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

A boundary value problem is treated for a second order elliptic equation with variable coefficients in a multidimensional domain perforated by small cavities closely spaced along a given manifold. The sizes of the cavities are assumed to be of the same smallness order, while their shapes and the distribution along the manifold are arbitrary. A nonlinear Robin condition is imposed on the boundaries of the cavities. It is proved that the solution of the perturbed problem converges to that of the homogenized problem in the L 2 L_2 - and W 2 1 W_2^1 -norms uniformly with respect to the L 2 L_2 -norm of the right-hand side of the equation. Estimates of the convergence rates are also obtained.