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

Filonov, N.

doi:10.1090/S1061-0022-2013-01282-9

Weyl asymptotics for the spectrum of the Maxwell operator in Lipschitz domains of arbitrary dimension
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by N. Filonov
Translated by: S. Kislyakov

St. Petersburg Math. J. 25 (2014), 117-149

DOI: https://doi.org/10.1090/S1061-0022-2013-01282-9

Published electronically: November 20, 2013

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Abstract:

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

References

A. B. Alekseev, Spectral asymptotics of elliptic boundary value problems with solvable constraints, Kand. diss., Leningrad. Gos. Univ., Leningrad, 1977. (Russian)
A. B. Alekseev and M. Š. Birman, Asymptotic behavior of the spectrum of elliptic boundary value problems with solvable constraints, Dokl. Akad. Nauk SSSR 230 (1976), no. 3, 505–507 (Russian). MR 0420027
A. B. Alekseev, M. Sh. Birman, and N. D. Filonov, Asymptotics of the spectrum of a “nonsmooth” variational problem with a solvable constraint, Algebra i Analiz 18 (2006), no. 5, 1–22 (Russian); English transl., St. Petersburg Math. J. 18 (2007), no. 5, 681–697. MR 2301038, DOI 10.1090/S1061-0022-07-00968-5
M. Š. Birman, L. S. Koplienko, and M. Z. Solomjak, Estimates of the spectrum of a difference of fractional powers of selfadjoint operators, Izv. Vysš. Učebn. Zaved. Matematika 3(154) (1975), 3–10 (Russian). MR 0385597
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, DOI 10.1090/trans2/220/02
M. Sh. Birman and M. Z. Solomjak, Spectral theory of selfadjoint operators in Hilbert space, Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, 1987. Translated from the 1980 Russian original by S. Khrushchëv and V. Peller. MR 1192782
M. Š. Birman and M. Z. Solomjak, Spectral asymptotics of nonsmooth elliptic operators. I, II, Trudy Moskov. Mat. Obšč. 27 (1972), 3–52; ibid. 28 (1973), 3–34 (Russian). MR 0364898
M. Š. Birman and M. Z. Solomjak, Quantitative analysis in Sobolev’s imbedding theorems and applications to spectral theory, Tenth Mathematical School (Summer School, Kaciveli/Nalchik, 1972) Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1974, pp. 5–189 (Russian). MR 0482138
M. Sh. Birman and M. Z. Solomyak, The Maxwell operator in domains with a nonsmooth boundary, Sibirsk. Mat. Zh. 28 (1987), no. 1, i, 23–36 (Russian). MR 886850
M. Sh. Birman and M. Z. Solomyak, Weyl asymptotics of the spectrum of the Maxwell operator for domains with a Lipschitz boundary, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 3 (1987), 23–28, 127 (Russian, with English summary). MR 928156
M. Sh. Birman and M. Z. Solomyak, The selfadjoint Maxwell operator in arbitrary domains, Algebra i Analiz 1 (1989), no. 1, 96–110 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 1, 99–115. MR 1015335
M. E. Bogovskiĭ, Solutions of some problems of vector analysis, associated with the operators $\textrm {div}$ and $\textrm {grad}$, Theory of cubature formulas and the application of functional analysis to problems of mathematical physics, Trudy Sem. S. L. Soboleva, No. 1, vol. 1980, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1980, pp. 5–40, 149 (Russian). MR 631691
Jean Bourgain and Haïm Brezis, On the equation $\textrm {div}\, Y=f$ and application to control of phases, J. Amer. Math. Soc. 16 (2003), no. 2, 393–426. MR 1949165, DOI 10.1090/S0894-0347-02-00411-3
A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289–309. MR 84633, DOI 10.2307/2372517
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, DOI 10.1090/trans2/225/05
N. D. Filonov, On a bounded solution of the equation $\textrm {div}\,u=f$ in plane domains [1 766 023], Proceedings of the St. Petersburg Mathematical Society, Vol. 6 (Russian), Tr. St.-Peterbg. Mat. Obshch., vol. 6, Nauchn. Kniga, Novosibirsk, 1998, pp. 231–241 (Russian). MR 1768334
K. O. Friedrichs, Differential forms on Riemannian manifolds, Comm. Pure Appl. Math. 8 (1955), 551–590. MR 87763, DOI 10.1002/cpa.3160080408
V. M. Gol′dshtein, V. I. Kuz′minov, and I. A. Shvedov, Differential forms on a Lipschitz manifold, Sibirsk. Mat. Zh. 23 (1982), no. 2, 16–30, 215 (Russian). MR 652220
Lars Hörmander, The analysis of linear partial differential operators. III, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. Pseudodifferential operators. MR 781536
Björn Jawerth and Marius Mitrea, Higher-dimensional electromagnetic scattering theory on $C^1$ and Lipschitz domains, Amer. J. Math. 117 (1995), no. 4, 929–963. MR 1342836, DOI 10.2307/2374954
O. A. Ladyzhenskaya and N. N. Ural′tseva, Lineĭnye i kvazilineĭnye uravneniya èllipticheskogo tipa, Izdat. “Nauka”, Moscow, 1973 (Russian). Second edition, revised. MR 0509265
Rolf Leis, Initial-boundary value problems in mathematical physics, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986. MR 841971, DOI 10.1007/978-3-663-10649-4
William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312
Marius 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, DOI 10.1215/S0012-7094-04-12322-1
R. Picard, An elementary proof for a compact imbedding result in generalized electromagnetic theory, Math. Z. 187 (1984), no. 2, 151–164. MR 753428, DOI 10.1007/BF01161700
Rainer Picard, Norbert Weck, and Karl-Josef Witsch, Time-harmonic Maxwell equations in the exterior of perfectly conducting, irregular obstacles, Analysis (Munich) 21 (2001), no. 3, 231–263. MR 1855908, DOI 10.1524/anly.2001.21.3.231
Yu. G. Safarov, Asymptotics of the spectrum of the Maxwell operator, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 127 (1983), 169–180 (Russian, with English summary). Boundary value problems of mathematical physics and related questions in the theory of functions, 15. MR 702849
N. A. Veniaminov, Estimate for the remainder in the Weyl asymptotics of the spectrum of the Maxwell operator in Lipschitz domains, J. Math. Sci. (N.Y.) 169 (2010), no. 1, 46–63. Problems in mathematical analysis. No. 47. MR 2839003, DOI 10.1007/s10958-010-0038-3
C. Weber, Regularity theorems for Maxwell’s equations, Math. Methods Appl. Sci. 3 (1981), no. 4, 523–536. MR 657071, DOI 10.1002/mma.1670030137
N. Weck, Maxwell’s boundary value problem on Riemannian manifolds with nonsmooth boundaries, J. Math. Anal. Appl. 46 (1974), 410–437. MR 343771, DOI 10.1016/0022-247X(74)90250-9
H. Weyl, Über das Spectrum der Hohlraumstrahlung, J. Reine Angew. Math. 141 (1912), 163–181.
Hermann Weyl, Die natürlichen Randwertaufgaben im Aussenraum für Strahlungsfelder beliebiger Dimension und beliebigen Ranges, Math. Z. 56 (1952), 105–119 (German). MR 54524, DOI 10.1007/BF01175027