Crossing thresholds in high contrast boundary homogenization problems: a review

Abstract

In this paper we provide a review of the state of the art in two topics of the homogenization theory: boundary homogenization for rapidly alternating boundary conditions on grill-type walls and homogenization of linear or nonlinear Robin boundary conditions in perforate media. High contrasts on the Robin boundary conditions are introduced. These contrasts are represented by new parameters and, along with the period and the sizes of "grill-crossing points"/perforations, play an important role in determining "thresholds" marking changes in the homogenized boundary conditions, either with "averaged terms" or "strange terms" or, somehow, "extreme" asymptotic behaviors for the solutions. Taking into account the advances of the last years in the topics, we also overview vector problems (cf. Winkler-Robin boundary conditions) and models in perforated media of interest in material sciences among others.