在隨機介質中橢圓方程之均質化

Abstract

本論文旨在探討在隨機介質中，橢圓方程如何均質化之問題。藉由多維度空間以隨機介質上的機率空間所引導出來的動態系統，來研究如何得到均質化的方程式係數矩陣。我們論述了有關於遍歷動態系統的特性以及遍歷定理，進而以此為主要假設證明了在某些條件之下，一個隨機介質中的橢圓方程可以確實得到均質化的結果。並且舉例說明如何將此均質化應用於某些例子。

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.