Geodesic
On a spectral problem for a system of~differential equations of mixed type
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 21 (2017) no. 4, pp. 611-632.

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

This article explores the spectral characteristics of the differential operator generated by the boundary problem for linear systems of differential equations of mixed type. The simplest example of a classical system of equations that falls within the field of our consideration is the system of equations of mixed type: $$D_tu_1-{\rm sign}D_xu_2-\varepsilon u_2=f_1, \quad D_tu_2+D_xu_1+\varepsilon u_1=f_2$$ Elliptic for $t>0$ and hyperbolic for $t0$.
Keywords: boundary-value problems, the spectrum of the operator of the system of differential equations, periodic problems, the system of equations of mixed type, basis. 
@article{VSGTU_2017_21_4_a1,
     author = {D. V. Kornienko},
     title = {On a spectral problem for a system of~differential equations of mixed type},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {611--632},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2017_21_4_a1/}
}
TY  - JOUR
AU  - D. V. Kornienko
TI  - On a spectral problem for a system of~differential equations of mixed type
JO  - Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
PY  - 2017
SP  - 611
EP  - 632
VL  - 21
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VSGTU_2017_21_4_a1/
LA  - ru
ID  - VSGTU_2017_21_4_a1
ER  - 
%0 Journal Article
%A D. V. Kornienko
%T On a spectral problem for a system of~differential equations of mixed type
%J Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
%D 2017
%P 611-632
%V 21
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VSGTU_2017_21_4_a1/
%G ru
%F VSGTU_2017_21_4_a1
D. V. Kornienko. On a spectral problem for a system of~differential equations of mixed type. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 21 (2017) no. 4, pp. 611-632. http://geodesic.mathdoc.fr/item/VSGTU_2017_21_4_a1/

[1] Dezin A. A., Obshchie voprosy teorii granichnykh zadach [General questions of the theory of boundary value problems], Nauka, Moscow, 1980, 208 pp. (In Russian) | MR | Zbl

[2] Kornienko D. V., “On the spectrum of the Dirichlet problem for systems of operator-differential equations”, Differ. Equ., 42:8 (2006), 1124–1133 | DOI | MR | Zbl

[3] Dezin A. A., Differential operator equations. A method of model operators in the theory of boundary value problems, Tr. Mat. Inst. Steklova, 229, ed. V. S. Vladimirov, E. F. Mishchenko, Nauka, Moscow, 2000, 176 pp. (In Russian)

[4] Romanko V. K., “Mixed boundary value problems for a system of equations”, Dokl. Akad. Nauk SSSR, 286:1 (1986), 47–50 (In Russian) | MR | Zbl

[5] Dunford N., Schwartz J. T., Linear operators. Part II. Spectral theory. Selfadjoint operators in Hilbert space. Reprint of the 1963 original, Wiley Classics Library. A Wiley-Interscience Publication, John Wiley Sons, Inc., New York, 1988, i–x, 859–1923 pp. | MR

[6] Moiseev E. I., “The basis property for systems of sines and cosines”, Dokl. Akad. Nauk SSSR, 275:4 (1984), 794–798 (In Russian) | MR | Zbl

[7] Moiseev E. I., “On the basis property of a system of sines”, Differ. Uravn., 23:1 (1987), 177–179 (In Russian) | MR | Zbl

[8] Moiseev E. I., Uravneniia smeshannogo tipa so spektral'nym parametrom [Mixed type equations with spectral parameter], Moscow State Univ., Moscow, 1988, 150 pp. (In Russian) | MR

[9] Moiseev E. I., Some questions in spectral theory of the mixed type equations, Dr. Phys. Math. Sci. Thesis, Moscow State Univ., Moscow, 1979 (In Russian)

[10] Romanko V. K., “Eigenvalues of boundary value problems for certain equations which change type”, Differ. Uravn., 19:10 (1983), 1759–1764 (In Russian) | MR | Zbl

[11] Kalmenov T. Sh., Regular boundary-value problems and range for equations of hyperbolic and mixed types, Dr. Phys. Math. Sci. Thesis, Moscow State Univ., Moscow, 1982 (In Russian)

[12] Soldatov A. P., “Riemann–Hilbert problems for the Lavrent'ev–Bitsadze system in a mixed domain with a characteristic part of the boundary”, Differ. Equ., 38:12 (2002), 1753–1763 | DOI | MR | Zbl

[13] Kaczmarz S., Steinhaus H., Theorie der Orthogonalreihen, Monografie Matematyczne, 6, Chelsea Publ., New York, 1951, viii+296 pp. | Zbl

[14] Kornienko D. V., “On a spectral problem for two hyperbolic systems of equations”, Differ. Equ., 42:1 (2006), 101–111 | DOI | MR | Zbl