The Maxwell system in nonhomogeneous anisotropic waveguides with slowly stabilizing characteristics of filling medium

B. A. Plamenevskii ; A. S. Poretskii

Algebra i analiz, Tome 34 (2022) no. 4, pp. 107-187

Résumé

In a region with a finite number of cylindrical exits to infinity, a stationary Maxwell system with ideally conducting boundary conditions is investigated. It is assumed that the dielectric and magnetic permittivity are arbitrary positive definite matrix functions with slow stabilization at infinity. A scattering matrix is introduced, a unique solvability of the problem with radiation conditions at infinity is established, and the asymptotics of solutions are described.

Détail

Keywords: waveguide, scattering matrix, asymptotics of a solution
Mots-clés : radiation conditions, elliptic extension.

Bibliographie

[1] Plamenevskii B. A., Poretskii A. S., “Sistema Maksvella v volnovodakh s neskolkimi tsilindricheskimi vykhodami na beskonechnost i neodnorodnym anizotropnym zapolneniem”, Algebra i analiz, 29:2 (2017), 89–126

[2] Ilinskii A. S., Kravtsov V. V., Sveshnikov A. G., Matematicheskie modeli elektrodinamiki, Uch. posob., Vyssh. shk., M., 1991

[3] Galishnikova T. N., Ilinskii A. S., Metod integralnykh uravnenii v zadachakh difraktsii voln, MAKS Press, M., 2013

[4] Bogolyubov A. N., Delitsyn A. L., Sveshnikov A. G., “O zadache vozbuzhdeniya volnovoda s neodnorodnym zapolneniem”, Zh. vychisl. mat. i mat. fiz., 39:11 (1999), 1869–1888 | MR | Zbl

[5] Bogolyubov A. N., Delitsyn A. L., Sveshnikov A. G., “Ob usloviyakh razreshimosti zadachi vozbuzhdeniya radiovolnovoda”, Dokl. AN SSSR, 370:4 (2000), 453–456 | MR

[6] Delitsyn A. L., “O postanovke kraevykh zadach dlya sistemy uravnenii Maksvella v tsilindre i ikh razreshimosti”, Izv. RAN. Ser. mat., 71:3 (2007), 61–112 | MR | Zbl

[7] Krasnushkin P. E., Moiseev E. I., “O vozbuzhdenii vynuzhdennykh kolebanii v sloistom radiovolnovode”, Dokl. AN SSSR, 264:5 (1982), 1123–1127 | MR

[8] Plamenevskii B. A., Poretskii A. S., “O sisteme Maksvella v volnovodakh s neskolkimi tsilindricheskimi vykhodami na beskonechnost”, Algebra i analiz, 25:1 (2013), 94–155

[9] Plamenevskii B. A., Poretskii A. S., “Izluchenie i rasseyanie v elektromagnitnykh volnovodakh vblizi porogov”, Algebra i analiz, 32:4 (2020), 234–270

[10] Plamenevskii B. A., Poretskii A. S., “O povedenii volnovodnykh matrits rasseyaniya v okrestnosti porogov”, Algebra i analiz, 30:2 (2018), 188–237

[11] Gudovich I. S., Krein S. G., Kraevye zadachi dlya pereopredelennykh sistem uravnenii v chastnykh proizvodnykh. Differentsialnye uravneniya i ikh primeneniya, Tr. semin., 9, In-t fiz. i mat. AN Lit. CCR, Vilnyus, 1974

[12] Birman M. Sh., Solomyak M. Z., “Samosopryazhennyi operator Maksvella v proizvolnykh oblastyakh”, Algebra i analiz, 1:1 (1989), 96–110

[13] Picard R., “On the low frequency asymptotics in electromagnetic theory”, J. Reine Angew. Math., 354 (1984), 50–73 | MR | Zbl

[14] Nazarov S. A., Plamenevskii B. A., Ellipticheskie zadachi v oblastyakh s kusochno gladkoi granitsei, Nauka, M., 1991

[15] Plamenevskii B., “On spectral properties of problems in domains with cylindrical ends”, Amer. Math. Soc. Transl. Ser. 2, 220 (2007), 123–139 ; Adv. Math. Sci., 59, Amer. Math. Soc., Providence, RI | MR | Zbl

[16] Mazya V. G., Plamenevskii B. A., “Otsenki v $L_p$ i v klassakh Geldera i printsip maksimuma Miranda–Agmona dlya reshenii ellipticheskikh kraevykh zadach v oblastyakh s osobymi tochkami na granitse”, Math. Nachr., 81 (1978), 25–82 | DOI | Zbl

[17] Nazarov S. A., Plamenevskii B. A., “Samosopryazhennye zadachi s usloviyami izlucheniya na rebrakh granitsy”, Algebra i analiz, 4:3 (1992), 196–225 | Zbl

[18] Colton D., Kress R., Inverse acoustic and electromagnetic scattering theory, Appl. Math. Sci., 93, Third ed., Springer-Verlag, New York, 2013 | DOI | MR | Zbl

[19] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965

[20] Agranovich M. S., Vishik M. I., “Ellipticheskie zadachi s parametrom i parabolicheskie zadachi obschego vida”, Uspekhi mat. nauk, 19:3 (1964), 53–160 | MR

[21] Birman M. Sh., Solomyak M. Z., Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, 2-e izd., Lan, SPb, 2010

[22] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971

[23] Filonov N. D., “Operator Maksvella v tsilindre s koeffitsientami, ne zavisyaschimi ot poperechnykh peremennykh”, Algebra i analiz, 32:1 (2020), 187–207

[24] Edward J., “Eigenfunction decay and eigenvalue accumulation for the Laplacian on asymptotically perturbed waveguides”, J. London Math. Soc. (2), 59:2 (1999), 620–636 | DOI | MR | Zbl