Homogenization of a mixed boundary-value problem for the Laplace operator in the case of an insoluble 'limit' problem

In this paper, the asymptotic behaviour of the solution of a mixed boundary-value problem for the Laplace operator in a domain with equal and periodically located stuck regions (with homogeneous Dirichlet data) is studied in two cases: the stuck regions are dispersed over the domain, or they are placed on the boundary. The period of the structure and the size of a stuck region compared with the period are small parameters. In the limit, the stuck regions disappear, and the formal limit problem (the averaged problem) does not necessarily have solutions. In particular, this means that zero is an eigenvalue of the Laplace operator with corresponding boundary conditions. Several terms of the asymptotic expansion of the solution with respect to the small parameters are obtained. Since the limit problem is insoluble, the asymptotics constructed contain terms that increase unboundedly.