# American Institute of Mathematical Sciences

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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**

 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.
Citation: Alfredo Lorenzi. Identification problems related to cylindrical dielectrics **in presence of polarization**. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [10] D. Sheen, A generalized Green's theorem, Appl. Math. Lett., 5 (1992), 95-98. doi: 10.1016/0893-9659(92)90096-R.

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