Homogenization of elliptic equations in random media

• Main Authors: Lin, Hong-Miao, 林鴻淼
• Other Authors: Yeh, Li-Ming
• Format: Others
• Language: en_US
• Published: 2010
• Online Access: http://ndltd.ncl.edu.tw/handle/87050132268235131827

碩士 === 國立交通大學 === 應用數學系所 === 98 === In the most general sense, a heterogeneous material is one that is composed of domains of different materials (or phases), such as a composite, or the same material in different states, such as a polycrystal. In many instances, the mi- crostructures can be characterized only statistically, and therefore are referred to as random heterogeneous materials(or random media), the chief of this study. Consider an elliptic equation :   −div(A(ε−1 x, ω)∇uε (x, ω)) = f (x) on Q,  uε (x, ω)| = 0 on ∂Q; where A, f, and u are in suitable function spaces , ω ∈ Ω and (Ω, Σ, μ) is a suitable probability space. In this study we introduce the ergodic dynamical systems on the probability space to describe the random media; we show the matrix A(x, ω) above admits homogenization( see Definition.4.2) and the ho- mogenized matrix is independent of ω ∈ Ω. We give definitions, examples, and proofs about ergodic dynamical systems in section two. Section three is about definition of realizations, and the ergodic theorem. In section four, we recall the definition of homogenization of ellip- tic equations for individual cases and statistical cases, and use the auxiliary equations to define the homogenized matrix, and prove the main convergence theorem through the div-curl lemma. In section five, we define the random sets of the percolation, consider the existence of the effective conductivity, and state the theorem of the existence of the effective conductivity of such random media.