High contrasting diffusion in Heisenberg group and homogenization of optimal control problems via unfolding

Abstract: After the development of multi-scale convergence in the 1990s, the periodic unfolding approach is one of the most effective methods for studying multi-scale problems like homogenization in the Euclidean setup. It provides a thorough understanding of the different micro-scales involved in the problem, which can be applied to determine the asymptotic limit.

This talk will discuss the periodic unfolding operator in the Heisenberg group.  Analogous to the Euclidean unfolding operator, we prove the integral equality, $L^p$-weak compactness, unfolding gradient convergence, and other related properties. Moreover, we have the adjoint operator for the unfolding operator, which can be recognized as an average operator. We apply the unfolding operator to homogenize an optimal control problem subject to a state equation having high contrast diffusive coefficients.