Geodesic
On the solvability of some boundary value problems with involution
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 26 (2020) no. 3, pp. 7-16
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This article is devoted to the study of the solvability of some boundary value problems with involution. In the space $ R ^ {n} $, the map $ Sx = -x $ is introduced. Using this mapping, a nonlocal analogue of the Laplace operator is introduced, as well as a boundary operator with an inclined derivative. Boundary-value problems are studied that generalize the well-known problem with an inclined derivative. Theorems on the existence and uniqueness of the solution of the problems under study are proved. In the Helder class, the smoothness of the solution is also studied. Using well-known statements about solutions of a boundary value problem with an inclined derivative for the classical Poisson equation, exact orders of smoothness of a solution to the problem under study are found.
Keywords: involution, nonlocal problem, smoothness, uniqueness.
Mots-clés : nonlocal equation, oblique derivative, Poisson equation, existence 
@article{VSGU_2020_26_3_a0,
     author = {K. Zh. Nazarova and B. Kh. Turmetov and K. I. Usmanov},
     title = {On the solvability of some boundary value problems with involution},
     journal = {Vestnik Samarskogo universiteta. Estestvennonau\v{c}na\^a seri\^a},
     pages = {7--16},
     year = {2020},
     volume = {26},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGU_2020_26_3_a0/}
}
TY  - JOUR
AU  - K. Zh. Nazarova
AU  - B. Kh. Turmetov
AU  - K. I. Usmanov
TI  - On the solvability of some boundary value problems with involution
JO  - Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ
PY  - 2020
SP  - 7
EP  - 16
VL  - 26
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VSGU_2020_26_3_a0/
LA  - ru
ID  - VSGU_2020_26_3_a0
ER  - 
%0 Journal Article
%A K. Zh. Nazarova
%A B. Kh. Turmetov
%A K. I. Usmanov
%T On the solvability of some boundary value problems with involution
%J Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ
%D 2020
%P 7-16
%V 26
%N 3
%U http://geodesic.mathdoc.fr/item/VSGU_2020_26_3_a0/
%G ru
%F VSGU_2020_26_3_a0
K. Zh. Nazarova; B. Kh. Turmetov; K. I. Usmanov. On the solvability of some boundary value problems with involution. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 26 (2020) no. 3, pp. 7-16. http://geodesic.mathdoc.fr/item/VSGU_2020_26_3_a0/

[1] T. Carleman, “Sur la theorie des equations integrales et ses applications”, Verhandlungen des Internationalen Mathematiker-Kongresse Zurich, v. 1, 1932, 132–151 https://studylibfr.com/doc/1054399/sur-la-th

[2] N. Karapetiants, S. Samko, Equations with Involutive Operators, Birkhäuser, Boston, 2001, 427 pp. | DOI | Zbl

[3] G. S. Litvinchuk, Boundary value problems and singular integral equations with shift, Nauka, M., 1977, 448 pp. (In Russ.)

[4] A. A. Andreev, “Analogs of Classical Boundary Value Problems for a Second-Order Differential Equation with Deviating Argument”, Differential Equations, 40 (2004), 1192–1194 | DOI | Zbl

[5] A. A. Andreev, E. N. Ogorodnikov, “On the formulation and substantiation of the well-posedness of the initial boundary value problem for a class of nonlocal degenerate equations of hyperbolic type”, Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, 2006, no. 43, 44–51 (In Russ.) | DOI

[6] A. G. Baskakov, N. B. Uskova, “Fourier method for first order differential equations with involution and groups of operators”, Ufa Mathematical Journal, 10:3 (2018), 11–34 | DOI | Zbl

[7] M. Sh. Burlutskaya, A. P. Khromov, “Classical solution of a mixed problem with involution”, Doklady Mathematics, 82 (2010), 865–868 | DOI | Zbl

[8] M. Sh. Burlutskaya, A. P. Khromov, “Substantiation of Fourier method in mixed problem with involution”, Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics, 11:4 (2011), 3–12 (In Russ.) | DOI

[9] A. V. Linkov, “Substantiation of a method the Fourier for boundary value problems with an involute deviation”, Vestnik of Samara State University. Natural Science Series, 12:2 (1999), 60–65 (In Russ.)

[10] A. V. Bitsadze, A. A. Samarskii, “Some elementary generalizations of linear elliptic boundary value problems”, Dokl. Akad. Nauk SSSR, 185:4 (1969), 739–740 (In Russ.) | Zbl

[11] A. L. Skubachevskii, “Nonclassical boundary value problems. I”, Journal of Mathematical Sciences, 155 (2008), 199 | DOI | Zbl

[12] A. L. Skubachevskii, “Nonclassical boundary value problems. II”, Journal of Mathematical Sciences, 166:4 (2010), 377–561 | DOI | Zbl

[13] V. V. Karachik, A. Sarsenbi, B. Kh. Turmetov, “On the solvability of the main boundary value problems for a nonlocal Poisson equation”, Turkish Journal of Mathematics, 43:3 (2019), 1604–1625 | DOI | Zbl

[14] V. V. Karachik, B. Kh. Turmetov, “Solvability of one nonlocal Dirichlet problem for the Poisson equation”, Novi Sad Journal of Mathematics, 50:1 (2020), 67–88 | DOI | Zbl

[15] D. Przeworska-Rolewicz, “Some boundary value problems with transformed argument”, Commentationes Mathematicae, 17 (1974), 451–457 http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.ojs-doi10_14708_cm_v17i2_5790/c/5790-5331.pdf | Zbl

[16] V. V. Karachik, B. Kh. Turmetov, “On solvability of some nonlocal boundary value problems for polyharmonic equation”, Kazakh Mathematical Journal, 19:1 (2019), 39–49 | Zbl

[17] Sh. A. Alimov, “On a problem with an oblique derivative”, Differential Equations, 17:10 (1981), 1738–1751 (In Russ.)

[18] Sh. A. Alimov, “On a boundary value problem”, Doklady Mathematics, 252:5 (1980), 1033–1034 (In Russ.) | Zbl