Homogenization and boundary layers (Fall 2008)

Please note the first class will be on wednesday September 10th.

The plan is to cover the following 4 parts :
1) In the first part, we will review classical results about periodic homogenization of elliptic operators. In particular we will look at : Darcy law, compensated-compactness thechniques, two scalce convergence methods, homogenization of Euler equation.
2) We will also treat few aspects of quasi-periodic, almost periodic and stochastic theories
3) Then, we will try to understand problems related to the boundary when we want to get higher order approximation. This requires a precise understanding of boundary layers
4) Finally, we will also study some rugosity problems.

Books :
1) Cioranescu, Doina; Donato, Patrizia An introduction to homogenization. Oxford Lecture Series in Mathematics and its Applications, 17. The Clarendon Press, Oxford University Press, New York, 1999. x+262 pp.
2) Jikov, V. V.; Kozlov, S. M.; Oleuinik, O. A. Homogenization of differential operators and integral functionals.. Translated from the Russian by G. A. Yosifian [G. A. Iosifprime yan]. Springer-Verlag, Berlin, 1994. xii+570 pp.

Articles :
1) Allaire, Grégoire; Amar, Micol Boundary layer tails in periodic homogenization. ESAIM Control Optim. Calc. Var. 4 (1999), 209--243
2) Moskow, Shari; Vogelius, Michael First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof. Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 6, 1263--1299.
3) David G\'erard-Varet and Nader Masmoudi, Homogenization in polygonal domains (in preparation)
4) Lions, P.-L.; Masmoudi, N. Homogenization of the Euler system in a 2D porous medium. J. Math. Pures Appl. (9) 84 (2005), no. 1, 1--20.