Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain

This paper is aimed at a homogenization problem for a parabolic variational inequality with unilateral constraints. The constraints on solutions are imposed on disk-shaped subsets belonging to the boundary of the domain and forming a periodic structure, so that one has a problem with rapidly oscillating boundary conditions on a part of the boundary. Under certain conditions on the relation between the period of the structure and the radius of the disks, the homogenized problem is obtained. With the help of special auxiliary functions, the solutions of the original variational inequalities are shown to converge to the solution of the homogenized problem in Sobolev space as the period of the structure tends to zero.