偏微分方程講義

前言

　　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?。　he cause of this degeneration of an important general mathematical the-ory into an endless stream of papers bearing titles like "On a property ofa solution of a boundary-value problem tor an equation" is most likely theattempt to create a unified， all-encompassing， superabstract "theory of every-thing."　　The principal source of partial differential equations is found in thecontinuous-medium models of mathematical and theoretical physics. Attemptsto extend the remarkable achievements of mathematical physics to systemsthat match its models only formally lead to complicated theories that aredifficult to visualize as a whole， just as attempts to extend the geometry ofsecond-order surfaces and the algebra of quadratic forms to objects of higherdegrees quickly leads to the detritus of algebraic geometry with its discourag-ing hierarchy of complicated degeneracies and answers that can be computedonly theoretically.　　The situation is even worse in the theory of partial differential equations：here the difficulties of conunutative algebraic geometry are inextricably boundup with noncomnutative differential algebra， in addition to which the topo-logical and analytic problems that arise arc profoundly nontrivial.

內(nèi)容概要

　　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　）

書籍目錄

Preface to the Second Russian Edition1.  The General Theory for One First-Order Equation Literature  2.  The General Theory for One First-Order Equation(Continued)Literature3.  Huygens' Principle in the Theory of Wave Propagation.4.  The Vibrating String (d'Alembert's Method)　4.1. The General Solution　4.2. Boundary-Value Problems and the Ca'uchy Problem　4.3. The Cauehy Problem for an Infinite Strifig. d'Alembert's Formula  　4.4. The Semi-Infinite String  　4.5. The Finite String. Resonance　4.6. The Fourier Method5.  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' Phenomenon6.  The Theory of Oscillations. The Variational Principle Literature7.  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 Case9.  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 Potential10. The Double-Layer Potential　10.1. Properties of the Double-Layer Potential11 Spherical Functions. Maxwell's Theorem. The Removable　Singularities Theorem12. Boundary-Value Problems for Laplaee's Equation. Theory of Linear Equations and Systems　12.1. Four Boundary-Value Problems for Laplace's Equation　12.2. Existence and Uniqueness of Solutions　12.3. Linear Partial Differential Equations and Their SymbolsA. The Topological Content of Maxwell's 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. Maxwell's Theorem and Cp2/con≈S4　A.6. The History of Maxwell's Theorem　LiteratureB. Problems　B.1. Material frorn the Seminars 　B.2. Written Examination Problems.

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