Differential Equations : Homogenization and Variational Methods for Partial Differential Equations

The main interest is on Aysmptotic Analysis of partial differential equations. This is a technique to understand the macroscopic behaviour of a composite medium through its microscopic properties. The technique is commonly used for PDE with highly oscillating coefficients. The idea is to replace a given heterogeneous medium by a fictitious homogeneous one (the `homogenized' material) for numerical computations. The technique is also known as ``Multi scale analysis''. The known and unknown quantities in the study of physical or mechanical processes in a medium with micro structure depend on a small parameter $\varepsilon$. The study of the limit as $ \varepsilon \rightarrow0 $, is the aim of the mathematical theory of homogenization. The notion of $G$-convergence, $H$-convergence, two-scale convergence are some examples of the techniques employed for specific cases. The variational characterization of the technique for problems in calculus of variations is given by $\Gamma$-convergence.