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

Abstract

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.