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 L2- and W21-norms uniformly with respect to the L2-norm of the right-hand side of the equation. Estimates of the convergence rates are also obtained. © 2024 American Mathematical Society