Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

Request Permissions   Purchase Content 
 
 

 

Weyl asymptotics for the spectrum of the Maxwell operator in Lipschitz domains of arbitrary dimension


Author: N. Filonov
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 25 (2013), nomer 1.
Journal: St. Petersburg Math. J. 25 (2014), 117-149
MSC (2010): Primary 35P20
DOI: https://doi.org/10.1090/S1061-0022-2013-01282-9
Published electronically: November 20, 2013
MathSciNet review: 3113431
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The eigenvalues of the multidimensional Maxwell operator in a domain in the Euclidean space are shown to obey the Weyl asymptotics.


References [Enhancements On Off] (What's this?)

  • 1. A. B. Alekseev, Spectral asymptotics of elliptic boundary value problems with solvable constraints, Kand. diss., Leningrad. Gos. Univ., Leningrad, 1977. (Russian)
  • 2. A. B. Alekseev and M. Sh. Birman, Asymptotic behavior of the spectrum of elliptic boundary value problems with solvable constraints, Dokl. Akad. Nauk SSSR 230 (1976), no. 3, 505-507; English transl., Soviet Math. Dokl. 17 (1976), no. 5, 1319-1322. MR 0420027 (54:8044)
  • 3. A. B. Alekseev, M. Sh. Birman, and N. D. Filonov, Spectrum asymptotics for one ``nonsmooth'' variational problem with solvable constraint, Algebra i Analiz 18 (2006), no. 5, 1-22; English transl., St. Petersburg Math. J. 18 (2007), no. 5, 681-697. MR 2301038 (2008b:35193)
  • 4. M. Sh. Birman, L. S. Koplienko, and M. Z. Solomyak, Estimates of the spectrum of a difference of fractional powers of selfadjoint operators, Izv. Vyssh. Uchebn. Zaved. Mat. 1975, no. 3, 3-10; English transl., Soviet Math. (Iz. VUZ) 19 (1975), no. 3, 1-6 (1976). MR 0385597 (52:6458)
  • 5. M. Sh. Birman and N. D.Filonov, Weyl asymptotics of the spectrum of the Maxwell operator with non-smooth coefficients in Lipschitz domains, Nonlinear Equations and Spectral Theory, Amer. Math. Soc. Transl. Ser. 2, vol. 220, Amer. Math. Soc., Providence, RI, 2007, pp. 27-44. MR 2343605 (2009a:35243)
  • 6. M. Sh. Birman and M. Z. Solomyak, Spectral theory of selfadjoint operators in Hilbert space, 2nd ed., Lan', St. Petersburg, 2010; English transl. of 1st ed., Math. Appl. (Soviet Series), D. Reidel Publ. Co., Dordrecht, 1987. MR 1192782 (93g:47001)
  • 7. -, Spectral asymptotics of nonsmooth elliptic operators. II, Trudy Moskov. Mat. Obshch. 28 (1973), 3-34; English transl., Transl. Moscow Math. Soc. 28 (1973), 1-32 (1975) MR 0364898 (51:1152)
  • 8. -, Quantitative analysis in Sobolev's imbedding theorems and applications to spectral theory, Tenth Mathematical Schood (Summer School, Kaciveli/Nal'chik, 1972), Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1974, pp. 5-189. (Russian) MR 0482138 (58:2224)
  • 9. -, The Maxwell operator in domains with a nonsmooth boundary, Sibirsk. Mat. Zh. 28 (1987), no. 1, 23-36; English transl., Siberian Math. J. 28 (1987), no. 1, 12-24. MR 0886850 (88g:35095)
  • 10. -, Weyl asymptotics of the spectrum of the Maxwell operator for domains with a Lipschitz boundary, Vestnik Leningrad. Univ. Ser. 1 1987, vyp. 3, 23-28; English transl., Vestnik Leningrad Univ. Math. 20 (1987), no. 3, 15-21. MR 0928156 (89h:35253)
  • 11. -, The selfadjoint Maxwell operator in arbitrary domains, Algebra i Analiz 1 (1989), no. 1, 96-110; English transl., Leningrad Math. J. 1 (1990), no. 1, 99-115. MR 1015335 (91e:35197)
  • 12. M. E. Bogovskiĭ, Solutions of some problems of vector analysis, associated with the operators $ \operatorname {div}$ and $ \operatorname {grad}$, Trudy Sem. S. L. Soboleva 1980, no. 1, 5-40. (Russian) MR 0631691 (82m:26014)
  • 13. J. Bourgain and H. Brezis, On the equation $ \operatorname {div}\,Y=f$ and application to control of phases, J. Amer. Math. Soc. 16 (2003), 393-426. MR 1949165 (2004d:35032)
  • 14. A. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289-309. MR 0084633 (18:894a)
  • 15. M. N. Demchenko and N. D. Filonov, Spectral asymptotics of the Maxwell operator on Lipschitz manifolds with boundary, Spectral Theory of Differential Operators, Amer. Math. Soc. Transl. Ser. 2, vol. 225, Amer. Math. Soc., Providence, RI, 2008, pp. 73-90. MR 2509776 (2010k:35335)
