This book provides a short introduction to partial differential equations (PDEs). It is primarily addressed to graduate students and researchers, who are new to PDEs. The book offers a user-friendly approach to the analysis of PDEs, by combining elementary techniques and fundamental modern methods.
The author focuses the analysis on four prototypes of PDEs, and presents two approaches for each of them. The first approach consists of the method of analytical and classical solutions, and the second approach consists of the method of weak (variational) solutions.
In connection with the approach of weak solutions, the book also provides an introduction to distributions, Fourier transform and Sobolev spaces. The book ends with an appendix chapter, which complements the previous chapters with proofs, examples and remarks.
This book can be used for an intense one-semester, or normal two-semester, PDE course. The reader is expected to have knowledge of linear algebra and of differential equations, a good background in real and complex calculus and a modest background in analysis and topology. The book has many examples, which help to better understand the concepts, highlight the key ideas and emphasize the sharpness of results, as well as a section of problems at the end of each chapter.
Author(s): Arian Novruzi
Series: CMS/CAIMS Books in Mathematics 11
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 219
City: Cham
Tags: Partial Differential Equations, Distributions, Weak Solutions, Sobolev Spaces
Preface
Contents
1 Notations and review
1.1 Continuous differentiable functions
1.2 Domains and Ck (∂Ω)
spaces
1.2.1 Partition of unity
1.2.2 Domains and Ck (∂Ω)
spaces
1.3 Review of some important results
1.3.1 Some results from Lp(Ω) spaces
1.3.2 Some results from (Functional) Analysis
1.3.3 An application: Ordinary Differential Equations
Problems
2 Partial differential equations
2.1 Some prototypes of PDEs
Problems
3 First-order PDEs: classical and weak solutions
3.1 Method of characteristics
3.2 Classical local solutions to first-order PDEs
3.2.1 Classical local solutions: flat boundary
3.2.1.1 Boundary and initial conditions
3.2.1.2 Classical local solutions
3.2.2 Classical local solutions: non-flat boundary
3.3 Conservation laws and weak solutions
Problems
4 Second-order linear elliptic PDEs: maximum principle and classical solutions
4.1 Laplace equation and the method of separation of variables
4.2 Dirichlet problem in a ball
4.3 Maximum principle for Laplacian
4.4 Solution to the Dirichlet problem
4.4.1 Sub(super) harmonic functions and sub(super) solutions
4.4.2 Solution to the Dirichlet problem
Problems
5 Distributions
5.1 Motivation
5.2 Distributions
5.2.1 Test functions
5.2.2 Distributions
5.2.3 Derivatives of distributions
5.3 Convolution of distributions and fundamental solutions
5.4 Tempered distributions and Fourier transform
5.4.1 Fourier transform
5.4.2 Tempered distributions and Fourier transform
Problems
6 Sobolev spaces
6.1 Definitions and some first properties
6.1.1 Density of D in Wk,p
6.1.2 Some applications
6.2 Hs spaces and Fourier transform: Ws,p and Ws,p0 spaces
6.3 Continuous, compact, and dense embedding theorems in Hs(Ω)
6.3.1 Case Ω = RN
6.3.2 Case Ω
RN
6.4 Boundary traces in Sobolev spaces
6.5 Poincaré inequality
6.6 H−s(Ω) and W−s,q(Ω) spaces
Problems
7 Second-order linear elliptic PDEs: weak solutions
7.1 Introduction
7.2 Existence and uniqueness of weak solutions
7.2.1 Preliminary results
7.2.2 Dirichlet problem
7.2.3 Neumann problem
7.3 Nonlinear second-order elliptic PDEs
Problems
8 Second-order parabolic and hyperbolic PDEs
8.1 Heat and wave equations and the method of separation of variables
8.1.1 Heat equation and the method of separation of variables
8.1.2 Wave equation and the method of separation of variables
8.2 Some preliminary results
8.3 Weak solution to the heat equation
8.4 Weak solution to the wave equation
Problems
9 Annex
9.1 Annex: Chapter 1
9.1.1 Continuous differentiable functions
9.1.2 Some results from Lp(Ω) spaces
9.1.3 An application: Ordinary Differential Equations
9.2 Annex: Chapter 3
9.2.1 Classical local solutions to first-order PDEs
9.2.2 Conservation laws and weak solutions
9.3 Annex: Chapter 4
9.3.1 Dirichlet problem in a ball
9.3.2 Maximum principle for second-order linear elliptic PDEs
9.3.3 Solution to the Dirichlet problem
9.3.3.1 Sub(super) harmonic functions and sub(super) solutions
9.3.3.2 Some auxiliary results from Analysis
9.3.3.3 Proof of Theorem 4.4.7
9.4 Annex: Chapter 5
9.4.1 Some useful inequalities
9.4.2 More operations with distributions. Examples
9.4.3 Convergence of distributions. Distributions of finite order
9.4.4 Convolution of distributions
9.4.5 Tempered distributions and Fourier transform
9.4.6 Tempered distributions and convolution
9.5 Annex: Chapter 6
9.5.1 Continuous and compact embeddings
9.5.2 Extension and density results in Sobolev spaces
9.5.3 Boundary traces in Sobolev spaces
9.6 Annex: Chapter 7
9.6.1 Regularity of weak solutions
9.6.1.1 Regularity in the interior
9.6.1.2 Regularity near the boundary
