Homogenization of a Parabolic Equation in a Perforated Domain with a Unilateral Dynamic Boundary Condition: Critical Case

The present paper studies the homogenization of the parabolic equation stated 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 to the variational problem involving an ordinary differential operator.