American Institute of Mathematical Sciences

Advanced
January  2012, 17(1): 1-31. doi: 10.3934/dcdsb.2012.17.1

Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions

Grégoire Allaire 1, , Yves Capdeboscq 2, and  Marjolaine Puel 3,

1. 

CMAP, CNRS UMR 7641, École Polytechnique, Route de Saclay, Palaiseau F91128, France
2. 

Mathematical Institute, 24-29 St Giles’, OXFORD OX1 3LB, United Kingdom
3. 

Institut de Mathématiques, Université de Toulouse and CNRS, Université Paul Sabatier, 31062 Toulouse Cedex 9, France

Received  April 2011 Revised  July 2011 Published  October 2011

We study the asymptotic behavior of the first eigenvalue and eigenfunction of a one-dimensional periodic elliptic operator with Neumann boundary conditions. The second order elliptic equation is not self-adjoint and is singularly perturbed since, denoting by $\epsilon$ the period, each derivative is scaled by an $\epsilon$ factor. The main difficulty is that the domain size is not an integer multiple of the period. More precisely, for a domain of size $1$ and a given fractional part $0\leq\delta<1$, we consider a sequence of periods $\epsilon_n=1/(n+\delta)$ with $n\in \mathbb{N}$. In other words, the domain contains $n$ entire periodic cells and a fraction $\delta$ of a cell cut by the domain boundary. According to the value of the fractional part $\delta$, different asymptotic behaviors are possible: in some cases an homogenized limit is obtained, while in other cases the first eigenfunction is exponentially localized at one of the extreme points of the domain.
Keywords: Homogenization, localization., spectral problem.
Mathematics Subject Classification: Primary: 35B27; Secondary: 74Q0.
Citation: Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1
References:
[1]

G. Allaire and Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion, Comput. Methods Appl. Mech. Engrg., 187 (2000), 91-117. doi: 10.1016/S0045-7825(99)00112-7.  Google Scholar

[2]

G. Allaire and Y. Capdeboscq, Homogenization and localization for a 1-D eigenvalue problem in a periodic medium with an interface, Ann. Math. Pura Appl. (4), 181 (2002), 247-282.  Google Scholar

[3]

G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis, J. Math. Pures et Appli. (9), 77 (1998), 153-208. doi: 10.1016/S0021-7824(98)80068-8.  Google Scholar

[4]

G. Allaire and R. Orive, Homogenization of periodic non self-adjoint problems with large drift and potential, ESAIM COCV, 13 (2007), 735-749. doi: 10.1051/cocv:2007030.  Google Scholar

[5]

G. Allaire and A. Piatnistki, Uniform spectral asymptotics for singularly perturbed locally periodic operators, Comm. PDE, 27 (2002), 705-725. doi: 10.1081/PDE-120002871.  Google Scholar

[6]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," North-Holland, Amsterdam, 1978. Google Scholar

[7]

Y. Capdeboscq, Homogenization of a diffusion equation with drift, C. R. Acad. Sci. Paris Série I Math., 327 (1998), 807-812.  Google Scholar

[8]

G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Ann. Sci. École Norm. Sup. Sér. (2), 12 (1883), 47-89.  Google Scholar

[9]

T. Kato, "Perturbation Theory for Linear Operators," Second edition, Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[10]

S. Kozlov, Reducibility of quasiperiodic differential operators and averaging, Trudy Moskov. Mat. Obshch., 46 (1983), 99-123.  Google Scholar

[11]

W. Magnus and S. Winkler, "Hill's Equation," Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966.  Google Scholar

[12]

S. Moskow and M. Vogelius, First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1263-1299.  Google Scholar

[13]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, On the limiting behaviour of a sequence of operators defined in different Hilbert's spaces, Upsekhi Math. Nauk., 44 (1989), 157-158. Google Scholar

[14]

B. Perthame and P. Souganidis, Asymmetric potentials and motor effect: A homogenization approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2055-2071. doi: 10.1016/j.anihpc.2008.10.003.  Google Scholar

[15]

F. Santosa and M. Vogelius, First-order corrections to the homogenized eigenvalues of a periodic composite medium, SIAM J. Appl. Math., 53 (1993), 1636-1668. doi: 10.1137/0153076.  Google Scholar

[16]

M. Vanninathan, Homogenization of eigenvalue problems in perforated domains, Proc. Indian Acad. Sci. Math. Sci., 90 (1981), 239-271. doi: 10.1007/BF02838079.  Google Scholar

show all references

References:
[1]

G. Allaire and Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion, Comput. Methods Appl. Mech. Engrg., 187 (2000), 91-117. doi: 10.1016/S0045-7825(99)00112-7.  Google Scholar

[2]

G. Allaire and Y. Capdeboscq, Homogenization and localization for a 1-D eigenvalue problem in a periodic medium with an interface, Ann. Math. Pura Appl. (4), 181 (2002), 247-282.  Google Scholar

[3]

G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis, J. Math. Pures et Appli. (9), 77 (1998), 153-208. doi: 10.1016/S0021-7824(98)80068-8.  Google Scholar

[4]

G. Allaire and R. Orive, Homogenization of periodic non self-adjoint problems with large drift and potential, ESAIM COCV, 13 (2007), 735-749. doi: 10.1051/cocv:2007030.  Google Scholar

[5]

