Spectral Asymptotics in Porous Media
نویسنده
چکیده
This thesis consists of two papers devoted to the asymptotic analysis of eigenvalue problems in perforated domains. The first paper investigates by means of the two-scale convergence method 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. In the second paper, reiterated homogenization of linear elliptic Neuman eigenvalue problems in multiscale perforated domains is considered beyond the periodic setting. The classical periodicity hypothesis on the coefficients of the operator is there substituted on each microscale by an abstract hypothesis covering a large set of concrete behaviors such as the periodicity, the almost periodicity, the weakly almost periodicity and many more besides. Furthermore, the usual double periodicity is generalized by considering a type of structure where the perforations on each scale follow not only the periodic distribution but also more complicated but realistic ones.
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