某方向周期椭圆算子均匀化的谱方法
Spectral approach to homogenization of an elliptic operator periodic in some directions
摘要：
在$L_{2}({\mathbb{R}}^{2})$中考虑算子\[ A_{\varepsilon}= D_{1} g_{1}(x_{1}/\varepsilon, x_{2}) D_{1} + D_{2} g_{2}(x_{1}/\varepsilon, x_{2}) D_{2} \]，其中$g_{j}(x_{1},x_{2})$，$j=1,2,$在$x_{1}$中是周期的，周期为1，有界正定.设函数$Q(x_{1},x_{2})$在$x_{1}$中是有界的、正定的和周期的，周期为1。让$Q^{\varepsilon}(x_{1},x_{2})= Q(x_{1}/\varepsilon, x_{2})$。研究了算子$(A_{\varepsilon}+ Q^{\varepsilon}%)^{-1}$作为$\varepsilon\to0$的行为。证明了算子$(A_{\varepsilon}+ Q^{\varepsilon})^{-1}$在$L_{2}(\mathbb{R}^{2})$的算子范数下趋于$(A^{0} + Q^{0})^{-1}$.这里$A^{0}$是其系数仅依赖于$x_{2}$的有效算子，$Q^{0}$是$x_{1}$中$Q$的平均值。得到了差分$(A_{\varepsilon}+ Q^{\varepsilon})^{-1}- (A^{0} + Q^{0})^{-1}$范数的一个锐阶估计。结果适用于均匀化的薛定谔算子与奇异位周期在一个方向上。
英文摘要：
The operator \[ A_{\varepsilon}= D_{1} g_{1}(x_{1}/\varepsilon, x_{2}) D_{1} + D_{2} g_{2}(x_{1}/\varepsilon, x_{2}) D_{2} \] is considered in $L_{2}({\mathbb{R}}^{2})$, where $g_{j}(x_{1},x_{2})$, $j=1,2,$ are periodic in $x_{1}$ with period 1, bounded and positive definite. Let function $Q(x_{1},x_{2})$ be bounded, positive definite and periodic in $x_{1}$ with period 1. Let $Q^{\varepsilon}(x_{1},x_{2})= Q(x_{1}/\varepsilon, x_{2})$. The behavior of the operator $(A_{\varepsilon}+ Q^{\varepsilon}%)^{-1}$ as $\varepsilon\to0$ is studied. It is proved that the operator $(A_{\varepsilon}+ Q^{\varepsilon})^{-1}$ tends to $(A^{0} + Q^{0})^{-1}$ in the operator norm in $L_{2}(\mathbb{R}^{2})$. Here $A^{0}$ is the effective operator whose coefficients depend only on $x_{2}$, $Q^{0}$ is the mean value of $Q$ in $x_{1}$. A sharp order estimate for the norm of the difference $(A_{\varepsilon}+ Q^{\varepsilon})^{-1}- (A^{0} + Q^{0})^{-1}$ is obtained. The result is applied to homogenization of the Schrödinger operator with a singular potential periodic in one direction.
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