偏微分方程引論

內(nèi)容概要

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.

作者簡介

作者：（美國）勒納迪（Michael Renardy） （美國）羅杰斯（Robert C.Rogers）

書籍目錄

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 Equatioas 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 WeierstraB 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
6  Function Spaces
7  Sobolev Spaces
8  Operator Theory
9  Linear Elliptic Equations
10  Nonlinear Elliptic Equations
11  Energy Methods for Evolution Problems
12  Semigroup Methods
A References
Index

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