Aram Karakhanyan: On a conjecture of De Giorgi related to homogenisation

In this talk a will discuss some problems related to homogenisation of first order systems of ODEs with oscillating structures. A typical example would be \(y' = F(y/\epsilon)\) where F is a smooth, periodic vector field and \(\epsilon > 0\) a small parameter. In 1994 E. De Giorgi conjectured that \(y_0\), the limit of y as \(\epsilon\to 0\), must be a linear function. This is proved in 1 and 2 dimensions for general flow generated by F and in higher dimensions for the shear flow. The results to be presented are from a joint work with H. Shahgholian.