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September  2013, 12(5): 2229-2266. doi: 10.3934/cpaa.2013.12.2229

Energy decay for Maxwell's equations with Ohm's law in partially cubic domains

Kim Dang Phung 1,

1. 

Université d'Orléans, Laboratoire MAPMO, CNRS UMR 7349, Fédération Denis Poisson, FR CNRS 2964, Bâtiment de Mathématiques, B.P. 6759, 45067 Orléans Cedex 2, France

Received  January 2012 Revised  June 2012 Published  January 2013

We prove a polynomial energy decay for the Maxwell's equations with Ohm's law in partially cubic domains with trapped rays. We extend the results of polynomial decay for the scalar damped wave equation in partially rectangular or cubic domain. Our approach have some similitude with the construction of reflected gaussian beams.
Keywords: Maxwell's equations, trapped ray., decay estimates.
Mathematics Subject Classification: Primary: 35B40, 35Q61; Secondary: 93D1.
Citation: Kim Dang Phung. Energy decay for Maxwell's equations with Ohm's law in partially cubic domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2229-2266. doi: 10.3934/cpaa.2013.12.2229
References:
[1]

C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains,, Math. Methods Appl. Sci., 21 (1998), 823.   Google Scholar

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C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024.  doi: 10.1137/0330055.  Google Scholar

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P. Boissoles, "Problèmes mathématiques et numériques issus de l'imagerie par résonance magnétique nucléaire,", Ph.D thesis, (2005).   Google Scholar

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N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains,, Math. Res. Lett., 14 (2007), 35.   Google Scholar

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M. Cessenat, "Mathematical Method in Electromagnetism, Linear Theory and Applications,", World Scientific, (1996).   Google Scholar

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R. Dautray and J.-L. Lions, "Analyse mathématique et calcul numérique pour les sciences et les techniques, Volume 5, Spectre des opérateurs,", Masson, (1988).   Google Scholar

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G. Duvaut and J.-L. Lions, "Les inéquations en mécanique et en physique,", Dunod, (1972).   Google Scholar

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S. S. Krigman and C. E. Wayne, Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, I: Spectral controllability,, J. Math. Anal. Appl., 329 (2007), 1375.  doi: 10.1016/j.jmaa.2006.06.101.  Google Scholar

[9]

J.-L. Lions, "Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués I,", Masson, (1988).   Google Scholar

[10]

H. Nishiyama, Polynomial decay for damped wave equations on partially rectangular domains,, Math. Res. Lett., 16 (2009), 881.   Google Scholar

[11]

K. D. Phung, Contrôle et stabilisation d'ondes électromagnétiques,, ESAIM Control Optim. Calc. Var., 5 (2000), 87.  doi: 10.1051/cocv:2000103.  Google Scholar

[12]

K. D. Phung, Polynomial decay rate for the dissipative wave equation,, J. Diff. Eq., 240 (2007), 92.  doi: 10.1016/j.jde.2007.05.016.  Google Scholar

[13]

J. Ralston, Gaussian beams and propagation of singularities,, in, 23 (1982), 206.   Google Scholar

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W. Wei, H-M. Yin and J. Tang, An optimal control problem for microwave heating,, Nonlinear Analysis, 75 (2012), 2024.  doi: 10.1016/j.na.2011.10.003.  Google Scholar

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R. Ziolkowski, Exact solutions of the wave equation with complex source locations,, J. Math. Phys., 26 (1985), 861.  doi: 10.1063/1.526579.  Google Scholar

show all references

References:
[1]

C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains,, Math. Methods Appl. Sci., 21 (1998), 823.   Google Scholar

[2]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024.  doi: 10.1137/0330055.  Google Scholar

[3]

P. Boissoles, "Problèmes mathématiques et numériques issus de l'imagerie par résonance magnétique nucléaire,", Ph.D thesis, (2005).   Google Scholar

[4]

N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains,, Math. Res. Lett., 14 (2007), 35.   Google Scholar

[5]

M. Cessenat, "Mathematical Method in Electromagnetism, Linear Theory and Applications,", World Scientific, (1996).   Google Scholar

[6]

R. Dautray and J.-L. Lions, "Analyse mathématique et calcul numérique pour les sciences et les techniques, Volume 5, Spectre des opérateurs,", Masson, (1988).   Google Scholar

[7]

G. Duvaut and J.-L. Lions, "Les inéquations en mécanique et en physique,", Dunod, (1972).   Google Scholar

[8]

S. S. Krigman and C. E. Wayne, Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, I: Spectral controllability,, J. Math. Anal. Appl., 329 (2007), 1375.  doi: 10.1016/j.jmaa.2006.06.101.  Google Scholar

[9]

J.-L. Lions, "Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués I,", Masson, (1988).   Google Scholar

[10]

H. Nishiyama, Polynomial decay for damped wave equations on partially rectangular domains,, Math. Res. Lett., 16 (2009), 881.   Google Scholar

[11]

K. D. Phung, Contrôle et stabilisation d'ondes électromagnétiques,, ESAIM Control Optim. Calc. Var., 5 (2000), 87.  doi: 10.1051/cocv:2000103.  Google Scholar

[12]

K. D. Phung, Polynomial decay rate for the dissipative wave equation,, J. Diff. Eq., 240 (2007), 92.  doi: 10.1016/j.jde.2007.05.016.  Google Scholar

[13]

J. Ralston, Gaussian beams and propagation of singularities,, in, 23 (1982), 206.   Google Scholar

[14]

W. Wei, H-M. Yin and J. Tang, An optimal control problem for microwave heating,, Nonlinear Analysis, 75 (2012), 2024.  doi: 10.1016/j.na.2011.10.003.  Google Scholar

[15]

R. Ziolkowski, Exact solutions of the wave equation with complex source locations,, J. Math. Phys., 26 (1985), 861.  doi: 10.1063/1.526579.  Google Scholar

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