Homagenization of partial differential equations in perforated domain

Abstract

Theory of homogenization in Partial Differential Equations has a variety of applications in many branches of science. In the present thesis, we study the homogenization of certain boundary value problems in a periodically perforated domain. This thesis is divided into four chapters. In the introductory chapter, a brief overview of homogenization and the description of various chapters are presented. It ends with a list of open problems related to the results discussed in the thesis. In the second chapter, we study the homogenization of the steady-state and evolution Stokes equation with non-homogeneous, non-zero data on the boundary of the holes of a porous domain ??, obtained from a domain ? by removing a large number of holes of size ? (? > 0, a small parameter), periodically distributed in the domain with period ?. In the homogenization process, we obtain a system which is a well-defined system of equations involving both the 'slow' variable x and the 'fast' variable y = x/?. We also derive Darcy’s law, which contains an extra term, and this additional term is the contribution from the non-homogeneous data. The homogenization of Stokes, Laplacian, and Bi-Laplacian eigenvalue problems in a periodically (period ? > 0) perforated domain is studied in Chapter 3. We study these problems when the size of the holes (say, a) is much smaller than the period ? and obtain the homogenized system. We show that for the Stokes and Laplacian problems, the critical size of the holes is: • a=C??N/(N-2)a = C \cdot \varepsilon^{N/(N-2)}a=C??N/(N?2) if N>3N > 3N>3, and • a=exp?(-C/?)a = \exp(-C/\varepsilon)a=exp(-C/?) if N=2N = 2N=2, where C is a constant and N is the dimension of the space. For the Bi-Laplacian case, the corresponding critical size is: • a=C??N/(N-4)a = C \cdot \varepsilon^{N/(N-4)}a=C??N/(N?4) if N>5N > 5N>5, and • a=exp?(-C/?)a = \exp(-C/\varepsilon)a=exp(?C/?) if N=4N = 4N=4. Corrector results for the eigenvalues and eigenvectors have also been studied. Chapter 4 is devoted to the study of the homogenization of the eigenvalue problem associated with an elasticity system in a periodically perforated domain with tiny holes. The critical size of the holes is the same as in the Stokes and Laplacian cases. We study the above eigenvalue problem as ??0\varepsilon \to 0??0 and obtain the corresponding homogenized system. We also study the correctors for the eigenvalues and eigenvectors.