See the syllabus on my web page.

Classes and Reading:

- Week 1: Introduction, L^p spaces

- Week 2: 2.1, beginning of 2.2
- Week 3: 2.2
- Week 4: End of 2.2

Midterm: Wednesday 29th at 12:00.

Final Exam: It will be in 2 parts. First an exercise sheet distributed in class on Dec. 5th and due Dec. 12th. You can find the sheet here. And second a written exam in class on Dec. 12th at 12:00.

Projects assignments:

Students must choose a project and complete it by the assigned time. Examples of projects that may be chosen are given below. It is however possible to choose another project, please see with me if you wish to do so.

Please confirm to me your choice for the 1st project by email by Oct. 1st. The project is due by November 5th.

Please confirm your choice for the 2nd project by email by Nov. 13th. The second project is due by December 3rd.

A tentative list of possible projects follows:

- Understand and explain the derivation of the linear elasticity
equations.

- Find explicit solutions to incompressible Stokes equations and Maxwell
equations in dimensions 2 and 3. Use this to understand some properties
of the solutions.

- Study the notion of weak solutions to incompressible Navier-Stokes
equations by Leray. See for instance P-L Lions, Mathematical topic in
fluid mechanic: Incompressible models.

- Study the "porous medium equation" du/dt = laplacian (u^2). In
particular try to understand the modeling assumptions behind this
equation (you can check chapter 2 in Vazquez). Check that
u=(t^{-n/(n+2)}-k |x|^2/t)_+ where ()_+ is the
positive part and k=1/(4(d+2)) is a weak solution. Use this explicit
weak solution to deduce some general properties of the equation. For
further, one can check the book by J-L Vazquez, The porous medium
equation: Mathematical Theory.

- Study the existence of solution to the elliptic equation -laplacian u=f(u) for a smooth non linear function of u, using the variationnal approach. You can see section 8.1.2 in Evans (and try to understand example 3). In particular try to determine which kind on conditions on f are needed (does f(u)=u^2 work for instance?).
- Write a finite difference code for the Poisson equation -laplacian u=f(x) in the box [0, 1]^2 with a given Dirichlet condition on the boundary.
- Write a finite difference code for the heat equation in dimension 2 in
the box [0, 1]^2 with a given Dirichlet condition on the boundary.

For the 2nd assignment, a tentative list of projects is:

- Understand and explain the notion of viscosity solutions for Hamilton-Jacobi equation with convex Hamiltonian. See for instance Bardi M., Capuzzo Dolcetta I. Optimal control and viscosity solutions of Hamilton-Jacobi Bellman equations.
- Write a finite difference code for the Burgers equation using Lax-Friedrichs scheme (or another one if you prefer).
- Implement numerically the Hopf-Lax formula for the eikonal equation
|nabla u|=1 in dimension 2 in the square with boundary condition f. Can
you find (numerically) the location of the shocks?

- Let M be a Riemanian manifold with some metric g. Explain the notion of gradient and use that to pose the eikonal |nabla u|=1/c(x) on M. Generalize the notion of characteristics. Find the relation between g and c s.t. the characteristics are also geodesics.
- Explain the kinetic formulation for entropy solutions of the Burgers equation. See for example Perthame, kinetic formulation of conservation laws.
- Write the 1d wave equation as the system partial_t v=\partial_x w;
partial_t w=partial_x v. Use a finite difference upwind scheme to solve
the system and compare the solution to the explicit formula for various
initial data. You can try to do the same in dimension 2.

- Study the 1d wave equation over the interval [0, 1]. Try to use that to explain the difference in sound between a piano and a harpsichord (1 string per key).
- Study the multi-d wave equation over a torous using Fourier series.