偏微分方程导论（英文版 第二版）
作者：福兰德 著
出版时间：2011年版
内容简介
《偏微分方程导论（第2版）》是一部数学专业研究生的偏微分方程教程。其旨在让读者更好地了解偏微分方程的经典基础结果，为读者更深层次学习这方面的专著和教程提供现代理论观点。这是第二版，较第一版增加了不少练习，专门增加了一章讲述拟微分算子，增加了不少材料，内容更加丰富。书中的前五章讲述经典理论，如一阶方程，局部存在性定理，数学物理基础偏微分方程，适时地运用现代物理技巧解释长期研究的话题。最后三章专注于现代理论，索伯列夫空间，椭？边界值问题和拟微分算子。
目录
chapter 0
preliminaries
a. notations and definitions
b. results from advanced calculus
c. convolutions
d. the fourier transform
e. distributions
f. compact operators
chapter 1
local existence theory
a. basic concepts
b. real first order equations
c. the general cauchy problem
d. the cauchy-kowalevski theorem
e. local solvability: the lewy example
f. constant-coeffcient operators: fundamental solutions
chapter 2
the laplace operator
a. symmetry properties of the laplacian
b. basic properties of harmonic functions
c. the fundamental solution
d. the dirichlet and neumann problems
e. the green's function
f. dirichlet's principle
g. the dirichlet problem in a half-space
h. the dirichlet problem in a ball
i. more about harmonic functions
chapter 3
layer potentials
a. the setup
b. integral operators
c. double layer potentials
d. single layer potentials
e. solution of the problems
f. further remarks
chapter 4
the heat operator
a. the gaussian kernel
b. functions of the laplacian
c. the heat equation in bounded domains
chapter 5
the wave operator
a. the cauchy problem
b. solution of the cauchy problem
c. the inhomogeneous equation
d. fourier analysis of the wave operator
e. the wave equation in bounded domains
f. the radon transform
chapter 6
the l2 theory of derivatives
a. sobolev spaces on r
b. further results on sobolev spaces
c. local regularity of elliptic operators
d. constant-coefficient hypoelliptic operators
e. sobolev spaces on bounded domains
chapter 7
elliptic boundary value problems
a. strong ellipticity
b. on integration by parts
c. dirichlet forms and boundary conditions
d. the coercive estimate
e. existence, uniqueness, and eigenvalues
f. regularity at the boundary: the second order case
g. further results and techniques
h. epilogue: the return of the green's function
chapter 8
pseudodifferential operators
a. basic definitions and properties
b. kernels of pseudodifferential operators
c. asymptotic expansions of symbols
d. amplitudes, adjoints, and products
e. sobolev estimates
f. elliptic operators
g. introduction to microlocal analysis
h. change of coordinates
bibliography
index of symbols
index
