  In the work we consider a boundary value problem for a second order equation
with variable coefficients in a multi-dimensional domain perforated by small
cavities closely spaced along a given manifold. We assume that the linear sizes
of all cavities are of a same order of smallness, while their shapes and
distributions are arbitrary. The boundaries of the cavities are subject to a
nonlinear Robin condition. We prove that the solution of the perturbed problem
converges to that of the homogenized problem in norm $L_2$ and $W_2^1$
uniformly in $L_2$-norm of the right hand side in the equation. We also
establish the estimates for the convergence rates.
