Browsing by Subject "Elliptic partial differential equations"

Stochastic homogenization consists of qualitative and quantitative homogenization. It studies the solutions of certain elliptic partial differential equations that exhibit rapid random oscillations in some heterogeneous physical system. Our aim is to homogenize these perturbations to some regular large-scale limiting function by utilizing particular corrector functions and homogenizing matrices. This thesis mainly considers elliptic qualitative homogenization and it is based on a research article by Scott Armstrong and Tuomo Kuusi. The purpose is to elaborate the topics presented there by viewing some other notable references in the literature of stochastic homogenization written throughout the years. An effort has been made to explain further details compared to the article, especially with respect to the proofs of some important results. Hopefully, this thesis can serve as an accessible introduction to the qualitative homogenization theory. In the first chapter, we will begin by establishing some notations and preliminaries, which will be utilized in the subsequent chapters. The second chapter considers the classical case, where every random coefficient field is assumed to be periodic. We will examine the general situation later that does not require periodicity. However, the periodic case still provides useful results and strategies for the general situation. Stochastic homogenization theory involves multiple random elements and hence, it heavily applies probability theory to the theory of partial differential equations. For this reason, the third chapter assembles the most important probability aspects and results that will be needed. Especially, the ergodic theorems for R^d and Z^d will play a central part later on. The fourth chapter introduces the general case, which does not require periodicity anymore. The only assumption needed for the random coefficient fields is stationarity, that is, the probability measure P is translation invariant with respect to translations in Zd. We will state and prove important results such as the homogenization for the Dirichlet problem and the qualitative homogenization theorem for stationary random coefficient fields. In the fifth chapter, we will briefly consider another approach to qualitative homogenization. This so-called variational approach was discovered in the 1970s and 1980s, when Ennio De Giorgi and Sergio Spagnolo alongside with Gianni Dal Maso and Luciano Modica studied qualitative homogenization. We will provide a second proof for the qualitative homogenization theorem that is based on their work. An additional assumption regarding the symmetricity of the random coefficient fields is needed. The last chapter is dedicated to the large-scale regularity theory of the solutions for the uniformly elliptic equations. We will concretely see the purpose of the stationarity assumption as it turns out that it guarantees much greater regularity properties compared to non-stationary coefficient fields. The study of large-scale regularity theory is very important, especially in the quantitative side of stochastic homogenization.