Abstract
We consider a waveguide modeled by the Laplacian in a straight planar strip. The Dirichlet boundary condition is taken on the upper boundary, while on the lower boundary we impose periodically alternating Dirichlet and Neumann condition assuming the period of alternation to be small. We study the case when the homogenization gives the Neumann condition instead of the alternating ones. We establish the uniform resolvent convergence and the estimates for the rate of convergence. It is shown that the rate of the convergence can be improved by employing a special boundary corrector. Other results are the uniform resolvent convergence for the operator on the cell of periodicity obtained by the Floquet–Bloch decomposition, the two terms asymptotics for the band functions, and the complete asymptotic expansion for the bottom of the spectrum with an exponentially small error term.
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Birman M.S., Suslina T.A.: Homogenization with corrector term for periodic elliptic differential operators. St. Petersburg Math. J. 17, 897–973 (2006)
Birman T.A., Suslina M.S.: Homogenization with corrector for periodic differential operators. Approximation of solutions in the Sobolev class \({H^1{\mathbb{R}}^d}\) . St. Petersburg Math. J. 18, 857–955 (2007)
Borisov, D., Cardone, G.: Homogenization of the planar waveguide with frequently alternating boundary conditions. J. Phys. A. 42, id365205 (21 pp) (2009)
Borisov D.I.: On a model boundary value problem for Laplacian with frequently alternating type of boundary condition. Asymptotic Anal. 35, 1–26 (2003)
Borisov D., Krejčiřík D.: \({\mathcal{PT}}\)-symmetric waveguide. Integr. Equat. Oper. Theor. 62, 489–515 (2008)
Borisov D.: Asymptotics for the solutions of elliptic systems with rapidly oscillating coefficients. St. Petersburg Math. J. 20, 175–191 (2009)
Borisov D.: On the spectrum of two quantum layers coupled by a window. J. Phys. A 40, 5045–5066 (2007)
Borisov D.: Discrete spectrum of a pair of non-symmetric waveguides coupled by a window. Sb. Math. 197, 475–504 (2006)
Borisov D., Exner P., Gadyl’shin R.: J. Math. Phys. 43, 6265–6278 (2002)
Borisov D.I.: Asymptotics and estimates for the eigenelements of the Laplacian with frequently alternating nonperiodic boundary conditions. Izv. Math. 67, 1101–1148 (2003)
Borisov, D., Cardone, G.: Complete asymptotic expansions for eigenvalues of Dirichlet Laplacian in thin three-dimensional rods. ESAIM: Contr. Op. Ca. Va. (to appear)
Bulla W., Gesztesy F., Renger W., Simon B.: Proc. Am. Math. Soc. 12, 1487–1495 (1997)
Chechkin G.A.: Averaging of boundary value problems with singular perturbation of the boundary conditions. Russ. Acad. Sci. Sb. Math. 79, 191–220 (1994)
Chechkin, G.A.: On boundary value problems for a second order elliptic equation with oscillating boundary conditions in book. In: Nonclassical Partial Differential Equations, Collection of Science Works, Novosibirsk, pp. 95–104. (1988) (in Russian)
Dittrich J., Kříž J.: Bound states in straight quantum waveguide with combined boundary condition. J. Math. Phys. 43, 3892–3915 (2002)
Doronina E.I., Chechkin G.A.: On the asymptotics of the spectrum of a boundary value problem with nonperiodic rapidly alternating boundary conditions. Funct. Differ. Eqs. 8, 111–122 (2001)
Doronina E.I., Chechkin G.A.: On the averaging of solutions of a second order elliptic equation with nonperiodic rapidly changing boundary conditions. Mosc. Univ. Math. Bull. 56, 14–19 (2001)
Exner P., Šeba P., Tater M., Vaněk D.: Bound states and scattering in quantum waveguides coupled laterally through a boundary window. J. Math. Phys. 37, 4867–4887 (1996)
Exner P., Vugalter S.: Bound-state asymptotic estimate for window-coupled Dirichlet strips and layers. J. Phys. A. Math. Gen. 30, 7863–7878 (1997)
