偏微分方程讲义

Preface to the Second Russian Edition

1. The General Theory for One First-Order Equation Literature

2. The General Theory for One First-Order Equation(Continued)Literature

3. Huygens Principle in the Theory of Wave Propagation.

4. The Vibrating String (dAlemberts Method)

4.1. The General Solution

4.2. Boundary-Value Problems and the Cauchy Problem

4.3. The Cauehy Problem for an Infinite Strifig. dAlemberts Formula

4.4. The Semi-Infinite String

4.5. The Finite String. Resonance

4.6. The Fourier Method

5. The Fourier Method (for the Vibrating String)

5.1. Solution of the Problem in the Space of Trigonometric Polynomials

5.2. A Digression

5.3. Formulas for Solving the Problem of Section 5.1

5.4. The General Case

5.5. Fourier Series

5.6. Convergence of Fourier Series

5.7. Gibbs Phenomenon

6. The Theory of Oscillations. The Variational Principle Literature

7. The Theory of Oscillations. The Variational Principle(Continued)

8. Properties of Harmonic Functions

8.1. Consequences of the Mean-Value Theorem

8.2. The Mean-Value Theorem in the Multidimensional Case

9. The Fundamental Solution for the Laplacian. Potentials

9.1. Examples and Properties

9.2. A Digression. The Principle of Superposition

9.3. Appendix. An Estimate of the Single-Layer Potential

10. The Double-Layer Potential

10.1. Properties of the Double-Layer Potential

11 Spherical Functions. Maxwells Theorem. The Removable

Singularities Theorem

12. Boundary-Value Problems for Laplaees Equation. Theory of Linear Equations and Systems

12.1. Four Boundary-Value Problems for Laplaces Equation

12.2. Existence and Uniqueness of Solutions

12.3. Linear Partial Differential Equations and Their Symbols

A. The Topological Content of Maxwells Theorem on the Multifield Representation of Spherical Functions

A.1. The Basic Spaces and Groups

A.2. Some Theorems of Real Algebraic Geometry

A.3. From Algebraic Geometry to Spherical Functions

A.4. Explicit Formulas

A.5. Maxwells Theorem and Cp2/con≈S4

A.6. The History of Maxwells Theorem

Literature

B. Problems

B.1. Material frorn the Seminars

B.2. Written Examination Problems.

In the mid-twentieth century the theory of partial differential equations wasconsidered the summit of mathematics， both because of the difficulty andsignificance of the problems it solved and because it came into existence laterthan most areas of mathematics.
　　Nowadays many are inclined to look disparagingly at this remarkable areaof mathematics as an old-fashioned art of juggling inequalities or as a testingground for applications of functional analysis. Courses in this subject haveeven disappeared from the obligatory program of many universities （for ex-ample， in Paris）. Moreover， such remarkable textbooks as the classical three-volume work of Goursat have been removed as superfluous from the library ofthe University of Paris-7 （and only through my own intervention was it possi-ble to save them， along with the lectures of Klein， Picard， Hermite， Darboux，Jordan ）