Mathematics – Functional Analysis
Scientific paper
2009-10-02
Mathematical Methods in the Applied Sciences, vol. 34 (2011) p. 1075-1096
Mathematics
Functional Analysis
30
Scientific paper
10.1002/mma.1424
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 Schr\"odinger operator with a singular potential periodic in one direction.
Bunoiu Renata
Cardone Giuseppe
Suslina T.
No associations
LandOfFree
Spectral approach to homogenization of an elliptic operator periodic in some directions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Spectral approach to homogenization of an elliptic operator periodic in some directions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Spectral approach to homogenization of an elliptic operator periodic in some directions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-26712