated singularities. A removable singularity theorem for harmonic functions 113132
- 5.7. Estimates of derivatives of a solution of a hypoelliptic equation 114133
- 5.8. Fourier transform of tempered distributions 116135
- 5.9. Applying the Fourier transform to find fundamental solutions 120139
- 5.10. Liouville’s theorem 121140
- 5.11. Problems 123142

- Chapter 6. Harmonic functions 127146
- 6.1. Mean-value theorems for harmonic functions 127146
- 6.2. The maximum principle 129148
- 6.3. Dirichlet’s boundary value problem 131150
- 6.4. Hadamard’s example 133152
- 6.5. Green’s function for the Laplacian 136155
- 6.6. Hölder regularity 140159
- 6.7. Explicit formulas for Green’s functions 143162
- 6.8. Problems 147166

- Chapter 7. The heat equation 151170
- 7.1. Physical meaning of the heat equation 151170
- 7.2. Boundary value problems for the heat and Laplace equations 153172
- 7.3. A proof that the limit function is harmonic 155174
- 7.4. A solution of the Cauchy problem for the heat equation and Poisson’s integral 156175
- 7.5. The fundamental solution for the heat operator. Duhamel’s formula 161180
- 7.6. Estimates of derivatives of a solution of the heat equation 164183
- 7.7. Holmgren’s principle. The uniqueness of solution of the Cauchy problem for the heat equation 165184
- 7.8. A scheme for solving the first and second initial-boundary value problems by the Fourier method 168187
- 7.9. Problems 170189

- Chapter 8. Sobolev spaces. A generalized solution of Dirichlet’s problem 171190
- Chapter 9. The eigenvalues and eigenfunctions of the Laplace operator 193212
- 9.1. Symmetric and selfadjoint operators in Hilbert space 193212
- 9.2. The Friedrichs extension 197216
- 9.3. Discreteness of spectrum for the Laplace operator in a bounded domain 201220
- 9.4. Fundamental solution of the Helmholtz operator and the analyticity of eigenfunctions of the Laplace operator at the interior points. Bessel’s equation 202221
- 9.5. Variational principle. The behavior of eigenvalues under variation of the domain. Estimates of eigenvalues 209228
- 9.6. Problems 212231

- Chapter 10. The wave equation 215234
- 10.1. Physical problems leading to the wave equation 215234
- 10.2. Plane, spherical, and cylindric waves 220239
- 10.3. The wave equation as a Hamiltonian system 222241
- 10.4. A spherical wave caused by an instant flash and a solution of the Cauchy problem for the three-dimensional wave equation 228247
- 10.5. The fundamental solution for the three-dimensional wave operator and a solution of the nonhomogeneous wave equation 234253
- 10.6. The two-dimensional wave equation (the descent method) 236255
- 10.7. Problems 239258

- Chapter 11. Properties of the potentials and their computation 241260
- Chapter 12. Wave fronts and short-wave asymptotics for hyperbolic equations 257276
- 12.1. Characteristics as surfaces of jumps 257276
- 12.2. The Hamilton-Jacobi equation. Wave fronts, bicharacteristics, and rays 262281
- 12.3. The characteristics of hyperbolic equations 269288
- 12.4. Rapidly oscillating solutions. The eikonal equation and the transport equations 271290
- 12.5. The Cauchy problem with rapidly oscillating initial data 281300
- 12.6. Problems 287306

- Chapter 13. Answers and hints. Solutions 289308
- Bibliography 311330
- Index 315334
- Back Cover Back Cover1341