Volume 51, Number 4, July-August 2017
|Page(s)||1317 - 1342|
|Published online||21 July 2017|
On the Steklov problem in a domain perforated along a part of the boundary∗
1 Department of Differential Equations, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow 119991, Russia.
2 Department of Mathematics and Statistics, Faculty of Physics and Mathematics, Bashkir State Pedagogical University, Ufa 450000, Russia.
3 Dipartimento di Ingegneria dell’Informazione e Matematica Applicata, Università degli Studi di Salerno, via Ponte don Melillo, 1, 84084 Fisciano (SA), Italia.
4 Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II, Complesso Monte S.Angelo – Edificio “T”, via Cintia 80126 Napoli, Italia.
Received: 19 August 2014
Revised: 13 July 2016
Accepted: 8 September 2016
We study the asymptotic behavior of solutions and eigenelements to a 2-dimensional and 3-dimensional boundary value problem for the Laplace equation in a domain perforated along part of the boundary. On the boundary of holes we set the homogeneous Dirichlet boundary condition and the Steklov spectral condition on the mentioned part of the outer boundary of the domain. Assuming that the boundary microstructure is periodic, we construct the limit problem and prove the homogenization theorem.
Mathematics Subject Classification: 35B40 / 35D05 / 35G30 / 35Q35
Key words: Homogenization / the Steklov spectral problem / asymptotic methods
GAC was partially supported by RFBR, research project No. 12-01-00445. The work of RRG was performed as a basic part of the state program in the field of scientific activity of the Russian Ministry of Education and Science. The work of the author UDM was partially supported by F.A.R.O. (project 2012) “Metodi matematici per la modellizzazione di fenomeni naturali” University of Naples Federico II.
© EDP Sciences, SMAI 2017
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