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The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$

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  • In this paper, for the second order elliptic problems with small periodic coefficients of the form $\frac{\partial}{\partial x_{i}} (a^{i j}(\frac{x}{\varepsilon})\frac{\partial u^{\varepsilon}(x)}{\partial x_{j}})=f(x)$, we shall discuss the multi-scale homogenization theory for Green's function $G_{y}^{\varepsilon}$ at point $y\in\Omega$ on Sobolev space $W^{1,q}(\Omega)$. Assume that $B(y,d)=\{x\in\Omega|dist(x,y)\leq d\},$ ${G}_{y}$ and $\theta_{G,y}^{\varepsilon}$ are the 1-order approximation and the boundary corrector of $G_{y}^{\varepsilon}$, respectively. We present an estimate for $\left\|G_{y}^{\varepsilon}-{G}_{y}-\theta_{G,y}^{\varepsilon}\right\|_{W^{1,q}(\Omega\ B(y,d))}$.
    Mathematics Subject Classification: 35J25.

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