Homogenization and asymptotics for a waveguide with an infinite number of closely located small windows

被引：0

作者：

Borisov D. ^{[1
]}

Bunoiu R. ^{[2
]}

Cardone G. ^{[3
]}

机构：

[1] Bashkir State Pedagogical University, Ufa 450000, 3a, October Revolution St

[2] LMAM, UMR 7122, Université Paul Verlaine Ile du Saulcy

[3] Department of Engineering, University of Sannio, Benevento 82100

来源：

Journal of Mathematical Sciences | 2011年 / 176卷 / 6期

基金：

俄罗斯基础研究基金会;

关键词：

Asymptotic Expansion; Dirichlet Boundary Condition; Essential Spectrum; Neumann Condition; Robin Boundary Condition;

D O I：

10.1007/s10958-011-0435-2

中图分类号：

学科分类号：

摘要：

We consider a planar waveguide modeled by the Laplacian in a straight infinite strip with the Dirichlet boundary condition on the upper boundary and with frequently alternating boundary conditions (Dirichlet and Neumann) on the lower boundary. The homogenized operator is the Laplacian subject to the Dirichlet boundary condition on the upper boundary and to the Dirichlet or Neumann condition on the lower one. We prove the uniform resolvent convergence for the perturbed operator in both cases and obtain the estimates for the rate of convergence. Moreover, we construct the leading terms of the asymptotic expansions for the first band functions and the complete asymptotic expansion for the bottom of the spectrum. Bibliography: 17 titles. Illustrations: 3 figures. © 2011 Springer Science+Business Media, Inc.

引用

页码：774 / 785

页数：11

共 17 条

[1]

Borisov D., Bunoiu R., Cardone G., On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition, Ann. Henri Poincaré, 11, 8, pp. 1591-1627, (2010)

[2]

Borisov D., Bunoiu R., Cardone G., On a waveguide with an infinite number of small windows, C. R., Math., Acad. Sci. Paris, 349, 1-2, pp. 53-56, (2011)

[3]

Borisov D., Cardone G., Homogenization of the planar waveguide with frequently alternating boundary conditions, J. Phys. A, Math. Theor, 42, (2009)

[4]

Borisov D., On a model boundary value problem for Laplacian with frequently alternating type of boundary condition, Asymptotic Anal., 35, 1, pp. 1-26, (2003)

[5]

Chechkin G.A., Averaging of boundary value problems with a singular perturbation of the boundary conditions [in Russian], Russ. Acad. Sci. Sb., Math., 79, 1, pp. 191-222, (1994)

[6]

Friedman A., Huang C., Yong J., Effective permeability of the boundary of a domain, Commun. Partial Differ. Equations, 20, 1-2, pp. 59-102, (1995)

[7]

Gadyl'shin R.R., Boundary value problem for the Laplacian with rapidly oscillating boundary conditions [in Russian], Dokl. Math., 58, 2, pp. 293-296, (1998)

[8]

Gadyl'shin R.R., Asymptotics of the eigenvalues of a boundary value problem with rapidly oscillating boundary conditions [in Russian], Differ. Equations, 35, 4, pp. 540-551, (1999)

[9]

Borisov D., The spectrum of two quantum layers coupled by a window, J. Phys. A, Math. Theor., 40, 19, pp. 5045-5066, (2007)

[10]

Bulla W., Gesztesy F., Renger W., Simon B., Weakly coupled bound states in quantum waveguides, Proc. Amer. Math. Soc., 12, pp. 1487-1495, (1997)

← 1 2 →