|
Spectrum asymptotics for one ``nonsmooth'' variational problem with solvable constraint
Author(s):
A.
B.
Alekseev;
M.
Sh.
Birman;
N.
D.
Filonov
Translated by:
N. D. Filonov
Original publication:
Algebra i Analiz,
tom 18
(2006),
nomer 5.
Journal:
St. Petersburg Math. J.
18
(2007),
681-697.
MSC (2000):
Primary 35P20
Posted:
August 9, 2007
MathSciNet review:
2301038
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
In a previous paper by Birman and Filonov, the spectrum of the Maxwell operator with nonsmooth coefficients in Lipschitz domains was investigated. The claim that its eigenvalues obey the Weyl asymptotics was proved up to a statement about the spectrum of an auxiliary problem with constraint. The proof of that statement is given in the present paper.
References:
-
- 1.
- A. B. Alekseev, Spectral asymptotics of elliptic boundary value problems with solvable constraints, Candidate 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), 1319-1322 (1977). MR 0420027 (54:8044)
- 3.
- 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 (154), 3-10; English transl., Soviet Math. (Iz. VUZ) 19 (1975), no. 3, 1-6 (1976). MR 0385597 (52:6458)
- 4.
- M. Sh. Birman and M. Z. Solomyak, Spectral asymptotics of nonsmooth elliptic operators. I, II, Trudy Moskov. Mat. Obshch. 27 (1972), 3-52; ibid. 28 (1973), 3-34; English transl., Trans. Moscow Math. Soc. 27 (1972), 1-52 (1975); ibid. 28 (1973), 1-32 (1975). MR 0364898 (51:1152)
- 5.
- -, Quantitative analysis in Sobolev imbedding theorems and applications to spectral theory, Tenth Mathematical School (Summer School, Kaciveli/Nalchik, 1972), Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1974, pp. 5-189; English transl., Amer. Math. Soc. Transl. (2), vol. 114, Amer. Math. Soc., Providence, RI, 1980. MR 0482138 (58:2224); MR 0562305 (80m:46026)
- 6.
- -, Weyl asymptotics of the spectrum of the Maxwell operator for domains with a Lipschitz boundary, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1987, vyp. 3, 23-28; English transl., Vestnik Leningrad Univ. Math. 20 (1987), no. 3, 15-21. MR 0928156 (89h:35253)
- 7.
- -, Spectral theory of selfadjoint operators in Hilbert space, Leningrad. Gos. Univ., Leningrad, 1980; English transl., Math. Appl. (Soviet Ser.), D. Reidel Publishing Co., Dordrecht, 1987. MR 0609148 (82k:47001); MR 1192782 (93g:47001)
- 8.
- -, On the main singularities of the electric component of the electro-magnetic field in regions with screens, Algebra i Analiz 5 (1993), no. 1, 143-159; English transl., St. Petersburg Math. J. 5 (1994), no. 1, 125-139. MR 1220492 (94f:35137)
- 9.
- 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.
- 10.
- M. Cwikel, Weak type estimates for singular values and the number of bound states of Schrödinger operators, Ann. of Math. (2) 106 (1977), 93-100. MR 0473576 (57:13242)
- 11.
- H. Weyl, Über das Spektrum der Hohlraumstrahlung, J. Reine Angew. Math. 141 (1912), 163-181.
Similar Articles:
Retrieve articles in St. Petersburg Mathematical Journal
with MSC
(2000):
35P20
Retrieve articles in all Journals with MSC
(2000):
35P20
Additional Information:
A.
B.
Alekseev
Affiliation:
Department of Physics, St. Petersburg State University, Ul{'}yanovskaya 1, Petrodvorets, St. Petersburg 198504, Russia
M.
Sh.
Birman
Affiliation:
Department of Physics, St. Petersburg State University, Ul{'}yanovskaya 1, Petrodvorets, St. Petersburg 198504, Russia
Email:
mbirman@list.ru
N.
D.
Filonov
Affiliation:
Department of Physics, St. Petersburg State University, Ul{'}yanovskaya 1, Petrodvorets, St. Petersburg 198504, Russia
Email:
laugre@mail.ru
DOI:
10.1090/S1061-0022-07-00968-5
PII:
S 1061-0022(07)00968-5
Keywords:
Weyl spectrum asymptotics,
resonator with perfectly conductive boundary,
spectrum of the quotient of quadratic forms,
problem with constraint
Received by editor(s):
4/AUG/2006
Posted:
August 9, 2007
Copyright of article:
Copyright
2007,
American Mathematical Society
|
