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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[6]	A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," North-Holland, Amsterdam, 1978.
[7]	Y. Capdeboscq, Homogenization of a diffusion equation with drift, C. R. Acad. Sci. Paris Série I Math., 327 (1998), 807-812.
[8]	G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Ann. Sci. École Norm. Sup. Sér. (2), 12 (1883), 47-89.
[9]	T. Kato, "Perturbation Theory for Linear Operators," Second edition, Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976.
[10]	S. Kozlov, Reducibility of quasiperiodic differential operators and averaging, Trudy Moskov. Mat. Obshch., 46 (1983), 99-123.
[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.
[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.
[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.
[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.
[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.
[16]	M. Vanninathan, Homogenization of eigenvalue problems in perforated domains, Proc. Indian Acad. Sci. Math. Sci., 90 (1981), 239-271. doi: 10.1007/BF02838079.

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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.
[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.
[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.
[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.
[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.
[6]	A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," North-Holland, Amsterdam, 1978.
[7]	Y. Capdeboscq, Homogenization of a diffusion equation with drift, C. R. Acad. Sci. Paris Série I Math., 327 (1998), 807-812.
[8]	G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Ann. Sci. École Norm. Sup. Sér. (2), 12 (1883), 47-89.
[9]	T. Kato, "Perturbation Theory for Linear Operators," Second edition, Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976.
[10]	S. Kozlov, Reducibility of quasiperiodic differential operators and averaging, Trudy Moskov. Mat. Obshch., 46 (1983), 99-123.
[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.
[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.
[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.
[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.
[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.
[16]	M. Vanninathan, Homogenization of eigenvalue problems in perforated domains, Proc. Indian Acad. Sci. Math. Sci., 90 (1981), 239-271. doi: 10.1007/BF02838079.

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