Homogenization of the Neumann problem for a differential equation in a periodically broken multidimensional cylinder leads to a second-order ordinary differential equation. We study asymptotics for the coefficient of the averaged operator in the case of small transverse cross-sections. The main asymptotic term depends on the “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 asymptotics is based on Friedrichs’s inequality with a coefficient independent of both small parameters: the period of fractures and the relative diameter of cross-sections.

周期性破碎的多维圆柱体中微分方程的诺伊曼问题的齐次化产生二阶常微分方程。我们研究了小横截面情况下平均算子系数的渐进性。主要渐近项取决于连杆横截面的“面积”、其长度以及原始算子的系数矩阵。我们发现了影响校正项的扭结区的特征，而渐近余数则呈指数级变小。渐进的论证基于弗里德里希不等式，其系数独立于两个小参数：断裂周期和横截面的相对直径。