Resources

T. Muthukumar, PhD (Institute of Mathematical Sciences)

Assistant Professor

Research Interests: Homogenization and Variational Methods for PDE's, Elliptic PDE's, Optimal controls
Email: tmk[AT]iitk.ac.in
Ph:+91-512-259-7911
Website: http://home.iitk.ac.in/~tmk
The theory of homogenization is, in a nut-shell, the asymptotic analysis of a system. A layman motivation towards homogenization can be given from the material science point of view. It is a mathematical concept that incorporates the study of the macroscopic behaviour of a composite medium through its microscopic properties. The known and unknown quantities in the study of physical or mechanical processes in a medium with microstructure depend on a small parameter $\varepsilon = \frac{l}{L}$, where $L$ is the macroscopic scale length of the dimension of a specimen of the medium and $l$ is the characteristic length of the medium configuration. The study of the limit, as $\varepsilon\rightarrow 0$, is the aim of the mathematical theory of homogenization. The case $\varepsilon\rightarrow 0$ is important as a tool for numerical computations. Over the time, the theory of homogenization has found application in various branches. One of them being Shape optimization. Simply put, one may be interested in finding the optimal shape of a body that will have, for instance, maximal conductivity, rigidity etc. All these physical problems end up as a PDE with rapidly oscillation coefficients.