We study the asymptotic behavior of solutions and eigenelements to a 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.
Nous étudions le comportement asymptotique des solutions et des éléments propres à un problème aux limites pour l'équation de Laplace dans un domaine perforé le long d'une partie de la frontière. Sur la frontière de trous, nous posons la condition de Dirichlet homogène et la condition spectrale de Steklov sur la part mentionnée de la frontière extérieure du domaine. En supposant que la microstructure de la frontière est périodique, nous construisons le problème aux limites et prouvons le théorème d'homogénéisation.
Accepted:
Published online:
Mots-clés : Homogénéisation, Problème spectral de Steklov, Méthodes asymptotiques
Gregory A. Chechkin 1; Ciro D' Apice 2; Umberto De Maio 3; Rustem R. Gadyl'shin 4, 5
@article{CRMECA_2016__344_1_12_0,
author = {Gregory A. Chechkin and Ciro D' Apice and Umberto De Maio and Rustem R. Gadyl'shin},
title = {On a singularly perturbed {Steklov} problem in a domain perforated along the boundary},
journal = {Comptes Rendus. M\'ecanique},
pages = {12--18},
publisher = {Elsevier},
volume = {344},
number = {1},
year = {2016},
doi = {10.1016/j.crme.2015.09.001},
language = {en},
}
TY - JOUR AU - Gregory A. Chechkin AU - Ciro D' Apice AU - Umberto De Maio AU - Rustem R. Gadyl'shin TI - On a singularly perturbed Steklov problem in a domain perforated along the boundary JO - Comptes Rendus. Mécanique PY - 2016 SP - 12 EP - 18 VL - 344 IS - 1 PB - Elsevier DO - 10.1016/j.crme.2015.09.001 LA - en ID - CRMECA_2016__344_1_12_0 ER -
%0 Journal Article %A Gregory A. Chechkin %A Ciro D' Apice %A Umberto De Maio %A Rustem R. Gadyl'shin %T On a singularly perturbed Steklov problem in a domain perforated along the boundary %J Comptes Rendus. Mécanique %D 2016 %P 12-18 %V 344 %N 1 %I Elsevier %R 10.1016/j.crme.2015.09.001 %G en %F CRMECA_2016__344_1_12_0
Gregory A. Chechkin; Ciro D' Apice; Umberto De Maio; Rustem R. Gadyl'shin. On a singularly perturbed Steklov problem in a domain perforated along the boundary. Comptes Rendus. Mécanique, Volume 344 (2016) no. 1, pp. 12-18. doi : 10.1016/j.crme.2015.09.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2015.09.001/
[1] Sur les problèmes fondamentaux de la physique mathématique, Ann. Sci. Éc. Norm. Super., Volume 19 (1902) no. 3, pp. 191-259 455–490 (in French)
[2] On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem, Discrete Contin. Dyn. Syst., Ser. B, Volume 7 (2007) no. 4, pp. 859-883
[3] On the spectrum of the Steklov problem in a domain with a peak, Vestn. St. Petersbg. Univ., Ser. 1, Volume 1 (2008), pp. 56-65 (English translation: Vestn. St. Petersbg. Univ., Math. 41 (1) (2008) 45–52)
[4] Convergence of solutions and eigenelements of Steklov type boundary value problems with boundary conditions of rapidly varying type, Probl. Mat. Anal., Volume 42 (2009), pp. 129-143 (English translation: J. Math. Sci. (N.Y.) 162 (3) (2009) 443–458)
[5] On estimate of eigenfunctions to the Steklov-type problem with small parameter in case of limit spectrum degeneration, Ufim. Mat. Zh., Volume 3 (2011) no. 3, pp. 127-139 (English translation: Ufa Math. J. 3 (3) (2011) 122–134)
[6] Boundary value problems in domains containing perforated walls, Paris ( 1980–1981 ), pp. 309-325
[7] Homogenization of mixed boundary value problem for the Poisson equation in a domain perforated along the boundary, Usp. Mat. Nauk, Volume 45 (1990) no. 4, p. 123 (in Russian)
[8] On homogenization of solutions of boundary value problems in domains perforated along manifolds, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 25 (1997) no. 3–4, pp. 611-629
[9] Homogenization in domains randomly perforated along the boundary, Discrete Contin. Dyn. Syst., Ser. B, Volume 12 (2009) no. 4, pp. 713-730
[10] On the Friedrichs inequality in a domain perforated along the boundary. Homogenization procedure. Asymptotics in parabolic problems, Russ. J. Math. Phys., Volume 16 (2009) no. 1, pp. 1-16
[11] On the convergence of solutions and eigenelements of a boundary value problem in a domain perforated along the boundary, Differ. Uravn., Volume 46 (2010) no. 5, pp. 665-677 (English translation: Differ. Equ. 46 (5) (2010) 667–680)
[12] A new weighted Friedrichs-type inequality for a perforated domain with a sharp constant, Eurasian Math. J., Volume 2 (2011) no. 1, pp. 81-103
[13] On the asymptotic behavior of a simple eigenvalue of a boundary value problem in a domain perforated along the boundary, Differ. Uravn., Volume 47 (2011) no. 6, pp. 819-828 (English translation: Differ. Equ. 47 (6) (2011) 822–831)
[14] On the spectral problem in a domain perforated along the boundary. Perturbation of multiple eigenvalue, Probl. Mat. Anal., Volume 73 (2013), pp. 31-45 (English translation: J. Math. Sci. (N.Y.) 196 (3) (2014) 276–292)
[15] Homogenization of the boundary value problem in a domain perforated along a part of the boundary, Probl. Mat. Anal., Volume 75 (2014), pp. 41-60 (English translation: J. Math. Sci. (N.Y.) 198 (6) (2014) 701–723)
[16] Partial Differential Equations, Mir, Moscow, 1978
[17] Equazioni della Fisica Matematica, Mir, Moscow, 1987 (in Italian), translated from the fourth Russian edition by Ernest Kozlov
[18] Perturbation Theory for Linear Operators, Springer, Heidelberg, Berlin, 1966
[19] Spectral Theory of Selfadjoint Operators in Hilbert Space, Mathematics and Its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, The Netherlands, 1987 (translated from Russian)
Cited by Sources:
Comments - Policy