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