偏微分方程導論 第2版（簡體書）

《偏微分方程導論(第2版)(英文版)》內容簡介：In 1975 I gave a course in partial difierential equations（PDE）at the Uni-versity of Washington to an audience consisting of graduate students whohad taken the standard first-year analysis courses but who had 1ittle back-ground in PDE.Accordingly.it focused on basic classical results in PDEbut aimed in the direction of the recent developments and made fairly ft。eeuse of the techniques of real and complex analysis.The roughly polishednotes for that course constituted the first edition of this book，which hasenjoyed some SUCCESS for the past two decades as a“modern”introductionto PDE.From time to time，however，my conscience ha.S nagged me tomake some revisions——to clean some things up，add more exercises，andinclude some material on pseudodifferential operators.

作者：（美國）福蘭德（Gerald B.Folland）

PREFACE
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-Coefficient Operators：Fundamental Solutions

Chapter 2
THE LAPLACE oPERATOR
A．Symmetry Properties of the Laplacian
B．Basic Properties of Harmonic Funotions
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．Funotions 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 THEOR．Y 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 Resuits 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