The first of three volumes on partial differential equations, this one introduces basic examples arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, in particular Fourier analysis, distribution theory, and Sobolev spaces. These tools are then applied to the treatment of basic problems in linear PDE, including the Laplace equation, heat equation, and wave equation, as well as more general elliptic, parabolic, and hyperbolic equations. The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.
The third edition further expands the material by incorporating new theorems and applications throughout the book, and by deepening connections and relating concepts across chapters. In includes new sections on rigid body motion, on probabilistic results related to random walks, on aspects of operator theory related to quantum mechanics, on overdetermined systems, and on the Euler equation for incompressible fluids. The appendices have also been updated with additional results, ranging from weak convergence of measures to the curvature of Kahler manifolds.
Author(s): Michael E. Taylor
Series: Applied Mathematical Sciences 115
Edition: 3
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 714
City: Cham
Tags: Linear Partial Differential Equations, Vector Fields, Sobolev Spaces, Fourier Analysis, Distributions
Contents of Volumes II and III
Preface
Acknowledgments
Introduction to the Second Edition
Introduction to the Third Edition
Contents
1 Basic Theory of ODE and Vector Fields
1 The derivative
2 Fundamental local existence theorem for ODE
3 Inverse function and implicit function theorems
4 Constant-coefficient linear systems; exponentiation of matrices
5 Variable-coefficient linear systems of ODE: Duhamel's principle
6 Dependence of solutions on initial data and on other parameters
7 Flows and vector fields
8 Lie brackets
9 Commuting flows; Frobenius's theorem
10 Hamiltonian systems
11 Geodesics
12 Variational problems and the stationary action principle
13 Differential forms
14 The symplectic form and canonical transformations
15 First-order, scalar, nonlinear PDE
16 Completely integrable hamiltonian systems
17 Examples of integrable systems; central force problems
18 Rigid body motion in Rn
19 Relativistic motion
20 Topological applications of differential forms
21 Critical points and index of a vector field
A Nonsmooth vector fields
References
2 The Laplace Equation and Wave Equation
1 Vibrating strings and membranes
2 The divergence of a vector field
3 The covariant derivative and divergence of tensor fields
4 The Laplace operator on a Riemannian manifold
5 The wave equation on a product manifold and energy conservation
6 Uniqueness and finite propagation speed
7 Lorentz manifolds and stress-energy tensors
8 More general hyperbolic equations; energy estimates
9 The symbol of a differential operator and a general Green–Stokes formula
10 The Hodge Laplacian on k-forms
11 Maxwell's equations
References
3 Fourier Analysis, Distributions,and Constant-Coefficient Linear PDE
1 Fourier series
2 Harmonic functions and holomorphic functions in the plane
3 The Fourier transform
4 Distributions and tempered distributions
5 The classical evolution equations
6 Radial distributions, polar coordinates, and Bessel functions
7 The method of images and Poisson's summation formula
8 Homogeneous distributions and principal value distributions
9 Elliptic operators
10 Local solvability of constant-coefficient PDE
11 The discrete Fourier transform
12 The fast Fourier transform
A The mighty Gaussian and the sublime gamma function
B The central limit theorem
C Natural extension of weak* convergence of measures
References
4 Sobolev Spaces
1 Sobolev spaces on Rn
2 The complex interpolation method
3 Sobolev spaces on compact manifolds
4 Sobolev spaces on bounded domains
5 The Sobolev spaces Hs0()
6 The Schwartz kernel theorem
7 Sobolev spaces on rough domains
References
5 Linear Elliptic Equations
1 Existence and regularity of solutions to the Dirichlet problem
2 The weak and strong maximum principles
3 The Poisson integral on the ball in Rn
4 The Riemann mapping theorem (smooth boundary)
5 The Dirichlet problem on a domain with a rough boundary
6 The Riemann mapping theorem (rough boundary)
7 The Neumann boundary problem
8 The Hodge decomposition and harmonic forms
9 Natural boundary problems for the Hodge Laplacian
10 Isothermal coordinates and conformal structures on 2D surfaces
11 General elliptic boundary problems
12 Operator properties of regular boundary problems
A Spaces of generalized functions on manifolds with boundary
B The Mayer–Vietoris sequence in de Rham cohomology
C Topological invariance of de Rham cohomology
References
6 Linear Evolution Equations
1 The heat equation and the wave equation on bounded domains
2 The heat equation and wave equation on unbounded domains
3 Maxwell's equations
4 The Cauchy–Kowalewsky theorem
5 Hyperbolic systems
6 Geometrical optics
7 The formation of caustics
8 Boundary layer phenomena for the heat semigroup
9 Schrödinger equations on Euclidean space
A Some Banach spaces of harmonic functions
B The stationary phase method
References
A Outline of Functional Analysis
1 Banach spaces
2 Hilbert spaces
3 Fréchet spaces; locally convex spaces
4 Duality
5 Linear operators
6 Compact operators
7 Fredholm operators
8 Unbounded operators
9 Semigroups
References
B Manifolds, Vector Bundles, and Lie Groups
1 Metric spaces and topological spaces
2 Manifolds
3 Vector bundles
4 Sard's theorem
5 Lie groups
6 The Campbell–Hausdorff formula
7 Representations of Lie groups and Lie algebras
8 Representations of compact Lie groups
9 Representations of SU(2) and related groups
References
Index
