The estimate of the multi-scale homogenization methodfor Green's function on Sobolev space $W^{1,q}(\Omega)$
Wen‐ming He, Junzhi Cui · Communications on Pure & Applied Analysis · 2012
In this paper, for the second order ellipticproblems with small periodic coefficients of the form$\frac{\partial}{\partial x_{i}}(a^{i j}(\frac{x}{\varepsilon})\frac{\partialu^{\varepsilon}(x)}{\partial x_{j}})=f(x)$, we shall discuss themulti-scale homogenization theory for Green's function$G_{y}^{\varepsilon}$ at point $y\in\Omega$ on Sobolev space$W^{1,q}(\Omega)$. Assume that $B(y,d)=\{x\in\Omega|dist(x,y)\leqd\},$ ${G}_{y}$ and $\theta_{G,y}^{\varepsilon}$ are the1-order approximation and the boundary corrector of$G_{y}^{\varepsilon}$, respectively. We present an estimate for$\left\|G_{y}^{\varepsilon}-{G}_{y}-\theta_{G,y}^{\varepsilon}\right\|_{W^{1,q}(\Omega\ B(y,d))}$.