2019/09/30 by Zhang, Yiping
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1909.13735
In this paper, we are interested in the reiterated homogenization of linear elliptic equations of the form -\frac∂∂ xi (ai j ((x)/(ε), \fracxε2) \frac∂ uε∂ xj)=f in Ω with Dirichlet boundary conditions. We obtain error estimates O(ε) for a bounded C1,1 domain for this equation as well as the interior Lipschitz estimates at (very) large scale. Compared to the general homogenization problems, the difficulty in the reiterated homogenization is that we need to handle different scales of x. To overcome this difficulty, we firstly introduce the Fourier transform in the homogenization theory to separate these different scales. We also note that this method may be adapted to the following reiterated homogenization problem: -\frac∂∂ xi (ai j ((x)/(ε),⋯, \fracxεN) \frac∂ uε∂ xj ) = f in Ω with Dirichlet boundary conditions. Moreover, our results may be extended to the related Neumann boundary problems without any real difficulty.