Homogenization of elliptic operators with periodic coefficients in dependence of the spectral parameter
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T. A. Suslina
Translated by: T. A. Suslina - St. Petersburg Math. J. 27 (2016), 651-708
- DOI: https://doi.org/10.1090/spmj/1412
- Published electronically: June 2, 2016
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Abstract:
Differential expressions of the form $b(\mathbf {D})^* g(\mathbf {x}/\varepsilon ) b(\mathbf {D})$, $\varepsilon >0$, are considered, where a matrix-valued function $g(\mathbf {x})$ in $\mathbb {R}^d$ is assumed to be bounded, positive definite, and periodic with respect to some lattice; $b(\mathbf {D})=\sum _{l=1}^d b_l D_l$ is a first order differential operator with constant coefficients. The symbol $b({\boldsymbol \xi })$ is subject to some condition ensuring strong ellipticity. The operator in $L_2(\mathbb {R}^d;\mathbb {C}^n)$ given by the expression $b(\mathbf {D})^* g(\mathbf {x}/\varepsilon ) b(\mathbf {D})$ is denoted by $\mathcal {A}_\varepsilon$. Let $\mathcal {O} \subset \mathbb {R}^d$ be a bounded domain of class $C^{1,1}$. The operators $\mathcal {A}_{D,\varepsilon }$ and $\mathcal {A}_{N,\varepsilon }$ under study are generated in the space $L_2(\mathcal {O};\mathbb {C}^n)$ by the above expression with the Dirichlet or Neumann boundary conditions. Approximations in various operator norms for the resolvents $(\mathcal {A}_\varepsilon - \zeta I)^{-1}$, $(\mathcal {A}_{D,\varepsilon }- \zeta I)^{-1}$, $(\mathcal {A}_{N,\varepsilon }-\zeta I)^{-1}$ are obtained with error estimates depending on $\varepsilon$ and $\zeta$.References
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Bibliographic Information
- T. A. Suslina
- Affiliation: St. Petersburg State University, Department of Physics, Petrodvorets, Ul′yanovskaya 3, 198504, St. Petersburg, Russia
- Email: suslina@list.ru
- Received by editor(s): December 10, 2014
- Published electronically: June 2, 2016
- Additional Notes: Supported by RFBR (grant no. 14-01-00760a) and by St. Petersburg State University (grant no. 11.38.263.2014)
- © Copyright 2016 American Mathematical Society
- Journal: St. Petersburg Math. J. 27 (2016), 651-708
- MSC (2010): Primary 35B27
- DOI: https://doi.org/10.1090/spmj/1412
- MathSciNet review: 3580194