This text on partial differential equations is intended for readers who want to understand the theoretical underpinnings of modern PDEs in settings that are important for the applications without using extensive analytic tools required by most advanced texts. The assumed mathematical background is at the level of multivariable calculus and basic metric space material, but the latter is recalled as relevant as the text progresses.
The key goal of this book is to be mathematically complete without overwhelming the reader, and to develop PDE theory in a manner that reflects how researchers would think about the material. A concrete example is that distribution theory and the concept of weak solutions are introduced early because while these ideas take some time for the students to get used to, they are fundamentally easy and, on the other hand, play a central role in the field. Then, Hilbert spaces that are quite important in the later development are introduced via completions which give essentially all the features one wants without the overhead of measure theory.
There is additional material provided for readers who would like to learn more than the core material, and there are numerous exercises to help solidify one's understanding.
Readership
The text should be suitable for advanced undergraduates or for beginning graduate students including those in engineering or the sciences and also Professors, graduate students, and others interested in teaching and learning partial differential equations.
Author(s): Andras Vasy
Series: Graduate Studies in Mathematics 169
Publisher: American Mathematical Society
Year: 2015
Language: English
Pages: C, X, 281, B
Preface
Chapter 1 Introduction
1. Preliminaries and notation
2. Partial differential equations
Additional material: More on normed vector spaces and metric spaces
Problems
Chapter 2 Where do PDE come from?
1. An example: Maxwell’s equations
2. Euler-Lagrange equations
Problems
Chapter 3 First order scalar semilinear equations
Additional material: More on ODE and the inverse function theorem
Problems
Chapter 4 First order scalar quasilinear equations
Problems
Chapter 5 Distributions and weak derivatives
Additional material: The space L 1
Problems
Chapter 6 Second order constant coefficient PDE: Types and d’Alembert’s solution of the wave equation
1. Classification of second order PDE
Problems
Chapter 7 Properties of solutions of second order PDE: Propagation, energy estimates and the maximum principle
1. Properties of solutions of the wave equation: Propagation phenomena
2. Energy conservation for the wave equation
3. The maximum principle for Laplace’s equation and the heat equation
4. Energy for Laplace’s equation and the heat equation
Problems
Chapter 8 The Fourier transform: Basic properties, the inversion formula and the heat equation
1. The definition and the basics
2. The inversion formula
3. The heat equation and convolutions
4. S y stem s o f P D E
5. Integral transforms
Additional material: A heat kernel proof of the Fourier inversion formula
Problems
Chapter 9 The Fourier transform:Tempered distributions, the wave equation and Laplace’s equation
1. Tempered distributions
2. The Fourier transform of tempered distributions
3. The wave equation and the Fourier transform
4. More on tempered distributions
Problems
Chapter 10 PDE and boundaries
1. The wave equation on a half space
2. The heat equation on a half space
3. More complex geometries
4. Boundaries and properties of solutions
5. PDE on intervals and cubes
Problems
Chapter 11 Duhamel’s principle
1. The inhomogeneous heat equation
2. The inhomogeneous wave equation
Problems
Chapter 12 Separation of variables
1. The general method
2. Interval geometries
3. Circular geometries
Problems
Chapter 13 Inner product spaces, symmetric operators, or thogonality
1. The basics of inner product spaces
2. Symmetric operators
3. Completeness of orthogonal sets and of the inner product space
Problems
Chapter 14 Convergence of the Fourier series and the Poisson formula on disks
1. Notions of convergence
2. Uniform convergence of the Fourier transform
3. What does the Fourier series converge to?
4. The Dirichlet problem on the disk
Additional material: The Dirichlet kernel
Problems
Chapter 15 Bessel functions
1. The definition of Bessel functions
2. The zeros of Bessel functions
3. Higher dimensions
Problems
Chapter 16 The method of stationary phase
Problems
Chapter 17 Solvability via duality
1. The general method
2. An example: Laplace’s equation
3. Inner product spaces and solvability
Problems
Chapter 18 Variational problems
1. The finite dimensional problem
2. The infinite dimensional minimization
Problems
Bibliography
Index
