偏微分方程引论

series preface

preface

1 introduction

1.1 basic mathematical questions

1.1.1 existence

1.1.2 multiplicity

1.1.3 stability

1.1.4 linear systems of odes and asymptotic stability

1.1.5 well-posed problems

1.1.6 representations

1.1.7 estimation

1.1.8 smoothness

1.2 elementary partial differential equations

1.2.1 laplace's equation

1.2.2 the heat equation

1.2.3 the wave equation

2 characteristics

2.1 classification and characteristics

2.1.1 the symbol of a differential expression

2.1.2 scalar equations of second order

2.1.3 higher-order equations and systems

2.1.4 nonlinear equations

2.2 the cauchy-kovalevskaya theorem

2.2.1 real analytic functions

2.2.2 majorization

2.2.3 statement and proof of the theorem

2.2.4 reduction of general systems

2.2.5 a pde without solutions

2.3 holmgren's uniqueness theorem

2.3.1 an outline of the main idea

2.3.2 statement and proof of the theorem

2.3.3 the weierstratβ approximation theorem

3 conservation laws and shocks

3.1 systems in one space dimension

3.2 basic definitions and hypotheses

3.3 blowup of smooth solutions

3.3.1 single conservation laws

3.3.2 the p system

3.4 weak solutions

3.4.1 the rankine-hugoniot condition

3.4.2 multiplicity

3.4.3 the lax shock condition

3.5 riemann problems

3.5.1 single equations

3.5.2 systems

3.6 other selection criteria

3.6.1 the entropy condition

3.6.2 viscosity solutions

3.6.3 uniqueness

4 maximum principles

4.1 maximum principles of elliptic problems

4.1.1 the weak maximum principle

4.1.2 the strong maximum principle

4.1.3 a priori bounds

4.2 an existence proof for the dirichlet problem

4.2.1 the dirichlet problem on a ball

4.2.2 subharmonic functions

4.2.3 the arzela-ascoli theorem

4.2.4 proof of theorem 4.13

4.3 radial symmetry

4.3.1 two auxiliary lemmas

4.3.2 proof of the theorem

4.4 maximum principles for parabolic equations

4.4.1 the weak maximum principle

4.4.2 the strong maximum principle

5 distributions

5.1 test functions and distributions

5.1.1 motivation

5.1.2 test functions

5.1.3 distributions

5.1.4 localization and regularization

5.1.5 convergence of distributions

5.1.6 tempered distributions

5.2 derivatives and integrals

5.2.1 basic definitions

5.2.2 examples

5.2.3 primitives and ordinary differential equations

5.3 convolutions and fundamental solutions

5.3.1 the direct product of distributions

5.3.2 convolution of distributions

5.3.3 fundamental solutions

5.4 the fourier transform

5.4.1 fourier transforms of test functions

5.4.2 fourier transforms of tempered distributions

5.4.3 the fundamental solution for the wave equation

5.4.4 fourier transform of convolutions

5.4.5 laplace transforms

5.5 green's functions

5.5.1 boundary-value problems and their adjoints

5.5.2 green's functions for boundary-value problems.

