Problems on Partial Differential Equations (Problem Books in Mathematics)

✍ Scribed by Maciej Borodzik, Paweł Goldstein, Piotr Rybka, Anna Zatorska-Goldstein

• Publisher: Springer
• Year: 2019
• Tongue: English
• Leaves: 260
• Series: Problem Books in Mathematics (Book 714)
• Edition: 1st ed. 2019
• Category: Library

✦ Synopsis

This book covers a diverse range of topics in Mathematical Physics, linear and nonlinear PDEs. Though the text reflects the classical theory, the main emphasis is on introducing readers to the latest developments based on the notions of weak solutions and Sobolev spaces.

In numerous problems, the student is asked to prove a given statement, e.g. to show the existence of a solution to a certain PDE. Usually there is no closed-formula answer available, which is why there is no answer section, although helpful hints are often provided.

 

This textbook offers a valuable asset for students and educators alike. As it adopts a perspective on PDEs that is neither too theoretical nor too practical, it represents the perfect companion to a broad spectrum of courses.

✦ Table of Contents

Preface
Introduction
The Content of the Chapters
About the Authors
Notation
Contents
1 Preliminaries
1.1 Integration by Parts
1.1.1 Theoretical Background
Basic Formulas
Surface Integrals
A Quick Review of Differential Forms
1.1.2 Worked-Out Problems
1.1.3 Problems
1.2 Convolutions
1.2.1 Theoretical Background
1.2.2 Worked-Out Problems
1.2.3 Problems
1.3 Bibliographical Remarks
2 Distributions, Sobolev Spaces and the Fourier Transform
2.1 The Fourier Transform
2.1.1 Theoretical Background
The Schwartz Space
The Fourier Transform
Properties of the Fourier Transform
Parseval's Identity and Plancherel's Theorem
2.1.2 Worked-Out Problems
2.1.3 Problems
2.2 The Theory of Distributions
2.2.1 Theoretical Background
Distributional Derivatives
Tempered Distributions
Convolutions
Fundamental Solutions
2.2.2 Worked-Out Problems
2.2.3 Problems
2.3 Sobolev Spaces
2.3.1 Theoretical Background
Sobolev Spaces Wm,p
Spaces Hs(Rn)
Spaces Hs(Ω)
Duality and the Spaces W-m,p'(Ω)
Trace of a Sobolev Function
Embedding Theorems
2.3.2 Worked-Out Problems
2.3.3 Problems
2.4 Bibliographical Remarks
3 Common Methods
3.1 Weak Convergence
3.1.1 Theoretical Background
Weak Convergence in Hilbert Spaces
Dual Spaces
Weak and Weak* Convergence in Banach Spaces
Banach Space-Valued Functions
3.1.2 Worked-Out Problems
3.1.3 Problems
3.2 The Separation of Variables Method
3.2.1 Theoretical Background
Hyperbolic Problems
Comments on the Regularity of u Given by Formula (3.21)
Hyperbolic Problems: A Non-homogeneous Case
Parabolic Problems
3.2.2 Worked-Out Problems
3.2.3 Problems
Hyperbolic Equations
Parabolic Equations
3.3 Galerkin Method
3.3.1 Theoretical Background
3.3.2 Worked-Out Problems
3.3.3 Problems
3.4 Bibliographical Remarks
4 Elliptic Equations
4.1 Classical Theory of Harmonic Functions
4.1.1 Theoretical Background
Definitions
Green's Functions
Holomorphic Functions and Harmonic Functions
Subharmonic Functions
General Elliptic Operators
4.1.2 Worked-Out Problems
The Laplace Operator in Different Coordinate Systems
Green's Functions
Harmonic and Holomorphic Functions
4.1.3 Problems
Harmonic Functions: Basic Properties
The Laplace Operator in Different Coordinate Systems
Green's Functions
Harmonic and Holomorphic Functions
Subharmonic Functions
4.2 Weak Solutions
4.2.1 Theoretical Background
Physical Motivation
Weak Formulation
Variational Approach on Hilbert Spaces
Lax–Milgram's Lemma
Regularity of Weak Solutions
4.2.2 Worked-Out Problems
4.2.3 Problems
4.3 Bibliographical Remarks
5 Evolution Equations
5.1 First-Order Equations and the Method of Characteristics
5.1.1 Theoretical Background
The Characteristic System
Admissible Boundary Conditions
Noncharacteristic Initial Conditions
Local Solutions
5.1.2 Worked-Out Problems
5.1.3 Problems
5.2 Hyperbolic Problems
5.2.1 Theoretical Background
Duhamel's Formula
5.2.2 Worked-Out Problems
The Wave Equation
Duhamel's Formula
Hyperbolic Systems of the First Order
The Use of the Fourier Transform
Energy Estimates and Uniqueness
5.2.3 Problems
The Wave Equation
Hyperbolic Systems of the First Order
Hyperbolic Equations of the Second Order, Considered More Generally
The Use of the Fourier Transform
Energy Estimates and Uniqueness
5.3 Parabolic Equations
5.3.1 Theoretical Background
Heat Equation
Linear Second-Order Parabolic Equations
Weak Solutions
5.3.2 Worked-Out Problems
5.3.3 Problems
5.4 Bibliographical Remarks
Bibliography

✦ Subjects

Mathematics;Calculus; Differential equations