Defects in Homogenization Theory

We review a series of works that address homogenization for partial differential equations with highly oscillatory coefficients. A prototypical setting is that of periodic coefficients that are locally, or more globally perturbed. We investigate the homogenization limits obtained, first for linear elliptic equations, both in conservative and non conservative forms, and next for nonlinear equations such as Hamilton-Jacobi type equations. The connection between the above theoretical endeavour and strategies for modeling actual materials and simulating them using computational mutiscale approaches will also be addressed.

The works presented have been completed in collaboration with a number of colleagues, in particular with Y. Achdou, X. Blanc, P. Cardaliaguet, P.-L. Lions, P. Souganidis, and R. Goudey.