Unfolding operator on Heisenberg Group and applications in Homogenization

TL;DR

This paper extends the periodic unfolding method to the Heisenberg group, enabling homogenization of PDEs with oscillating coefficients and applications in optimal control within a non-commutative setting.

Contribution

It develops the periodic unfolding operator for the Heisenberg group, including key properties and an adjoint operator, and applies it to homogenize elliptic PDEs and analyze optimal control problems.

Findings

Successfully defined the unfolding operator on the Heisenberg group.

Proved integral equality, compactness, and convergence properties.

Applied the method to homogenize elliptic PDEs and characterize optimal controls.

Abstract

The periodic unfolding method is one of the latest tool after multi-scale convergence to study multi-scale problems like homogenization problems. It provides a good understanding of various micro scales involved in the problem which can be conveniently and easily applied to get the asymptotic limit. In this article, we develop {\it the periodic unfolding} for the Heisenberg group which has a non-commutative group structure. In order to do this, the concept of the greatest integer part, fractional part for the Heisenberg group has been introduced corresponding to the periodic cell. Analogous to the Euclidean unfolding operator, we prove the integral equality, $L^2$-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. As an application of the unfolding operator, we have homogenized the standard elliptic PDE with oscillating coefficients. We have also considered an optimal control problem and characterized the interior periodic optimal control in terms of unfolding operator.