Elliptic Operators in Multidimensional Cylinders with Frequently Alternating Boundary Conditions Along a Given Curve

We consider a selfadjoint elliptic operator in an infinite multidimensional cylinder with the Dirichlet boundary condition which is replaced by the Robin condition on small sets located along a given line on the boundary. The shape and distribution of these sets are arbitrary. The characteristic linear size of these sets is a small parameter of the problem. It is shown that the resolvent of such an operator converges to the resolvent of the homogenized operator, and an estimate for the convergence rate is obtained. The homogenized operator is the same operator, but without alternating boundary conditions. The difference of resolvents is estimated in the norm of bounded operators acting from L2 to $$ {W}_2^1. $$ If the small sets are periodically distributed and have the same shape and the Robin condition is replaced by the Neumann condition, we derive two-sided asymptotic estimates for the lower band functions, which provides the possibility to estimate from below the length of the first band of the spectrum.