Unfolding operators for general oscillating domains and Homogenization

Abstract: Unfolding operators have been introduced and used to study homogenization problems. Initially, they were introduced for problems with rapidly oscillating coefficients and porous domains. Later, this has been developed for domains with oscillating boundaries, typically with rectangular or pillar type boundaries which are classified as non-smooth. We have successfully developed unfolding operators for general smooth domains. In this talk, I will present the unfolding operators that we have developed, where the oscillations can be smooth and hence they have wider applications. And, as an application, I will demonstrate the homogenization of a semi-linear problem posed on this domain.  We have used this new operator to characterize the optimal control on oscillating domains which will not be elaborated in this talk. I will also demonstrate the application of the unfolding operator on a circular domain which is more related to real life applications like heat radiators, jet engines etc.