Geodesic
Asymptotics of the spectrum of problems with constraints
Sbornik. Mathematics, Tome 57 (1987) no. 1, pp. 77-95 Cet article a éte moissonné depuis la source Math-Net.Ru

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By means of the method of an approximate spectral projection operator the classical asymptotic formula for the distribution function of eigenvalues with an estimate of the remainder is proved both for problems with an unsolvable constraint such as an incompressibility condition (the Navier–Stokes and Maxwell systems) and those with a solvable constraint (an example is the spectral problem of the theory of electroelasticity). Problems in a bounded Lipschitz domain are considered. We note that an estimate of the remainder for the linearized Navier–Stokes system was obtained earlier only for the case of a domain with boundary of class $C^\infty$, while for problems with solvable constraints only the leading term of the asymptotics was known; the asymptotics of the spectrum in the problem of the theory of electroelasticity has not been studied previously. Bibliography: 13 titles. 
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S. Z. Levendorskii. Asymptotics of the spectrum of problems with constraints. Sbornik. Mathematics, Tome 57 (1987) no. 1, pp. 77-95. http://geodesic.mathdoc.fr/item/SM_1987_57_1_a4/

[1] Belokon A. V., Vorovich I. I., “Nekotorye matematicheskie voprosy teorii elektrouprugikh tel”, Aktualnye problemy mekhaniki deformiruemykh sred, DGU, Dnepropetrovsk, 1979, 53–69

[2] Alekseev A. B., Birman M. Sh., “Asimptotika spektra ellipticheskikh granichnykh zadach s razreshimymi svyazyami”, DAN SSSR, 230:3 (1976), 505–507 | MR | Zbl

[3] Metivier G., “Valeurs propres d'operateurs definis par la restriction de systemes vatiationnels a des sous-espaces”, J. Math. Pures et appl., 57:2 (1978), 133–156 | MR | Zbl

[4] Babenko K. N., “Ob asimptotike sobstvennykh znachenii linearizovannykh uravnenii Nave–Stoksa”, DAN SSSR, 263:3 (1982), 521–525 | MR | Zbl

[5] Kozhevnikov A. N., “Ob ostatochnom chlene v asimptotike spektra zadachi Stoksa”, DAN SSSR, 272:4 (1983), 294–296 | MR | Zbl

[6] Tulovskii V. N., Shubin M. A., “Ob asimptoticheskom raspredelenii sobstvennykh znachenii psevdodifferentsialnykh operatorov v $R^n$”, Matem. sb., 92(134) (1973), 571–586 | MR

[7] Feigin V. I., “Asimptoticheskoe raspredelenie sobstvennykh chisel dlya gipoellipticheskikh sistem v $R^n$”, Matem. sb., 99(141) (1976), 594–614 | MR | Zbl

[8] Levendorskii S. Z., “Asimptoticheskoe raspredelenie sobstvennykh znachenii”, Izv. AN SSSR. Ser. matem., 46:4, 810–852 | MR | Zbl

[9] Levendorskii S. Z., “Metod priblizhennogo spektralnogo proektora kak obschii metod issledovaniya asimptotiki spektra”, DAN SSSR, 271:2 (1983), 287–291 | MR | Zbl

[10] Levendorskii S. Z., “Asimptotika spektra lineinykh operatornykh puchkov”, Matem. sb., 124(166) (1984), 251–271 | MR | Zbl

[11] Hörmander L., “The Weyl calculus of pseudodifferential operators”, Comm. Pure Appl. Math., 32 (1979), 259–443 | MR

[12] Levendorskii S. Z., “Asimptotika spektra zadach so svyazyami”, DAN SSSR, 276:2 (1984), 282–285 | MR

[13] Beals R., “Weighted distribution spaces and pseudodifferential operators”, J. Anal. Math., 39 (1981), 131–187 | DOI | MR | Zbl