Geodesic
Asymptotic Estimates of the Difference Between the Dirichlet and Neumann Counting Functions
Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 4, pp. 54-64.

Voir la notice de l'article provenant de la source Math-Net.Ru

The Dirichlet and Neumann problems for the Laplace operator in a bounded domain in Euclidean space are considered. Some estimates of the difference $N_\mathrm{N}(\lambda)-N_\mathrm{D}(\lambda)$ of counting functions are discussed.
Keywords: Dirichlet and Neumann eigenvalues, counting function, boundary value problem. 
@article{FAA_2010_44_4_a4,
     author = {Yu. G. Safarov and N. D. Filonov},
     title = {Asymptotic {Estimates} of the {Difference} {Between} the {Dirichlet} and {Neumann} {Counting} {Functions}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {54--64},
     publisher = {mathdoc},
     volume = {44},
     number = {4},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2010_44_4_a4/}
}
TY  - JOUR
AU  - Yu. G. Safarov
AU  - N. D. Filonov
TI  - Asymptotic Estimates of the Difference Between the Dirichlet and Neumann Counting Functions
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2010
SP  - 54
EP  - 64
VL  - 44
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2010_44_4_a4/
LA  - ru
ID  - FAA_2010_44_4_a4
ER  - 
%0 Journal Article
%A Yu. G. Safarov
%A N. D. Filonov
%T Asymptotic Estimates of the Difference Between the Dirichlet and Neumann Counting Functions
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2010
%P 54-64
%V 44
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2010_44_4_a4/
%G ru
%F FAA_2010_44_4_a4
Yu. G. Safarov; N. D. Filonov. Asymptotic Estimates of the Difference Between the Dirichlet and Neumann Counting Functions. Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 4, pp. 54-64. http://geodesic.mathdoc.fr/item/FAA_2010_44_4_a4/

[1] I. P. Kornfeld, Ya. G. Sinai, S. V. Fomin, Ergodicheskaya teoriya, Nauka, M., 1980 | MR

[2] N. Filonov, “Ob odnom neravenstve na sobstvennye chisla zadach Dirikhle i Neimana dlya operatora Laplasa”, Algebra i analiz, 16:2 (2004), 172–176 | MR | Zbl

[3] L. Friedlander, “Some inequalities between Dirichlet and Neumann eigenvalues”, Arch. Rational Mech. Anal., 116:2 (1991), 153–160 | DOI | MR | Zbl

[4] T. E. Gureev, Yu. G. Safarov, “Tochnaya asimptotika spektra operatora Laplasa na mnogoobrazii s periodicheskimi geodezicheskimi”, Kraevye zadachi matematicheskoi fiziki, 13, Trudy MIAN SSSR, 179, 1988, 36–53 | MR | Zbl

[5] V. Ya. Ivrii, “O vtorom chlene spektralnoi asimptotiki dlya operatora Laplasa–Beltrami na mnogoobraziyakh s kraem”, Funkts. analiz i ego pril., 14:2 (1980), 25–34 | MR | Zbl

[6] V. Ivrii, “Sharp spectral asymptotics for operators with irregular coefficients. II. Domains with boundary and degeneration”, Comm. Partial Differential Equations, 28:1-2 (2003), 103–128 | DOI | MR | Zbl

[7] Yu. Netrusov, “Sharp remainder estimates in the Weyl formula for the Neumann Laplacian on a class of planar regions”, J. Funct. Anal., 250:1 (2007), 21–41 | DOI | MR | Zbl

[8] Yu. Netrusov, Yu. Safarov, “Weyl asymptotic formula for the Laplacian on domains with rough boundaries”, Commun. Math. Phys., 253:2 (2005), 481–509 | DOI | MR | Zbl

[9] Yu. Safarov, “Fourier Tauberian theorems and applications”, J. Funct. Anal., 185:1 (2001), 111–128 | DOI | MR | Zbl

[10] Yu. Safarov, D. Vassiliev, The Asymptotic Distribution of Eigenvalues of Partial Differential Operators, Translation of Math. Monographs, 155, Amer. Math. Soc., Providence, RI, 1997 | MR

[11] D. G. Vasilev, “Dvuchlennaya asimptotika spektra kraevoi zadachi v sluchae kusochno gladkoi granitsy”, Dokl. AN SSSR, 286:5 (1986), 1043–1046 | MR

[12] D. G. Vasilev, “Dvuchlennaya asimptotika spektra kraevoi zadachi pri vnutrennem otrazhenii obschego vida”, Funkts. analiz i ego pril., 18:4 (1984), 1–13 | MR | Zbl

[13] Ya. B. Vorobets, “O mere mnozhestva periodicheskikh tochek bilyarda”, Matem. zametki, 55:5 (1994), 25–35 | MR | Zbl