Geodesic
On the Steklov problem in a domain perforated along a part of the boundary
Chechkin, Gregory A.1 ; Gadyl’shin, Rustem R.2 ; D’Apice, Ciro3 ; De Maio, Umberto4 
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.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1317-1342

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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.

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DOI : 10.1051/m2an/2016063
Classification : 35B40, 35D05, 35G30, 35Q35
Keywords: Homogenization, the Steklov spectral problem, asymptotic methods

Chechkin, Gregory A. 1 ; Gadyl’shin, Rustem R. 2 ; D’Apice, Ciro 3 ; De Maio, Umberto 4

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. 
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     title = {On the {Steklov} problem in a domain perforated along a part of the boundary},
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Chechkin, Gregory A.; Gadyl’shin, Rustem R.; D’Apice, Ciro; De Maio, Umberto. On the Steklov problem in a domain perforated along a part of the boundary. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1317-1342. doi: 10.1051/m2an/2016063

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