• DOI:10.1137/120901921
• Corpus ID: 3194018

Homogenization of the Neumann Problem for Elliptic Systems with Periodic Coefficients

@article{Suslina2013HomogenizationOT,
  title={Homogenization of the Neumann Problem for Elliptic Systems with Periodic Coefficients},
  author={Tatiana Suslina},
  journal={SIAM J. Math. Anal.},
  year={2013},
  volume={45},
  pages={3453-3493}
}

• T. Suslina
• Published 5 December 2012
• Mathematics
• SIAM J. Math. Anal.

Let ${\mathcal O} \subset {\mathbb R}^d$ be a bounded domain with the boundary of class $C^{1,1}$. In $L_2({\mathcal O};{\mathbb C}^n)$, a matrix elliptic second order differential operator ${\mathcal A}_{N,\varepsilon}$ with the Neumann boundary condition is considered. Here $\varepsilon>0$ is a small parameter; the coefficients of ${\mathcal A}_{N,\varepsilon}$ are periodic and depend on ${\mathbf x} /\varepsilon$. There are no regularity assumptions on the coefficients. It is shown that the resolvent $({\mathcal A}_{N,\varepsilon}-\lambda I)^{-1}$, $\lambda \in {\mathbb C} {\setminus} {\mathbb R}_+$, converges in the $L_2$-operator norm to the resolvent of the effective operator ${\mathcal A}_N^0$ with constant coefficients, as $\varepsilon \to 0$. A sharp order estimate $\|({\mathcal A}_{N,\varepsilon}-\lambda I)^{-1} - ({\mathcal A}_{N}^0 -\lambda I)^{-1}\|_{L_2\to L_2} \le C\varepsilon $ is obtained. Approximation for the resolvent $({\mathcal A}_{N,\varepsilon}-\lambda I)^{-1}$ in the norm of opera...