• 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

• 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.

Reading: McOwen 1.2

• 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

• 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

• Cauchy-Kovalevskaya theorem.
• Characteristic surfaces.

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

• Cauchy-Kovalevskaya theorem

Reading: John 3.1-3, notes

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

Reading: John 3.1-3, notes

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

Reading: John 3.1-3, notes

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

Reading: McOwen 3.1, John 2.4

• 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.

Reading: John 5.1, McOwen 3.2

• 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.

Reading: John 5.1, McOwen 3.2

• 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

Reading: McOwen 4.1, 4.4

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

Reading: McOwen 4.1, 4.4

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

Reading: McOwen 4.1.

• Weak elliptic maximum principle
• Strong elliptic maximum principle

Reading: McOwen 4.1 d, 8.3

• Adjoint operators, weak derivatives and distributions.

Reading: McOwen 8.3, 2.3 a, c

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

Reading: McOwen 2.3 d, 4.2 a.

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

Reading: McOwen 4.2

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

Reading: McOwen 4.2

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

Reading: McOwen 4.2 f, 6.1.

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

Reading: McOwen 6.2.

• Sobolev spaces.
• Lax-Milgram Theorem.

Reading: McOwen 6.4.

• Sobolev spaces, Sobolev inequalities and embeddings I.

Reading: McOwen 6.4.

• Sobolev spaces, Sobolev inequalities and embeddings II.

Reading: McOwen 6.4.

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

Reading: McOwen 5.1.

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

Reading: McOwen 5.1.

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

Reading: McOwen 7.1.

• Eigenvalues of the Laplacian.
• Viscosity solutions for the Hamilton-Jacobi equation.

Reading: McOwen 7.1, notes on the homepage.

• Viscosity solutions for the Hamilton-Jacobi equation.

Reading: notes on the homepage.