Homogenization of the Scalar Boundary Value Problem in a Thin Periodically Broken Cylinder

Homogenization of the Neumann problem for a differential equationin a periodically broken multidimensional cylinderleads to a second-order ordinary differential equation.We study asymptotics for the coefficient of the averaged operatorin the case of small transverse cross-sections.The main asymptotic term depends onthe “area” of cross-sections of the links,their lengths,and the coefficient matrix of the original operator.We find the characteristics of kink zones which affect correction terms,while the asymptotic remainder becomes exponentially small.The justification of the asymptoticsis based on Friedrichs’s inequalitywith a coefficient independent of both small parameters:the period of fractures and the relative diameter of cross-sections.