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Homogenization and norm-resolvent convergence for elliptic operators in a strip perforated along a curve

Part of: Spectral theory and eigenvalue problems Partial differential equations

Published online by Cambridge University Press:  08 November 2016

Denis Borisov ,
Giuseppe Cardone  and
Tiziana Durante
Denis Borisov
Affiliation:
Institute of Mathematics with Computer Centre, Ufa Scientific Centre, Russian Academy of Sciences, Chernyshevsky Street 112, Ufa 450008, Russia; Bashkir State Pedagogical University, October Street 3a, Ufa 450000, Russia and University of Hradec Králové 62, Rokitanského, Hradec Králové 50003, Czech Republic ([email protected])
Giuseppe Cardone
Affiliation:
University of Sannio, Department of Engineering, Corso Garibaldi 107, 82100 Benevento, Italy ([email protected])
Tiziana Durante
Affiliation:
University of Salerno, Department of Information and Electrical Engineering and Applied Mathematics, Via Ponte Don Melillo 1, 84084 Fisciano (SA), Italy ([email protected])
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Extract

We consider an infinite planar straight strip perforated by small holes along a curve. In such a domain, we consider a general second-order elliptic operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic and satisfies rather weak assumptions, we describe all possible homogenized problems. Our main result is the norm-resolvent convergence of the perturbed operator to a homogenized one in various operator norms and the estimates for the rate of convergence. On the basis of the norm-resolvent convergence, we prove the convergence of the spectrum.

Keywords

perforationelliptic operatorunbounded domainhomogenizationnorm-resolvent convergence

MSC classification

Secondary: 35B27: Homogenization; equations in media with periodic structure 35P05: General topics in linear spectral theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 146 , Issue 6 , December 2016 , pp. 1115 - 1158
Copyright
Copyright © Royal Society of Edinburgh 2016 

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