Mathematics

Dr. Nandakumaran A. K.
Indian Institute of Science Bangalore

The strong and weak convergence in a Hilbert space, in particular

L2space is well known. But when we take a limit of a sequence, more specifically weak limit, the limitwill loose lot of important information contained in the original sequence. For example,

the weak limit of the sequence sin nx is 0, and we do not see the oscillations present in the sequence. Further, the product of two weakly convergent sequences do not converge to the product of the limits. This causes trouble at various stages of studying problems and doing the analysis. In this talk, we introduce the notion of two-scale convergence to retrieve certain lost information in relevance to homogenization theory and present

the importance of two-scale convergence and a compactness theorem. Most of the time will spend on the general theory to cater to the general audience who are familiar with Functional Analysis. In the last part, we apply it to study a homogenization problem in composite media consisting of two highly contrasting materials which is modeled by a hyperbolic equation.