Operator estimates for non-periodically perforated domains with Dirichlet and nonlinear Robin conditions: Strange term

We consider a boundary value problem for a general second-order linear equation in a domain with a fine perforation. The latter is made by small cavities; both the shapes of the cavities and their distribution are arbitrary. The boundaries of the cavities are subject either to a Dirichlet or a nonlinear Robin condition. On the perforation, certain rather weak conditions are imposed to ensure that under the homogenization, we obtain a similar problem in a non-perforated domain with an additional potential in the equation usually called a strange term. Our main results state the convergence of the solution of the perturbed problem to that of the homogenized one in W21$$ {W}_2 circumflex 1 $$- and L2$$ {L}_2 $$-norms uniformly in L2$$ {L}_2 $$-norm of the right hand side in the equation. The estimates for the convergence rates are established, and their order sharpness is discussed.