Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Two-Scale Convergence of Stekloff Eigenvalue Problems in Perforated Domains | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 853717 15 pages doi 2010 853717 Research Article Two-Scale Convergence of Stekloff Eigenvalue Problems in Perforated Domains Hermann Douanla Department of Mathematical Sciences Chalmers University of Technology 41296 Gothenburg Sweden Correspondence should be addressed to Hermann Douanla douanla@ Received 31 July 2010 Accepted 11 November 2010 Academic Editor Gary Lieberman Copyright 2010 Hermann Douanla. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. By means of the two-scale convergence method we investigate the asymptotic behavior of eigenvalues and eigenfunctions of Stekloff eigenvalue problems in perforated domains. We prove a concise and precise homogenization result including convergence of gradients of eigenfunctions which improves the understanding of the asymptotic behavior of eigenfunctions. It is also justified that the natural local problem is not an eigenvalue problem. 1. Introduction We are interested in the spectral asymptotics as 0 of the linear elliptic eigenvalue problem N d yA-tj1 dxt x ati x du dxj 0 in Q N x du t y atj x d vt l u on dT 1 oxi u 0 on dQ U ị2dơ x 1 where Q is a bounded open set in the numerical space of variables x x1 . xN with integer N 2 with Lipschitz boundary dQ atj e C Q L RN 1 t j N with 2 Boundary Value Problems the symmetry condition Uji aij the periodicity hypothesis for each x e Q and for every k el N one has aij x y k aij x y almost everywhere in y el N and finally the ellipticity condition there exists a 0 such that for any x e Q N _ Re aij x y ềjềi a ị 2 i j 1 for all ị e CN and for almost all y el N where ị 2 ịi 2 N 2. The set QQ e 0 is a domain perforated as follows. Let T c Y 0 1 N be a compact subset in RN with smooth boundary dT S and nonempty .
