Regularity Estimates In Homogenization Theory Of Elliptic Systems

In this thesis, we study uniform regularities and convergence rates for general elliptic op-erators arising in homogenization theory. Our results are based on three aspects. First one, M. Avellaneda and F. Lin published a series of papers to show the uniform Holder estimates, Lp estimates with (1<p<∞), Lipschitz estimates, and non-tangential maximal function estimates, for elliptic operators with Dirichlet boundary condition in homogenization theory. In a direct view of elliptic homogenization problems, coefficients of a operator depend on a parameter ε. So by traditional methods such as the freezing coefficient method, the constant in the estimates depends on the smoothness of the coefficients, and therefore relies on the parameter ε. Hence the difficulty is how to obtain uniform estimate (which means the constant in the estimate does not depend on ε). The main contribution is that the above two authors develop a compactness method to overcome this difficulty. However, for Neumann boundary problem, it is not until the paper es-tablished by C. Kenig, F. Lin and Z. Shen in 2013 that there was no significant progress on this topic. They first obtain the Rellich estimate (which is actually more difficult than Lipschitz one) and then have the optimal estimate by using compactness method. This is the second aspect. The last aspect is the research of convergence rates. T. Suslina employs the Steklov smoothing operator coupled with the duality method to obtain the sharp convergence rates in L2 space on a smooth domain. For non-smooth domain, the main progress is made by C. Kenig, F. Lin, Z. Shen.Due to the above references, we study the following elliptic operator with Dirichlet boundary value, and Neumann boundary value. We obtain a series of results, which prove Wl,p estimates, Holder estimates, Lipschitz estimates, and non-tangential maximal function estimates (only archived for Dirichlet problem), as well as convergence rates in the sense of different norms. We note that we do not repeat compactness methods or the new one which has been developed by S. Armstrong and Z. Shen recently. Instead we plan to apply the acquired results of the elliptic operator only with main term to our cases as many as possible. Here we need to construct a new Dirichlet boundary corrector and a Neumann boundary corrector, which transform an unfamiliar form to a familiar one. Our method makes the proof concise and clear. Finally, we point out that the following example shows that our results are not very trivial as it seems to be. where W is referred to as the rapidly oscillating potential term. We can see that though main term is the Laplace operator, we can not figure out any estimate only when the coefficients satisfy some additional condition such as periodicity, as e goes to zero. The above operator is also our original model in the papers.