Semi-linear optimal control problems with strong contrasting diffusivity in rough domains

Abstract

This article explores the homogenization of semi-linear elliptic partial differential equations (PDEs) with strong contrasting diffusivity in domains with highly oscillating boundary. The oscillations considered are general and can occur in multiple directions, and the oscillating part is made up of two distinct materials with strong contrasting diffusivity. Also, corrector results are established within this framework. The tool of analysis is periodic unfolding method and to deal with the non-linear term, we use the Browder-Mindy method. Furthermore, the research extends to the analysis of optimal control problems governed by semi-linear elliptic PDEs, where the cost functional and PDEs exhibit strong contrasting diffusivity. Later in this article we also make the similar studies in a circular oscillating boundary domain, where the oscillations are in circular fashion.