Homogenization of a parabolic equation in a perforated domain with a unilateral dynamic boundary condition: critical case

In this paper, we study the homogenization of a parabolic equation given in a domain perforated by “tiny” balls. On the boundary of these perforations, a unilateral dynamic boundary constraints are specified. We address the so-called “critical” case that is characterized by a relation between the coefficient in the boundary condition, the period of the structure and the size of the holes. In this case, the homogenized equation contains a nonlocal “strange” term. This term is obtained as a solution of the variational problem involving ordinary differential operator.