5.5.3 boundary integral methods

6 function spaces

6.1 banach spaces and hilbert spaces

6.1.1 banach spaces

6.1.2 examples of banach spaces

6.1.3 hilbert spaces

6.2 bases in hilbert spaces

6.2.1 the existence of a basis

6.2.2 fourier series

6.2.3 orthogonal polynomials

6.3 duality and weak convergence

6.3.1 bounded linear mappings

6.3.2 examples of dual spaces

6.3.3 the hahn-banach theorem

6.3.4 the uniform boundedness theorem

6.3.5 weak convergence

7 sobolev spaces

7.1 basic definitions

7.2 characterizations of sobolev spaces

7.2.1 some comments on the domain ω

7.2.2 sobolev spaces and fourier transform

7.2.3 the sobolev imbedding theorem

7.2.4 compactness properties

7.2.5 the trace theorem

7.3 negative sobolev spaces and duality

7.4 technical results

7.4.1 density theorems

7.4.2 coordinate transformations and sobolev spaces on manifolds

7.4.3 extension theorems

7.4.4 problems

8 operator theory

8.1 basic definitions and examples

8.1.1 operators

8.1.2 inverse operators

8.1.3 bounded operators, extensions

8.1.4 examples of operators

8.1.5 closed operators

8.2 the open mapping theorem

8.3 spectrum and resolvent

8.3.1 the spectra of bounded operators

8.4 symmetry and self-adjointness

8.4.1 the adjoint operator

8.4.2 the hilbert adjoint operator

8.4.3 adjoint operators and spectral theory

8.4.4 proof of the bounded inverse theorem for hilbert spaces

8.5 compact operators

8.5.1 the spectrum of a compact operator

8.6 sturm-liouville boundary-value problems

8.7 the fredholm index

9 linear elliptic equations

9.1 definitions

9.2 existence and uniqueness of solutions of the dirichlet problem

9.2.1 the dirichlet problem——types of solutions

9.2.2 the lax-milgram lemma

9.2.3 garding's inequality

9.2.4 existence of weak solutions

9.3 eigenfunction expansions

9.3.1 fredholm theory

9.3.2 eigenfunction expansions

9.4 general linear elliptic problems

9.4.1 the neumann problem

9.4.2 the complementing condition for elliptic systems

9.4.3 the adjoint boundary-value problem

9.4.4 agmon's condition and coercive problems

9.5 interior regularity

9.5.1 difference quotients

9.5.2 second-order scalar equations

9.6 boundary regularity

10 nonlinear elliptic equations

10.1 perturbation results

10.1.1 the banach contraction principle and the implicit function theorem

10.1.2 applications to elliptic pdes

10.2 nonlinear variational problems

10.2.1 convex problems

10.2.2 nonconvex problems

10.3 nonlinear operator theory methods

10.3.1 mappings on finite-dimensional spaces

10.3.2 monotone mappings on banach spaces

10.3.3 applications of monotone operators to nonlinear pdes

10.3.4 nemytskii operators

10.3.5 pseudo-monotone operators

10.3.6 application to pdes

11 energy methods for evolution problems

11.1 parabolic equations

11.1.1 banach space valued functions and distributions

11.1.2 abstract parabolic initial-value problems

11.1.3 applications

11.1.4 regularity of solutions

11.2 hyperbolic evolution problems

11.2.1 abstract second-order evolution problems

11.2.2 existence of a solution

11.2.3 uniqueness of the solution

11.2.4 continuity of the solution

12 semigroup methods

12.1 semigroups and infinitesimal generators

12.1.1 strongly continuous semigroups

12.1.2 the infinitesimal generator

12.1.3 abstract odes

12.2 the hille-yosida theorem

12.2.1 the hille~yosida theorem

12.2.2 the lumer-phillips theorem

12.3 applications to pdes

12.3.1 symmetric hyperbolic systems

12.3.2 the wave equation

12.3.3 the schrsdinger equation

12.4 analytic semigroups

12.4.1 analytic semigroups and their generators

12.4.2 fractional powers

12.4.3 perturbations of analytic semigroups

12.4.4 regularity of mild solutions

a references

a.1 elementary texts

a.2 basic graduate texts

a.3 specialized or advanced texts

a.4 multivolume or encyclopedic works

a.5 other references

index

《偏微分方程引论（影印版）（第2版）》主要讲了：Partial differential equations are fundamental to the modeling of natural phenomena， arising in every field of science. Consequently， the desire to understand the solutions of these equations has always had a prominent place in the efforts of mathematicians; it has inspired such diverse fields as complex function theory， functional analysis and algebraic topology. Like algebra， topology， and rational mechanics， partial differential equations are a core area of mathematics. This book aims to provide the background necessary to initiate work on a Ph.D. thesis in PDEs for beginning graduate students. Prerequisites include a truly advanced calculus course and basic complex variables. Lebesgue integration is needed only in Chapter 10， and the necessary tools from functional analysis are developed within the course. The book can be used to teach a variety of different courses. This new edition features new problems throughout and the problems have been rearranged in each section from simplest to most difficult. New examples have also been added. The material on Sobolev spaces has been rearranged and expanded. A new section on nonlinear variational problems with "Young-measure" solutions appears. The reference section has also been expanded.