Friedman A., Huang Ch., Yong J.: Effective permeability of the boundary of a domain. Commun. Part. Diff. Eq. 20, 59–102 (1995)
Friedlander L., Solomyak M.: On the spectrum of the Dirichlet Laplacian in a narrow strip. Isr. J. Math. 170, 337–354 (2009)
Gadylshin R.R.: Ramification of a multiple eigenvalue of the Dirichlet problem for the Laplacian under singular perturbation of the boundary condition. Math. Notes. 52, 1020–1029 (1992)
Gadyl’shin R.R.: Boundary value problem for the Laplacian with rapidly oscillating boundary conditions. Dokl. Math. 58, 293–296 (1998)
Gadyl’shin R.R.: Homogenization and asymptotics for a membrane with closely spaced clamping points. Comp. Math. Math. Phys. 41, 1765–1776 (2001)
Gadylshin R.R.: Asymptotics of the eigenvalues of a boundary value problem with rapidly oscillating boundary conditions. Diff. Equ. 35, 540–551 (1999)
Gadyl’shin R.R.: On regular and singular perturbation of acoustic and quantum waveguides. 332, 647–652 (2004)
Il’in, A.M.: Matching of asymptotic expansions of solutions of boundary value problems. Translations of Mathematical Monographs, vol. 102, American Mathematical Society, Providence (1992)
Oleinik O.A., Sanchez-Hubert J., Yosifian G.A.: On vibrations of a membrane with concentrated masses. Bull. Sci. Math. Ser. 115, 1–27 (1991)
Olejnik, O.A., Shamaev, A.S., Yosifyan, G.A.: Mathematical problems in elasticity and homogenization. In: Studies in Mathematics and its Applications, vol. 26. North-Holland, Amsterdam (1992)
Pastukhova S.E., Tikhomirov R.N.: Operator estimates in reiterated and locally periodic homogenization. Dokl. Math. 76, 548–553 (2007)
Pastukhova S.E.: Some estimates from homogenized elasticity problems. Dokl. Math. 73, 102–106 (2006)
Reed M., Simon B.: Methods of modern mathematical physics I: functional analysis. Academic Press, New York (1980)
Reed M., Simon B.: Methods of modern mathematical physics IV: analysis of operators. Academic Press, New York (1978)
Suslina T.A.: Homogenization with corrector for a stationary periodic Maxwell system. St. Petersburg Math. J. 19, 455–494 (2008)
Suslina T.A.: Homogenization in Sobolev class \({H^1(\mathbb R^d)}\) for periodic elliptic second order differential operators including first order terms. Alg. Anal. 22, 108–222 (2010) (in Russian)
Suslina T.A., Kharin A.A.: Homogenization with corrector for a periodic elliptic operator near an edge of inner gap. J. Math. Sci. 159, 264–280 (2009)
Vishik M.I., Lyusternik L.A.: Regular degeneration and boundary layer for linear differential equations with small parameter. Transl. Ser. 2 Am. Math. Soc. 20, 239–364 (1962)
Zhikov V.V.: On operator estimates in homogenization theory. Dokl. Math. 72, 534–538 (2005)
Zhikov V.V.: Some estimates from homogenization theory. Dokl. Math. 73, 96–99 (2006)
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Communicated by Jan Derezinski.
This work was partially done during the visit of D.B. to the University of Sannio (Italy) and of G.C. to LMAM of University Paul Verlaine of Metz (France). They are grateful for the warm hospitality extended to them. D.B. was partially supported by RFBR (09-01-00530), by the Grants of the President of Russia for young scientists–doctors of sciences (MD-453.2010.1) and for Leading Scientific School (NSh-6249.2010.1), by Federal Task Program “Research and educational professional community of innovation Russia” (contract 02.740.11.0612), FCT (ptdc/mat/101007/2008), and by the project “Progetto ISA: Attività di Internazionalizzazione dell’Università degli Studi del Sannio”.
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Borisov, D., Bunoiu, R. & Cardone, G. On a Waveguide with Frequently Alternating Boundary Conditions: Homogenized Neumann Condition. Ann. Henri Poincaré 11, 1591–1627 (2010). https://doi.org/10.1007/s00023-010-0065-0
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DOI: https://doi.org/10.1007/s00023-010-0065-0