Lecture Notes.

Partial Differential Equations

Gerald Teschl

This manuscript provides a brief introduction to Partial Differential Equations. The first part is intended as a first introduction and does neither require any functional analytic tools nor Lebesgue or Sobolev spaces. The second part deals with advanced techniques from functional analysis.

MSC: 35-01
Keywords: Method of characteristics, Cauchy-Kovalevskaya theorem, Separation of variables, Fourier transform, Dispersion, Laplace equation, Harmonic functions, Dirichlet problem, Heat equation, Wave equation Sobolev spaces, Elliptic regularity, Operator semigroups, Reaction diffusion equations, Calculus of variations, Nonlinear Schrödinger equation, Strichartz estimates

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Table of contents

    Part 1: Classical Partial Differential Equations

  1. First order PDE
    1. The method of characteristics
    2. Semilinear equations
    3. Quasilinear equations
    4. Fully nonlinear equations
    5. Classification and canonical forms
  2. The Cauchy-Kovalevskaya theorem
    1. First order systems
    2. Second order equations
  3. Separation of variables
    1. The heat equation for a thin rod
    2. Outlook: The reaction diffusion equation
    3. The wave equation for a string
    4. The wave equation on a rectangle and on a disc
    5. The Laplace equation on a disc
  4. The Fourier transform and problems on the line
    1. Motivation
    2. The Fourier transform in one dimension
    3. The heat equation on the line
    4. The wave equation on the line
    5. Dispersion
    6. Symmetry groups
  5. The Laplace equation
    1. Harmonic functions
    2. Subharmonic functions
    3. The Newton potential and the Poisson equation on Rn
    4. Poisson equation on a domain and Green's function
    5. The Dirichlet principle
    6. Solution for a half space and for a ball
    7. The Perron method for solving the Dirichlet problem
    8. General elliptic equations
  6. The heat equation
    1. The Fourier transform
    2. The fundamental solution
    3. The heat equation on a bounded domain and the maximum principle
    4. Energy methods
    5. General parabolic equations
  7. The wave equation
    1. Solution via the Fourier transform
    2. Solution in arbitrary dimensions
    3. Energy methods

    Part 2: Advanced Partial Differential Equations

  8. A first look at weak derivatives and L2 based Sobolev spaces
    1. Motivation
    2. The Fourier transform on L2
    3. The Sobolev spaces Hr(Rn)
    4. Evolution problems
  9. General Sobolev spaces
    1. Basic properties
    2. Extension and trace operators
    3. Embedding theorems
  10. Elliptic equations
    1. The Poisson equation
    2. Elliptic equations
    3. Elliptic regularity
    4. The Poisson equation in C(U)
  11. Operator semigroups
    1. Single variable calculus in Banach spaces
    2. Uniformly continuous operator groups
    3. Strongly continuous semigroups
    4. Generator theorems
    5. Applications to parabolic equations
    6. Applications to hyperbolic equations
  12. Nonlinear equations
    1. Semilinear equations
    2. Reaction diffusion equations
  13. Calculus of Variations
    1. Differentiation in Banach spaces
    2. The direct method
    3. Constraints
  14. The nonlinear Schrödinger equation
    1. Local well-posedness in Hr for r>n/2
    2. Global solutions and blowup in H1
    3. Strichartz estimates
    4. Well-posedness in L2 and H1
    5. Standing waves
    6. Appendix: Radial Sobolev spaces

    Part 3: Appendices

  15. Appendix: Calculus facts
    1. Differentiation
    2. Integration
    3. Fourier series
  16. Appendix: Real and functional analysis
    1. Differentiable and Hölder continuous functions
    2. Lebesgue spaces
    3. Closed operators
    4. The Bochner integral
    Glossary of notations