Uniform Estimates of Resolvents in Homogenization Theory of Elliptic Systems

In this paper, we study the estimates of resolvents $R(\lambda ,{\mathcal{L}}_{\epsilon })=({\mathcal{L}}_{\epsilon }-\lambda I{)}^{-1}$, where $${\mathcal{L}}_{\epsilon }=-\mathrm{div}(A(x\mathrm{/}\epsilon )\mathrm{\nabla })$$ is a family of second elliptic operators with symmetric, periodic and oscillating coefficients defined on a bounded domain $\mathrm{\Omega }$ with $\epsilon >0$. For $1<p<\mathrm{\infty }$, we will establish uniform $L^p\to L^p$, $L^p\to W_0^{1,p}$, $W^{-1,p}\to L^p$ and $W^{-1,p}\to W_0^{1,p}$ estimates by using the real variable method. Meanwhile, we use Green functions for operators ${\mathcal{L}}_{\epsilon }-\lambda I$ to study the asymptotic behavior of $R(\lambda ,{\mathcal{L}}_{\epsilon })$ and obtain convergence estimates in $L^p\to L^p$, $L^p\to W_0^{1,p}$ norm.