Partial Differential Equations I: Basic Theory

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 H₀^s(Omega)
  6. The Schwartz kernel theorem
• 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
• 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