Geodesic
On the spectrum of the operator pencil generated by the Dirichlet problem in a cone
Sbornik. Mathematics, Tome 73 (1992) no. 1, pp. 27-48 
V. A. Kozlov; V. G. Maz'ya. On the spectrum of the operator pencil generated by the Dirichlet problem in a cone. Sbornik. Mathematics, Tome 73 (1992) no. 1, pp. 27-48. http://geodesic.mathdoc.fr/item/SM_1992_73_1_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The operator pencil whose eigenvalues determine singularities of solutions to the Dirichlet problem at the vertex of a cone is studied. First the Dirichlet–Sobolev problem with data on the ray is considered, and the eigenvalues and eigenfunctions of the corresponding operator pencil are described. Then this information is used to show that the result of [2] is best possible in a sense.

[1] Kondratev V. A., “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi ili uglovymi tochkami”, Tr. MMO, 16 (1967), 209–292

[2] Kozlov V. A., Mazya V. G., “Spektralnye svoistva operatornykh puchkov, porozhdennykh ellipticheskimi kraevymi zadachami v konuse”, Funktsion. analiz i ego pril., 22:2 (1988), 38–46 | MR | Zbl

[3] Nečas J., “Sur la coercivité des formes sesquilinéaires elliptiques”, Rev. Math. Pures Appl., 9 (1964), 47–69 | MR

[4] Mazya V. G., Nazarov S. A., Plamenevskii B. A., “Ob osobennostyakh reshenii zadachi Dirikhle vo vneshnosti tonkogo konusa”, Matem. sb., 122 (164) (1983), 435–457 | MR

[5] Kozlov V. A., “O spektre operatornogo puchka, porozhdennogo zadachei Dirikhle dlya ellipticheskogo uravneniya v ugle”, Matem. zametki, 45:5 (1989), 117–118 | MR | Zbl

[6] Ion F., Ploskie volny i sfericheskie srednie, Izd-vo IL, M., 1958

[7] Gelfand I. M., Obobschennye funktsii i deistviya nad nimi, Fizmatgiz, M., 1958

[8] Stein I., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974 | MR | Zbl

[9] Beitmen G., Erdeii A., Preobrazovaniya Besselya. Integraly ot spetsialnykh funktsii, Nauka, M., 1970

[10] Mazya V. G., Plamenevskii B. A., “Vesovye prostranstva s neodnorodnymi normami i kraevye zadachi v oblastyakh s konicheskimi tochkami”, Zap. nauchn. seminarov LOMI, 1978, 161–190 | MR

[11] Gokhberg I. Ts., Sigal E. I., “Operatornoe obobschenie teoremy o logarifmicheskom vychete i teoremy Rushe”, Matem. sb., 84 (126) (1971), 607–629 | Zbl