Well-posedness of the Drude-Born-Fedorov model for chiral media. (English) Zbl 1126.78002
Math. Models Methods Appl. Sci. 17, No. 3, 461-484 (2007); erratum ibid. 19, No. 1, 173-174 (2009).
Summary: We consider a chiral medium in a bounded domain, enclosed in a perfectly conducting material. We solve the transient Maxwell equations in this domain, when the medium is modeled by the Drude-Born-Fedorov constitutive equations. The input data is located on the boundary, in the form of given surface current and surface charge densities. It is proved that, except for a countable set of chirality admittance values, the problem is mathematically well-posed. This result holds for domains with non-smooth boundaries.
MSC:
| 78A25 | Electromagnetic theory (general) |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
References:
| [1] | Ammari, H.Bao, G., Math. Nachr.251, 3 (2002), DOI: 10.1002/mana.200310026. |
| [2] | Ammari, H.Laouadi, M.Nédélec, J.-C., SIAM J. Appl. Math.58, 1022 (1998), DOI: 10.1137/S0036139996310431. · Zbl 0959.78007 |
| [3] | Ammari, H.Nédélec, J.-C., Math. Meth. Appl. Sci.21, 327 (1998), DOI: 10.1002/(SICI)1099-1476(19980310)21:4<327::AID-MMA952>3.0.CO;2-6. |
| [4] | Ammari, H.Nédélec, J. C., SIAM J. Math. Anal.29, 395 (1998), DOI: 10.1137/S0036141096305504. |
| [5] | Amrouche, C.et al., Math. Meth. Appl. Sci.21, 823 (1998), DOI: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B. |
| [6] | Athanasiadis, C.Costakis, G.Stratis, I. G., IMA J. Appl. Math.64, 245 (2000), DOI: 10.1093/imamat/64.3.245. · Zbl 1064.78006 |
| [7] | Athanasiadis, C.Costakis, G.Stratis, I. G., Math. Meth. Appl. Sci.25, 927 (2002), DOI: 10.1002/mma.321. · Zbl 0998.78007 |
| [8] | Athanasiadis, C.Martin, P. A.Stratis, I. G., SIAM J. Appl. Math.59, 1745 (1999), DOI: 10.1137/S003613999833633X. |
| [9] | Athanasiadis, C.Roach, G. F.Stratis, I. G., Math. Nachr.250, 3 (2003), DOI: 10.1002/mana.200310018. |
| [10] | Bernardi, C.Dauge, M.Maday, Y., C. R. Acad. Sci. Paris, Ser. I331, 679 (2000). · Zbl 0976.52008 |
| [11] | Sh. Birman, M.Solomyak, M. Z., Siberian Math. J.28, 12 (1987), DOI: 10.1007/BF00970204. |
| [12] | Bonnet-Ben Dhia, A.-S.Hazard, C.Lohrengel, S., SIAM J. Appl. Math.59, 2028 (1999), DOI: 10.1137/S0036139997323383. |
| [13] | Boulmezaoud, T.-Z.Amari, T., Z. Angew. Math. Phys.51, 942 (2000), DOI: 10.1007/PL00001531. |
| [14] | Boulmezaoud, T.-Z.Maday, Y.Amari, T., ESAIM Math. Model. Numer. Anal.33, 359 (1999), DOI: 10.1051/m2an:1999121. |
| [15] | Buffa, A.Ciarlet, P., Math. Meth. Appl. Sci.24, 9 (2001), DOI: 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2. |
| [16] | Buffa, A.Ciarlet, P., Math. Meth. Appl. Sci.24, 31 (2001), DOI: 10.1002/1099-1476(20010110)24:1<31::AID-MMA193>3.0.CO;2-X. |
| [17] | Buffa, A.Costabel, M.Sheen, D., J. Math. Anal. Appl.276, 845 (2002), DOI: 10.1016/S0022-247X(02)00455-9. |
| [18] | Costabel, M., Math. Meth. Appl. Sci.12, 365 (1990), DOI: 10.1002/mma.1670120406. |
| [19] | Courilleau, P.Horsin Molinaro, T., C. R. Acad. Sci. Paris, Ser. I341, 665 (2005). · Zbl 1146.93310 |
| [20] | Fernandes, P.Gilardi, G., Math. Mod. Meth. Appl. Sci.7, 957 (1997), DOI: 10.1142/S0218202597000487. |
| [21] | Frantzeskakis, D. J.et al., Appl. Anal.82, 839 (2003), DOI: 10.1080/0003681031000151443. |
| [22] | Friedrichs, K. O., Comm. Pure Appl. Math.8, 551 (1955), DOI: 10.1002/cpa.3160080408. |
| [23] | Girault, V.; Raviart, P.-A., Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, 5, 1986, Springer-Verlag · Zbl 0585.65077 |
| [24] | Grisvard, P., Elliptic Problems in Nonsmooth Domains, 1985, Pitman · Zbl 0695.35060 |
| [25] | Jackson, J. D., Classical Electrodynamics, 1975, John Wiley & Sons · Zbl 0997.78500 |
| [26] | Lagnese, J. E., SIAM J. Control Optim.27, 374 (1989), DOI: 10.1137/0327019. |
| [27] | Lakhtakia, A., Beltrami Fields in Chiral Media, 2, 1994, World Scientific |
| [28] | Lions, J.-L., Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués; tome 1 Contrôlabilité Exacte, 8, 1988, Masson · Zbl 0653.93002 |
| [29] | Nicaise, S., SIAM J. Control Optim.38, 1145 (2000), DOI: 10.1137/S0363012998344373. |
| [30] | Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, 44, 1983, Springer-Verlag · Zbl 0516.47023 |
| [31] | Stratis, I. G.Yannacopoulos, A. N., Abstr. Appl. Anal.2004, 471 (2004), DOI: 10.1155/S1085337504306287. |
| [32] | Weber, C., Math. Meth. Appl. Sci.2, 12 (1980), DOI: 10.1002/mma.1670020103. |
| [33] | Yoshida, Z.Giga, Y., Math. Z.204, 235 (1990), DOI: 10.1007/BF02570870. |
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.