G. Allaire and A. Piatnistki, Uniform spectral asymptotics for singularly perturbed locally periodic operators, Comm. PDE, 27 (2002), 705-725. doi: 10.1081/PDE-120002871.  Google Scholar

[6]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," North-Holland, Amsterdam, 1978. Google Scholar

[7]

Y. Capdeboscq, Homogenization of a diffusion equation with drift, C. R. Acad. Sci. Paris Série I Math., 327 (1998), 807-812.  Google Scholar

[8]

G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Ann. Sci. École Norm. Sup. Sér. (2), 12 (1883), 47-89.  Google Scholar

[9]

T. Kato, "Perturbation Theory for Linear Operators," Second edition, Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[10]

S. Kozlov, Reducibility of quasiperiodic differential operators and averaging, Trudy Moskov. Mat. Obshch., 46 (1983), 99-123.  Google Scholar

[11]

W. Magnus and S. Winkler, "Hill's Equation," Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966.  Google Scholar

[12]

S. Moskow and M. Vogelius, First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1263-1299.  Google Scholar

[13]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, On the limiting behaviour of a sequence of operators defined in different Hilbert's spaces, Upsekhi Math. Nauk., 44 (1989), 157-158. Google Scholar

[14]

B. Perthame and P. Souganidis, Asymmetric potentials and motor effect: A homogenization approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2055-2071. doi: 10.1016/j.anihpc.2008.10.003.  Google Scholar

[15]

F. Santosa and M. Vogelius, First-order corrections to the homogenized eigenvalues of a periodic composite medium, SIAM J. Appl. Math., 53 (1993), 1636-1668. doi: 10.1137/0153076.  Google Scholar

[16]

M. Vanninathan, Homogenization of eigenvalue problems in perforated domains, Proc. Indian Acad. Sci. Math. Sci., 90 (1981), 239-271. doi: 10.1007/BF02838079.  Google Scholar

[1]

Eugenia Pérez. On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 859-883. doi: 10.3934/dcdsb.2007.7.859
[2]

Fanghua Lin, Xiaodong Yan. A type of homogenization problem. Discrete & Continuous Dynamical Systems, 2003, 9 (1) : 1-30. doi: 10.3934/dcds.2003.9.1
[3]

Renata Bunoiu, Claudia Timofte. Homogenization of a thermal problem with flux jump. Networks & Heterogeneous Media, 2016, 11 (4) : 545-562. doi: 10.3934/nhm.2016009
[4]

Sunghan Kim, Ki-Ahm Lee, Henrik Shahgholian. Homogenization of the boundary value for the Dirichlet problem. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 6843-6864. doi: 10.3934/dcds.2019234
[5]

Natalia O. Babych, Ilia V. Kamotski, Valery P. Smyshlyaev. Homogenization of spectral problems in bounded domains with doubly high contrasts. Networks & Heterogeneous Media, 2008, 3 (3) : 413-436. doi: 10.3934/nhm.2008.3.413
[6]

Ben Schweizer, Marco Veneroni. The needle problem approach to non-periodic homogenization. Networks & Heterogeneous Media, 2011, 6 (4) : 755-781. doi: 10.3934/nhm.2011.6.755
[7]

Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems & Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025
[8]

Germain Gendron. Uniqueness results in the inverse spectral Steklov problem. Inverse Problems & Imaging, 2020, 14 (4) : 631-664. doi: 10.3934/ipi.2020029
[9]

Leonid Berlyand, Petru Mironescu. Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain. Networks & Heterogeneous Media, 2008, 3 (3) : 461-487. doi: 10.3934/nhm.2008.3.461
[10]

T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks & Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675
[11]

Frédéric Legoll, William Minvielle. Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 1-27. doi: 10.3934/dcdss.2015.8.1
[12]

Grégoire Allaire, Alessandro Ferriero. Homogenization and long time asymptotic of a fluid-structure interaction problem. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 199-220. doi: 10.3934/dcdsb.2008.9.199
[13]

Aníbal Rodríguez-Bernal, Robert Willie. Singular large diffusivity and spatial homogenization in a non homogeneous linear parabolic problem. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 385-410. doi: 10.3934/dcdsb.2005.5.385
[14]

Grégoire Allaire, Zakaria Habibi. Second order corrector in the homogenization of a conductive-radiative heat transfer problem. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 1-36. doi: 10.3934/dcdsb.2013.18.1
[15]

Mamadou Sango. Homogenization of the Neumann problem for a quasilinear elliptic equation in a perforated domain. Networks & Heterogeneous Media, 2010, 5 (2) : 361-384. doi: 10.3934/nhm.2010.5.361
[16]

Arianna Giunti. Convergence rates for the homogenization of the Poisson problem in randomly perforated domains. Networks & Heterogeneous Media, 2021, 16 (3) : 341-375. doi: 10.3934/nhm.2021009
[17]

Weinan E, Weiguo Gao. Orbital minimization with localization. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 249-264. doi: 10.3934/dcds.2009.23.249
[18]

Radu Balan, Peter G. Casazza, Christopher Heil and Zeph Landau. Density, overcompleteness, and localization of frames. Electronic Research Announcements, 2006, 12: 71-86.

[19]

Luisa Berchialla, Luigi Galgani, Antonio Giorgilli. Localization of energy in FPU chains. Discrete & Continuous Dynamical Systems, 2004, 11 (4) : 855-866. doi: 10.3934/dcds.2004.11.855
[20]

Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences