Uniform convergence for problems with perforation alogn a given manifold and with a nonlinear Robin condition on the boundaries of cavities

In this work we consider a boundary value problem for a second order elliptic equation with variable coefficients in a multi-dimensional domain perforated by small cavities closely spaced along a given manifold. We suppose that the sizes of the cavities are of the same smallness order, while their shapes and distribution along the manifold are arbitrary. On the boundaries of the cavities we impose a nonlinear Robin condition. We prove the convergence of the solution of the perturbed problem to that of the homogenized proble in $L_2$- and $W_2^1$-norms uniformly in $L_2$-norm of the right hand side in the equation and we obtain estimates for the convergence rates.