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 bou ndary-value problem for the Laplace operator in a domain with equal an d periodically located stuck regions (with homogeneous Dirichlet data) is studied in two cases: the stuck regions are dispersed over the dom ain, or they are placed on the boundary. The period of the structure a nd the sire of a stuck region compared with the period are small param eters. In the limit, the stuck regions disappear, and the formal limit problem (the averaged problem) does not necessarily have solutions. I n particular, this means that zero is an eigenvalue of the Laplace ope rator with corresponding boundary conditions. Several terms of the asy mptotic expansion of the solution with respect to the small parameters are obtained. Since the limit problem is insoluble, the asymptotics c onstructed contain terms that increase unboundedly.