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September  2014, 19(7): 2247-2265. doi: 10.3934/dcdsb.2014.19.2247

Identification problems related to cylindrical dielectrics **in presence of polarization**

Alfredo Lorenzi 1,

1. 

Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Saldini 50, 20133, Milano

Received  April 2013 Revised  August 2013 Published  August 2014

We consider the problem of recovering a polarization kernel in an axially inhomogeneous cylindrical dielectric, the polarization depending on time and the axial variable, but being constant on each cross section of the cylinder.
    For this problem, under some additional measurement, we prove an existence and uniqueness result.
Keywords: Maxwell equations, existence and uniqueness results., identification problems, polarization.
Mathematics Subject Classification: Primary: 30R35; Secondary: 35K20, 34G10, 45N05, 45Q0.
Citation: Alfredo Lorenzi. Identification problems related to cylindrical dielectrics **in presence of polarization**. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2247-2265. doi: 10.3934/dcdsb.2014.19.2247
References:
[1]

H. T. Banks and G. A. Pinter, Maxwell-systems with nonlinear polarization, Nonlinear Anal. Real World Appl., 4 (2003), 483-501. doi: 10.1016/S1468-1218(02)00074-3.  Google Scholar

[2]

H. T. Banks and N. L. Gibson, Well-posedness in Maxwell systems with distributions of polarization relaxation parameters, Appl. Math. Lett., 18 (2005), 423-430. doi: 10.1016/j.aml.2004.02.008.  Google Scholar

[3]

M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Methods Appl. Sci., 12 (1990), 365-368. doi: 10.1002/mma.1670120406.  Google Scholar

[4]

V. Girault and P. A. Raviart, Finite Element Approximation Of The Navier-Stokes Equations, Lecture notes in Mathematics vol. 749, Springer Verlag, Berlin 1979.  Google Scholar

[5]

F. Jochmann, Energy decay of solutions to Maxwell's equations with conductivity and polarization, J. Differential Equations, 203 (2004), 232-254. doi: 10.1016/j.jde.2004.05.005.  Google Scholar

[6]

F. Jochmann, Exponential decay of solutions of Maxwell's equations coupled with a first-order ordinary differential equation for the polarization, J. Math. Anal. Appl., 288 (2003), 411-423. doi: 10.1016/j.jmaa.2003.08.052.  Google Scholar

[7]

A. Lorenzi and V. I. Priimenko, Identification problems related to electro-magneto-elastic interactions, J. Inverse Ill Posed Probl., 4 (1996), 115-143. doi: 10.1515/jiip.1996.4.2.115.  Google Scholar

[8]

A. L. Nazarov and V. G. Romanov, A uniqueness theorem in the inverse problem for the integrodifferential electrodynamics equations, Sibirskii Zhurnal Industrial'noi Mathematiki, 15 (2012), 77-86 (in Russian); English translation in Journal of Applied and Industrial Mathematics, 6 (2012), 460-468. doi: 10.1134/S1990478912040072.  Google Scholar

[9]

V. G. Romanov, Problem of kernel recovering for the viscoelasticity equation, Doklady Akademii Nauk, 446 (2012), 18-20; English transl. in Doklady Mathematics, 86 (2012), 608-610. doi: 10.1134/S1064562412050067.  Google Scholar

[10]

D. Sheen, A generalized Green's theorem, Appl. Math. Lett., 5 (1992), 95-98. doi: 10.1016/0893-9659(92)90096-R.  Google Scholar

show all references

References:
[1]

H. T. Banks and G. A. Pinter, Maxwell-systems with nonlinear polarization, Nonlinear Anal. Real World Appl., 4 (2003), 483-501. doi: 10.1016/S1468-1218(02)00074-3.  Google Scholar

[2]

H. T. Banks and N. L. Gibson, Well-posedness in Maxwell systems with distributions of polarization relaxation parameters, Appl. Math. Lett., 18 (2005), 423-430. doi: 10.1016/j.aml.2004.02.008.  Google Scholar

[3]

M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Methods Appl. Sci., 12 (1990), 365-368. doi: 10.1002/mma.1670120406.  Google Scholar

[4]

V. Girault and P. A. Raviart, Finite Element Approximation Of The Navier-Stokes Equations, Lecture notes in Mathematics vol. 749, Springer Verlag, Berlin 1979.  Google Scholar

[5]

F. Jochmann, Energy decay of solutions to Maxwell's equations with conductivity and polarization, J. Differential Equations, 203 (2004), 232-254. doi: 10.1016/j.jde.2004.05.005.  Google Scholar

[6]

F. Jochmann, Exponential decay of solutions of Maxwell's equations coupled with a first-order ordinary differential equation for the polarization, J. Math. Anal. Appl., 288 (2003), 411-423. doi: 10.1016/j.jmaa.2003.08.052.  Google Scholar

[7]

A. Lorenzi and V. I. Priimenko, Identification problems related to electro-magneto-elastic interactions, J. Inverse Ill Posed Probl., 4 (1996), 115-143. doi: 10.1515/jiip.1996.4.2.115.  Google Scholar

[8]

A. L. Nazarov and V. G. Romanov, A uniqueness theorem in the inverse problem for the integrodifferential electrodynamics equations, Sibirskii Zhurnal Industrial'noi Mathematiki, 15 (2012), 77-86 (in Russian); English translation in Journal of Applied and Industrial Mathematics, 6 (2012), 460-468. doi: 10.1134/S1990478912040072.  Google Scholar

[9]

V. G. Romanov, Problem of kernel recovering for the viscoelasticity equation, Doklady Akademii Nauk, 446 (2012), 18-20; English transl. in Doklady Mathematics, 86 (2012), 608-610. doi: 10.1134/S1064562412050067.  Google Scholar

[10]

D. Sheen, A generalized Green's theorem, Appl. Math. Lett., 5 (1992), 95-98. doi: 10.1016/0893-9659(92)90096-R.  Google Scholar

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