Geodesic
On the singularities of solutions of the Dirichlet problem in the exterior of a slender cone
Sbornik. Mathematics, Tome 50 (1985) no. 2, pp. 415-437 
V. G. Maz'ya; S. A. Nazarov; B. A. Plamenevskii. On the singularities of solutions of the Dirichlet problem in the exterior of a slender cone. Sbornik. Mathematics, Tome 50 (1985) no. 2, pp. 415-437. http://geodesic.mathdoc.fr/item/SM_1985_50_2_a7/
@article{SM_1985_50_2_a7,
     author = {V. G. Maz'ya and S. A. Nazarov and B. A. Plamenevskii},
     title = {On the singularities of solutions of the {Dirichlet} problem in the exterior of a~slender cone},
     journal = {Sbornik. Mathematics},
     pages = {415--437},
     year = {1985},
     volume = {50},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1985_50_2_a7/}
}
TY  - JOUR
AU  - V. G. Maz'ya
AU  - S. A. Nazarov
AU  - B. A. Plamenevskii
TI  - On the singularities of solutions of the Dirichlet problem in the exterior of a slender cone
JO  - Sbornik. Mathematics
PY  - 1985
SP  - 415
EP  - 437
VL  - 50
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1985_50_2_a7/
LA  - en
ID  - SM_1985_50_2_a7
ER  - 
%0 Journal Article
%A V. G. Maz'ya
%A S. A. Nazarov
%A B. A. Plamenevskii
%T On the singularities of solutions of the Dirichlet problem in the exterior of a slender cone
%J Sbornik. Mathematics
%D 1985
%P 415-437
%V 50
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1985_50_2_a7/
%G en
%F SM_1985_50_2_a7

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

The power singularities of solutions of the Dirichlet problem for strongly elliptic differential systems of order $2m$ outside a slender cone $k_\varepsilon$ are studied, where $\varepsilon$ is a small positive parameter which characterizes the angle vertex of the cone. In essence, the asymptotics as $\varepsilon\to0$ of the small eigenvalues $\lambda_j(\varepsilon)$ of the first boundary value problem on the unit sphere with a small hole are discussed for a differential operator depending polynomially on a complex parameter $\lambda$. As an application of the asymptotic formulas for $\lambda_j(\varepsilon)$, a theorem is obtained on the validity of a bound on the maximum modulus of a solution of the Dirichlet problem in a region with a slender conical notch. Bibliography: 22 titles.

[1] Kondratev V. A., “Kraevye zadachi dlya ellipticheskikh uravnenii v oblasti s konicheskimi ili uglovymi tochkami”, Trudy mosk. matem. ob-va, 16 (1967), 219–292

[2] Mazya V. G., Plamenevskii B. A., “O koeffitsientakh v asimptotike reshenii ellipticheskikh kraevykh zadach v oblasti s konicheskimi tochkami”, Mathematishe Nachrichten, 76 (1977), 29–60 | DOI

[3] Mazya V. G., Plamenevskii B. A., “Otsenki v $L_p$ i v klassakh Gëldera i printsip Miranda–Agmona dlya reshenii ellipticheskikh kraevykh zadach v oblastyakh s osobymi tochkami na granitse”, Mathematishe Nachrichten, 81 (1978), 25–82 | DOI

[4] Mazya V. G., Plamenevskii B. A., “Ob asimptotike fundamentalnykh reshenii ellipticheskikh kraevykh zadach v oblastyakh s konicheskimi tochkami”, Problemy matem. analiza. Kraevye zadachi. Spektralnaya teoriya, vyp. 7, izd-vo LGU, L., 1979, 100–145 | MR

[5] Mazya V. G., Plamenevskii B. A., “O printsipe maksimuma dlya bigarmonicheskogo uravneniya v oblasti s konicheskimi tochkami”, Izvestiya VUZ'ov. Matematika, 1981, no. 2, 52–59 | MR

[6] Mazya V. G., Plamenevskii B. A., “O svoistvakh reshenii trekhmernykh zadach teorii uprugosti i gidrodinamiki v oblastyakh s izolirovannymi osobennostyami”, Dinamika sploshnoi sredy, vyp. 50, Novosibirsk, 1981, 99–121 | MR

[7] Samarskii A. A., “O vliyanii zakrepleniya na sobstvennye chastoty zamknutykh ob'emov”, UMN, 5:3 (1950), 133–134

[8] Dnestrovskii Yu. N., “Ob izmenenii sobstvennykh znachenii pri izmenenii granitsy oblasti”, Vestnik MGU, 1954, no. 9, 61–74 | MR | Zbl

[9] Swanson C. A., “Asymptotic variational formulae for eigenvalues”, Canad. Math. Bull., 6:1 (1963), 15–25 | MR | Zbl

[10] Ozawa Shin., “Singular Hadamar's variation of domains and eigenvalues of the Laplacian. I”, Proc. Jap. Acad., A56 (1980), 351–357; “II”, Proc. Jap. Acad., A57 (1981), 242–246 | DOI | MR

[11] Mazya V. G., Nazarov S. A., Plamenevskii B. A., “Ob asimptotike reshenii ellipticheskikh kraevykh zadach pri neregulyarnom vozmuschenii oblasti”, Problemy matematicheskogo analiza. Kraevye zadachi. Spektralnaya teoriya, vyp. 8, izd-vo LGU, L., 1981, 72–153 | MR

[12] Mazya V. G., Nazarov S. A., Plamenevskii B. A., “Otsutstvie teoremy tipa De Dzhordzhi dlya silno ellipticheskikh uravnenii s kompleksnymi koeffitsientami”, Kraevye zadachi matematicheskoi fiziki i smezhnye voprosy teorii funktsii, 14 (Zap. nauch. semin. LOMI, t. 115), Nauka, L., 1982, 156–168 | MR

[13] De Giorgi E., “Sulla differenziabilita e l'analiticita multipli regolari”, Mem. Acc. Sci. Torino, 3 (1957), 1–19

[14] Nash J., “Continuity of solutions of parabolic and elliptic equations”, Amer. J. Math., 80:4 (1958), 931–954 | DOI | MR | Zbl

[15] Mazya V. G., Nazarov S. A., Plamenevskii B. A., “Ob odnorodnykh resheniyakh zadachi Dirikhle vo vneshnosti tonkogo konusa”, DAN SSSR, 266:2 (1982), 281–284 | MR | Zbl

[16] Agmon S., Douglis A., Nirenberg L., “Estimates near the boundary for solutions of elliptic equations satisfying general boundary conditions. II”, Comm. Pure Appl. Math., 17 (1964), 35–92 | DOI | MR | Zbl

[17] Solonnikov V. A., “Ob obschikh kraevykh zadachakh dlya ellipticheskikh po Duglisu–Nirenbergu sistem. I”, Izv. AN SSSR. Seriya matem., 28 (1964), 665–706 ; “II”, Труды МИАН, 92 (1966), 233–297 | MR | MR | Zbl

[18] Mikhlin S. G., Mnogomernye singulyarnye integraly i integralnye uravneniya, Fizmatgiz, M., 1962 | MR | Zbl

[19] Ion F., Ploskie volny i sfericheskie srednie, IL, M., 1958

[20] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965, 448 pp. | MR | Zbl

[21] Mazya V. G., “O zadache Dirikhle dlya ellipticheskikh uravnenii proizvolnogo poryadka v neogranichennykh oblastyakh”, DAN SSSR, 150:6 (1963), 1221–1224

[22] Gelfand I. M., Shilov G. E., Obobschennye funktsii i deistviya nad nimi, Fizmatgiz, M., 1958