  • 16. N. Filonov, On a bounded solution of the equation $ \operatorname {div}\, u = f$ in plane domains, Tr. S.-Peterburg. Mat. Obshch. 6 (1998), 231-245; English transl., Amer. Math. Soc. transl. Ser. 2, vol. 199. Proc. St. Petersburg Math. Soc. vol. 6, Amer. Math. Soc., Providence, RI, 2000, pp. 199-207. MR 1768334; MR 1766023 (2002b:35168)
  • 17. K. O. Friedrichs, Differential forms on Riemannian manifolds, Comm. Pure Appl. Math. 8 (1955), 551-590. MR 0087763 (19:407a)
  • 18. V. M. Gol'dshteĭn, V. I. Kuz'minov, and I. A. Shvedov, Differential forms on a Lipschitz manifold, Sibirsk. Mat. Zh. 23 (1982), no. 2, 16-30; English transl., Siberian Math. J. 23 (1982), no. 2, 151-161. MR 0652220 (83j:58004)
  • 19. L. Hörmander, Analysis of linear partial differential operators. III. Pseudodifferential operators, Grundlehren Math. Wiss., Bd. 274, Springer-Verlag, Berlin, 1985. MR 0781536 (87d:35002a)
  • 20. B. Jawerth and M. Mitrea, Higher-dimensional electromagnetic scattering theory on $ C^1$ and Lipschitz domains, Amer. J. Math. 117 (1995), 929-963. MR 1342836 (96h:35143)
  • 21. O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear equations of elliptic type, 2nd ed., Nauka, Moscow, 1973; English transl. of 1st ed., Linear and quasilinear elliptic equations, Acad. Press, New York-London, 1968. MR 0509265 (58:23009); MR 0244627 (39:5941)
  • 22. R. Leis, Initial-boundary value problems in mathematical physics, B. G. Teubner, Stutgart; J. Wiley and Sons, Ltd., Chichester, 1986. MR 0841971 (87h:35003)
  • 23. W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge Univ. Press, Cambridge, 2000. MR 1742312 (2001a:35051)
  • 24. M. Mitrea, Sharp Hodge decompositions, Maxwell's equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds, Duke Math. J. 125 (2004), no. 3, 467-547. MR 2166752 (2007g:35246)
  • 25. R. Picard, An elementary proof for a compact imbedding result in generalized electromagnetic theory, Math. Z. 187 (1984), 151-164. MR 0753428 (85k:35212)
  • 26. R. Picard, N. Weck, and K.-J. Witsch, Time-harmonic Maxwell equations in the exterior of perfectly conducting, irregular obstacles, Analysis (Munich) 21 (2001), 231-263. MR 1855908 (2002h:78009)
  • 27. Yu. G. Safarov, Asymptotics of the spectrum of the Maxwell operator, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 127 (1983), 169-180; English transl. in J. Soviet Math. 27 (1984), no. 2. MR 0702849 (85d:35088)
  • 28. N. A. Veniamoniv, Estimate for the remainder in the Weyl asymptotics of the spectrum of the Maxwell operator in Lipschitz domains, Probl. Mat. anal., No. 47, Novosibirsk, 2010, pp. 43-58; English transl., J. Math. Sci. (N. Y.) 169 (2010), no. 1, 46-63. MR 2839003 (2012k:35541)
  • 29. C. Weber, Regularity theorems for Maxwell's equations, Math. Methods Appl. Sci. 3 (1981), 523-536. MR 0657071 (83h:35113)
  • 30. N. Weck, Maxwell's boundary value problem on Riemannian manifolds with nonsmooth boundaries, J. Math. Anal. Appl. 46 (1974), 410-437. MR 0343771 (49:8511)
  • 31. H. Weyl, Über das Spectrum der Hohlraumstrahlung, J. Reine Angew. Math. 141 (1912), 163-181.
  • 32. -, Die natürlichen Randwertaufgaben im Aussenraum für Strahlungsfelder beliebiger Dimension und beliebiger Ranges, Math. Z. 56 (1952), 105-119. MR 0054524 (14:933g)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 35P20

Retrieve articles in all journals with MSC (2010): 35P20


Additional Information

N. Filonov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023; St. Petersburg State University, Universitetskii pr. 28, Staryi Peterhof, St. Petersburg 198504, Russia
Email: filonov@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-2013-01282-9
Keywords: Maxwell operator, Weyl spectral asymptotics, domains with Lipschitz boundary, differential forms, ratios of quadratic forms
Received by editor(s): September 21, 2012
Published electronically: November 20, 2013
Additional Notes: Supported by the RFBR grant 11-01-00458-a and by the grant NSh-357.2012.1
Dedicated: To the memory of M. Sh. Birman
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society