Affiliations: Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow Lomonosov State University, Moscow 119991, Russia. E-mail: [email protected] | Dipartimento di Ingegneria dell'Informazione e Matematica Applicata, Università degli Studi di Salerno, Via Ponte don Melillo, 1, Fisciano (SA) 84084, Italia. E-mail: [email protected] | Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Complesso Monte S. Angelo, Via Cintia, 80126 Napoli, Italy. E-mail: [email protected] | P.N. Lebedev Physical Institute RAS, Leninski pr., 53, Moscow 117924, Russia and Narvik University College, Postboks 385, 8505 Narvik, Norway. E-mail: [email protected]
Abstract: In the paper we deal with the homogenization problem for the Poisson equation in a singularly perturbed domain with multilevel oscillating boundary. This domain consists of the body, a large number of thin periodically situated cylinders joining to the body through thin random transmission zone with rapidly oscillating boundary. Inhomogeneous Fourier boundary conditions with perturbed coefficients are set on the boundaries of the thin cylinders and on the boundary of the transmission zone. We prove the homogenization theorems. Moreover we derive estimates of deviation of the solution to initial problem from the solution to the homogenized problem in different cases. It appears that depending on small parameters in Fourier boundary conditions of initial problem one can obtain Dirichlet, Neumann or Fourier boundary conditions in the homogenized problem. We estimate the convergence of solutions in these three cases.
Keywords: homogenization, estimates of convergence, rapidly oscillating boundary, singular perturbations, random structures
