Geodesic
On degenerate nonself-adjoint differential equations of fourth order
Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (2012), pp. 29-33.

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

We consider the degenerate nonself-adjoint differential equation of fourth order $Lu\equiv(t^{\alpha}u^{\prime\prime})^{\prime\prime}+au^{\prime\prime\prime}-pu^{\prime}+qu=f$ where $t\in(0, b), \ 0\leq\alpha\leq 2, \alpha\neq 1, a, p, q $ are the constant numbers and $a\neq0, p>0, f\in L_2(0, b)$. We prove that the statement of the Dirichlet problem for the above equation depends on the sign of the number $a$ (Keldysh Teorem).
Keywords: Dirichlet problem, degenerate equations, weighted Sobolev spaces, spectral theory of linear operators. 
@article{UZERU_2012_3_a5,
     author = {L. P. Tepoyan and H. S. Grigoryan},
     title = {On degenerate nonself-adjoint differential equations of fourth order},
     journal = {Proceedings of the Yerevan State University. Physical and mathematical sciences},
     pages = {29--33},
     publisher = {mathdoc},
     number = {3},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UZERU_2012_3_a5/}
}
TY  - JOUR
AU  - L. P. Tepoyan
AU  - H. S. Grigoryan
TI  - On degenerate nonself-adjoint differential equations of fourth order
JO  - Proceedings of the Yerevan State University. Physical and mathematical sciences
PY  - 2012
SP  - 29
EP  - 33
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/UZERU_2012_3_a5/
LA  - en
ID  - UZERU_2012_3_a5
ER  - 
%0 Journal Article
%A L. P. Tepoyan
%A H. S. Grigoryan
%T On degenerate nonself-adjoint differential equations of fourth order
%J Proceedings of the Yerevan State University. Physical and mathematical sciences
%D 2012
%P 29-33
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/UZERU_2012_3_a5/
%G en
%F UZERU_2012_3_a5
L. P. Tepoyan; H. S. Grigoryan. On degenerate nonself-adjoint differential equations of fourth order. Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (2012), pp. 29-33. http://geodesic.mathdoc.fr/item/UZERU_2012_3_a5/

[1] Amer. Math. Soc., 1988, no. 8, 1469–1470 | MR | Zbl

[2] M.V. Keldysh, “On Certain Cases of Degeneration of Equations of Elliptic Type on the Boundary of a Domain”, Doklady Akad. Nauk SSSR, 77 (1951), 181—183 (in Russian) | MR

[3] Math. USSR-Sb., 43:3 (1982), 287–298 | DOI | MR | Zbl | Zbl

[4] Math. Notes, 68:5 (2000), 576–587 | DOI | DOI | MR | Zbl

[5] G. Jaiani, “On a Generalization of the Keldysh Theorem”, Georgian Mat. Journal, 2:3 (1995), 291–297 | DOI | MR | Zbl

[6] G. Jaiani, “Theory of Cusped Euler-Bernoulli Beams and Kirchoff-Love Plates”, Lecture Notes of TICMI, 3 (2002) | MR

[7] E.V. Makhover, “Bending of a Plate of Variable Thickness with a Cusped Edge”, Scientific Notes of Leningrad State Ped. Institute, 17:2 (1957), 28–39 (in Russian)

[8] V.I. Burenkov, Sobolev Spaces on Domains, Teubner, 1998, 312 pp. | MR | Zbl

[9] L.P. Tepoyan, “On the Spectrum of a Degenerate Operator”, Izvestiya Natsionalnoi Akademii Nauk Armenii. Matematika, 38:5 (2003), 53–57 | MR | Zbl

[10] L.D. Kudryavtzev, “On Equivalent Norms in the Weight Spaces”, Trudy Mat. Inst. AN SSSR, 170, 1984, 161–190 (in Russian) | MR

[11] A.A. Dezin, Partial Degenerate Equations (An Introduction to a General Theory of Linear Boundary Vaiue Problems), Springer, 1987 | MR

[12] J. Weidmann, Lineare Operatoren in Hilberträumen, v. 1, Teubner Verlag, Stuttgart–Grundlagen, 2000 | MR | Zbl