×

Stability properties for a problem of light scattering in a dispersive metallic domain. (English) Zbl 1504.35530

Summary: In this work, we study the well-posedness and some stability properties of a PDE system that models the propagation of light in a metallic domain with a hole. This model takes into account the dispersive properties of the metal. It consists of a linear coupling between Maxwell’s equations and a wave type system. We prove that the problem is well posed for several types of boundary conditions. Furthermore, we show that it is polynomially stable and that the exponential stability is conditional on the exponential stability of the Maxwell system.

MSC:

35Q61 Maxwell equations
35Q60 PDEs in connection with optics and electromagnetic theory
35B35 Stability in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
78A45 Diffraction, scattering
78A40 Waves and radiation in optics and electromagnetic theory
74J20 Wave scattering in solid mechanics
74F15 Electromagnetic effects in solid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] L. · Zbl 1326.35037 · doi:10.1016/j.jde.2015.03.018
[2] \(C.%%C.%\) · Zbl 0914.35094 · doi:10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
[3] H. Barucq and B. Hanouzet, Étude asymptotique du système de Maxwell avec la condition aux limites absorbante de Silver-Müller II, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 1019-1024. · Zbl 0776.35073
[4] A. · Zbl 1118.47034 · doi:10.1002/mana.200410429
[5] C. J. K. · Zbl 1185.47043 · doi:10.1007/s00028-008-0424-1
[6] A. Boardman, Electromagnetic Surface Modes, John Wiley & Sons, 1972.
[7] A. · Zbl 1185.47044 · doi:10.1007/s00208-009-0439-0
[8] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. · Zbl 1220.46002
[9] A. · Zbl 1106.35304 · doi:10.1016/S0022-247X(02)00455-9
[10] C. Carle, Numerische Verfahren für Plasmonische Nanostrukturen (in German), Master’s thesis, Karlsruher Institut für Technologie, 2017.
[11] M. Costabel, A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains, Math. Methods Appl. Sci., 12, 365-368 (1990) · Zbl 0699.35028 · doi:10.1002/mma.1670120406
[12] M. Costabel, M. Dauge and S. Nicaise, Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth domains, 2010.
[13] M. Daoulatli, Energy decay rates for solutions of the wave equation with linear damping in exterior domain, Evol. Equ. Control Theory, 5, 37-59 (2016) · Zbl 1353.35058 · doi:10.3934/eect.2016.5.37
[14] M. Eller, J. E. Lagnese and S. Nicaise, Stabilization of heterogeneous Maxwell’s equations by linear or nonlinear boundary feedback, Electron. J. Differential Equations, (2002), No. 21, 26 pp. · Zbl 1030.93026
[15] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, vol. 5 of Springer Series in Computational Mathematics, Springer, Berlin, 1986. · Zbl 0585.65077
[16] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1, 43-56 (1985) · Zbl 0593.34048
[17] \(Y.%%J.%\) · Zbl 1359.78016 · doi:10.1016/j.camwa.2016.06.003
[18] Z. · Zbl 1100.47036 · doi:10.1007/s00033-004-3073-4
[19] S. Nicaise, Stabilization of a Drude/vacuum model, Z. Anal. Anwend., 37, 349-375 (2018) · Zbl 1398.35226 · doi:10.4171/ZAA/1618
[20] \(S.%%C. Nicaise%\) · Zbl 1445.78012 · doi:10.1016/j.camwa.2020.02.006
[21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Math. Sciences, Springer-Verlag, New York, 1983. · Zbl 0516.47023
[22] K. D. Phung, Contrôle et stabilisation d’ondes électromagnétiques, ESAIM Control Optim. Calc. Var., 5, 87-137 (2000) · Zbl 0942.93002 · doi:10.1051/cocv:2000103
[23] A. · doi:10.1364/JOSAB.36.002989
[24] J. Prüss, On the spectrum of \(C_0\)-semigroups, Trans. Amer. Math. Soc., 284, 847-857 (1984) · Zbl 0572.47030 · doi:10.2307/1999112
[25] N. Schmitt, High-Order Simulation and Calibration Strategies for Spatially Dispersive Metals in Nanophotonics, PhD thesis, Côte d’Azur University, 2018.
[26] Ch. Weber, A local compactness theorem for Maxwell’s equations, Math. Meth. Appl. Sci., 2, 12-25 (1980) · Zbl 0432.35032 · doi:10.1002/mma.1670020103
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.