- 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
- 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.
Reading: McOwen 1.2
- 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
- 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
- Lecture 5
- Cauchy-Kovalevskaya theorem.
- Characteristic surfaces.
Reading: John 3.1-3 and the notes on the homepage
- Lecture 6
- Cauchy-Kovalevskaya theorem
Reading: John 3.1-3, notes
- Lecture 7
- Cauchy-Kovalevskaya theorem
- Characteristic surfaces, symbols for differential operators ans classification of PDE's.
Reading: John 3.1-3, notes
- Lecture 8
- THE WAVE EQUATION I: one dimension
- Initial value problem.
- Weak solutions.
- Initial value problem. D'Alembert's formula.
Reading: John 3.1-3, notes
- Lecture 9
- Initial/boundary value problem. Method of parallelograms.
- The non-homogeneous wave-equation.
Reading: McOwen 3.1, John 2.4
- 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.
Reading: John 5.1, McOwen 3.2
- 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.
Reading: John 5.1, McOwen 3.2
- 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
Reading: McOwen 4.1, 4.4
- Lecture 13
- 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
- Lecture 14
- The Laplace and Poisson equation, II.
- Mean value theorem.
Reading: McOwen 4.1.
- Lecture 15
- Weak elliptic maximum principle
- Strong elliptic maximum principle
Reading: McOwen 4.1 d, 8.3
- Lecture 16
- Adjoint operators, weak derivatives and distributions.
Reading: McOwen 8.3, 2.3 a, c
- Lecture 17
- Distributions II.
- The fundamental solution for Laplace equation.
Reading: McOwen 2.3 d, 4.2 a.
- Lecture 18
- Potential theory for Laplace equation.
- Green's function. Green's function in a ball.
Reading: McOwen 4.2
- Lecture 19
- Green's function and Poisson integral formula.
- Properties of harmonic functions.
Reading: McOwen 4.2
- Lecture 20
- Functional spaces. Banach and Hilbert spaces.
- Sobolev spaces, Introduction.
- Representation theorems.
Reading: McOwen 4.2 f, 6.1.
- Lecture 21
- Weak solutions of the Poisson equation.
- Poincare inequality.
Reading: McOwen 6.2.
- Lecture 22
- Sobolev spaces.
- Lax-Milgram Theorem.
Reading: McOwen 6.4.
- Lecture 23
- Sobolev spaces, Sobolev inequalities and embeddings I.
Reading: McOwen 6.4.
- Lecture 24
- Sobolev spaces, Sobolev inequalities and embeddings II.
Reading: McOwen 6.4.
- Lecture 25
- The heat equation in a bounded domain.
- Separation of variable.
- Maximum principle for the heat equation.
Reading: McOwen 5.1.
- Lecture 26
- The initial value problem for the heat equation.
- Fourier transfrom.
- The non-homogoneous heat equation.
Reading: McOwen 5.1.
- Lecture 27
- Calculus of variations.
- Weak existence for Dirichlet and Nuemann problems.
- Lagrange multipliers.
Reading: McOwen 7.1.
- Lecture 28
- Eigenvalues of the Laplacian.
- Viscosity solutions for the Hamilton-Jacobi equation.
Reading: McOwen 7.1, notes on the homepage.
- Lecture 29
- Viscosity solutions for the Hamilton-Jacobi equation.
Reading: notes on the homepage.