Volume 1

Partial Differential Equations I: Basic Theory

This volume introduces basic examples of partial differential equations, arising in continuum mechanics, electromagnetism, complex analysis, and other areas. It develops a number of tools for their analysis, including Fourier analysis, distribution theory, Sobolev spaces, energy estimates, and maximum principles. There is also a basic account of the theory of vector fields and ODE. This serves to introduce the variational method of deriving differential equations in physics and geometry, in a simpler context, and it also provides valuable tools for the analysis of PDE.

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 theorem
    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. Relativistic motion
    19. Topological applications of differential forms
    20. Critical points and index of a vector field
  • 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
  • 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
  • 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 H0s(Omega)
    6. The Schwartz kernel theorem
    7. Sobolev spaces on rough domains
  • 5. Linear Elliptic Equations
    1. Existence and regularity of solutions to the Dirichlet problem
    2. The weak and strong maximum principles
    3. The Dirichlet problem on the ball in Rn
    4. The Riemann mapping theorem (smooth boundary)
    5. The Dirichlet problem on a domain with 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 surfaces
    11. General elliptic boundary problems
    12. Operator properties of regular boundary problems
  • 6. Linear Evolution Equations
    1. The heat equation and 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 equation
  • A. Outline of Functional Analysis
    1. Banach spaces
    2. Hilbert spaces
    3. Frechet spaces; locally convex spaces
    4. Duality
    5. Linear operators
    6. Compact operators
    7. Fredholm operators
    8. Unbounded operators
    9. Semigroups
  • 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