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Lectures on Partial Differential Equations

Vladimir I. Arnold
Publisher: 
Springer Verlag
Publication Date: 
2004
Number of Pages: 
157
Format: 
Paperback
Series: 
Universitext
Price: 
49.95
ISBN: 
3-540-40448-1
Category: 
Textbook
[Reviewed by
Victor Shubov
, on
01/20/2001
]

This book is written by a mathematician who is famous both for his fundamental contributions to modern mathematics and as an author of several textbooks and monographs on Mathematical Methods in Classical Mechanics, Ordinary Differential Equations, Singularities of Differentiable Maps, and Topological Methods in Hydrodynamics. The book is based on a short course of lectures delivered to the third year mathematics students of the Independent University of Moscow in the fall semester of 1994.

This introduction to PDEs is highly unusual and is very different from numerous textbooks on the subject. The book is not designed to serve as a systematic textbook for, e.g., science and engineering students. The reason is twofold. First, the reader is assumed to have a background in abstract differential geometry. Namely, such notions as differential forms,jets, symplectic and contact structures, cotangent bundle, Riemannian manifold, projective space etc. are used from the very beginning of the text. (Some of these notions are briefly defined but the reader should definitely be familiar with them.) Secondly, only a number of selected topics is discussed. For example, the wave equation and separation of variables are considered only in the case of a one -dimensional vibrating string. The heat equation is not considered at all. Some topics, e.g., boundary value problems for the Laplace equation, are described in a short verbal paragraphs. In fact, several topics whose description in standard texts involves a lot of important formulas are described in short verbal paragraphs. These descriptions are always very precise and go directly to the heart of the topic. However, the reader must definitely have sufficient mathematical maturity and technical skills to translate these short explanations to the language of formulas.

The topics that are selected for discussion are treated with great elegance and in a nonstandard manner, emphasizing both physical intuition and geometric insight. Some parts of the lectures contain material whose accessible exposition can hardly be found in any any other place. An example of this is an appendix devoted to topological content of the multifield representation of spherical functions (i. e., the fact that spherical functions can be obtained by calculating all partial derivatives of the Coulomb potential). The book contains many nontrivial problems.

The book can serve as a nonstandard, geometrically motivated introduction to PDEs for students having some background in pure abstract mathematics. It will also be of definite interest to specialists since it provides a fresh unusual approach to some standard topics. It is, probably, worth mentioning that the introduction contains some general philosophical views of the author on the subject of PDEs and modern mathematics as a whole and will be of interest to a broad mathematical audience.


Victor Shubov is Visiting Professor of Mathematics at Colby College.

 

Contents
Preface to the Second Russian Edition . . . . . . . . . . . . . . . . . . . . . . . . . V
1. The General Theory for One First-Order Equation . . . . . . . . 1
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2. The General Theory for One First-Order Equation
(Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3. Huygens' Principle in the Theory of Wave Propagation . . . . 21
4. The Vibrating String (d'Alembert's Method) . . . . . . . . . . . . . . 27
4.1. The General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2. Boundary-Value Problems and the Cauchy Problem . . . . . . . . . 28
4.3. The Cauchy Problem for an Infinite String. d'Alembert's
Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4. The Semi-Infinite String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.5. The Finite String. Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.6. The Fourier Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5. The Fourier Method (for the Vibrating String) . . . . . . . . . . . . 35
5.1. Solution of the Problem in the Space of Trigonometric
Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2. A Digression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3. Formulas for Solving the Problem of Section 5.1 . . . . . . . . . . . . . 36
5.4. The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.5. Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.6. Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.7. Gibbs' Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
X Contents
6. The Theory of Oscillations. The Variational Principle . . . . . . 41
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7. The Theory of Oscillations. The Variational Principle
(Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
8. Properties of Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 65
8.1. Consequences of the Mean-Value Theorem . . . . . . . . . . . . . . . . . . 67
8.2. The Mean-Value Theorem in the Multidimensional Case . . . . . . 73
9. The Fundamental Solution for the Laplacian. Potentials . . . 77
9.1. Examples and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
9.2. A Digression. The Principle of Superposition . . . . . . . . . . . . . . . . 79
9.3. Appendix. An Estimate of the Single-Layer Potential . . . . . . . . 89
10. The Double-Layer Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.1.Properties of the Double-Layer Potential . . . . . . . . . . . . . . . . . . . 94
11. Spherical Functions. Maxwell's Theorem. The Removable
Singularities Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
12. Boundary-Value Problems for Laplace's Equation. Theory
of Linear Equations and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 121
12.1. Four Boundary-Value Problems for Laplace's Equation . . . . . . . 121
12.2.Existence and Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . . 125
12.3.Linear Partial Differential Equations and Their Symbols . . . . . . 127
A. The Topological Content of Maxwell's Theorem on the
Multifield Representation of Spherical Functions. . . . . . . . . . . 135
A.1. The Basic Spaces and Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A.2. Some Theorems of Real Algebraic Geometry . . . . . . . . . . . . . . . . 137
A.3. From Algebraic Geometry to Spherical Functions . . . . . . . . . . . . 139
A.4. Explicit Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.5. Maxwell's Theorem and CP2/conj ~ S4 . . . . . . . . . . . . . . . . . . . . 144
A.6. The History of Maxwell's Theorem . . . . . . . . . . . . . . . . . . . . . . . . 145
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
B. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
B.1. Material from the Seminars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
B.2. Written Examination Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

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