Course Contents
1. Lecture 1
• Introduction, course information.
• Review of ordinary differential equations.
• Example of derivation of PDE.
• Characteristic curves for a linear equation.
• The Cauchy problem for a first order equation.

Reading: McOwen 1.1 or John 1.4-1.6

2. Lecture 2
• Elementary classification of PDE's.
• Examples of quasilinear equations.
• Conservation laws. The inviscid Burger equation.
• Method characteristics for Burger's equation. Shocks and jump conditions.

3. Lecture 3
• General first-order nonlinear partial differential equations in two variables.
• The method of characteristic strips.

Reading: McOwen 1.3 or John 1.7-1.8

4. Lecture 4
• Complete integrals and the method of envelopes.
• Introduction to higher-order PDE's.

Reading: McOwen 1.3 or John 1.9; John 3.1, 3.2

5. Lecture 5
• Cauchy-Kovalevskaya theorem.
• Characteristic surfaces.

Reading: John 3.1-3 and the notes on the homepage

6. Lecture 6
• Cauchy-Kovalevskaya theorem

7. Lecture 7
• Cauchy-Kovalevskaya theorem
• Characteristic surfaces, symbols for differential operators ans classification of PDE's.

8. Lecture 8
• THE WAVE EQUATION I: one dimension
• Initial value problem.
• Weak solutions.
• Initial value problem. D'Alembert's formula.

9. Lecture 9
• Initial/boundary value problem. Method of parallelograms.
• The non-homogeneous wave-equation.

10. Lecture 10
• Hyperbolic and elliptic PDE's.
• THE WAVE EQUATION II: higher dimensions
• Derivation of the wave equation in two spacial variables.
• Spherical means and the initial value problem.

11. Lecture 11
• Solution of the wave equation in three special variables. Kirchhoff's formula.
• Hadamard's method of descent and the initial value problem in two special variables.

12. Lecture 12
• The Laplace and Poisson equation, I: introduction.
• Examples in physiscs.
• Separation of variables and spherical harmonics.
• The Dirichlet and Neumann problems for the Laplace equation

13. Lecture 13
• The Dirichlet and Neumann problems for the Laplace equation
• Green's Identities.
• Uniqueness theorems for Dirichlet and Neumann problems.

14. Lecture 14
• The Laplace and Poisson equation, II.
• Mean value theorem.

15. Lecture 15
• Weak elliptic maximum principle
• Strong elliptic maximum principle

16. Lecture 16
• Adjoint operators, weak derivatives and distributions.

Reading: McOwen 8.3, 2.3 a, c

17. Lecture 17
• Distributions II.
• The fundamental solution for Laplace equation.

Reading: McOwen 2.3 d, 4.2 a.

18. Lecture 18
• Potential theory for Laplace equation.
• Green's function. Green's function in a ball.

19. Lecture 19
• Green's function and Poisson integral formula.
• Properties of harmonic functions.

20. Lecture 20
• Functional spaces. Banach and Hilbert spaces.
• Sobolev spaces, Introduction.
• Representation theorems.

21. Lecture 21
• Weak solutions of the Poisson equation.
• Poincare inequality.

22. Lecture 22
• Sobolev spaces.
• Lax-Milgram Theorem.

23. Lecture 23
• Sobolev spaces, Sobolev inequalities and embeddings I.

24. Lecture 24
• Sobolev spaces, Sobolev inequalities and embeddings II.

25. Lecture 25
• The heat equation in a bounded domain.
• Separation of variable.
• Maximum principle for the heat equation.

26. Lecture 26
• The initial value problem for the heat equation.
• Fourier transfrom.
• The non-homogoneous heat equation.

27. Lecture 27
• Calculus of variations.
• Weak existence for Dirichlet and Nuemann problems.
• Lagrange multipliers.