Operator estimates for non-periodic perforation along boundary: homogenized Dirichlet condition

Abstract

We consider a boundary value problem for a second--order elliptic equation with variable coefficients in a multidimensional domain perforated by small cavities along the boundary. We suppose that the sizes of all cavities are of the same order, and their shape and distribution along the boundary can be arbitrary. The cavities are arbitrarily divided into two sets. The Dirichlet condition is imposed on the boundaries of cavities in the first set, and a nonlinear Robin boundary condition is imposed on the boundaries of cavities in the second set. The Neumann condition is imposed on the boundary along which the perforation is arranged. It is assumed that the cavities with the Dirichlet condition are not too small and are located fairly closely. We shown that under such assumptions, the cavities disappear under the homogenization, and the Dirichlet condition arises on the boundary. Our main result is estimates for the difference between the solutions of the homogenized and perturbed problems in the $W_2^1$-norm uniformly in the $L_2$--norm of the right hand side.