#### For citation:

Aldashev S. A. Nonlocal Boundary-Value Problems in the Cylindrical Domain for the Multidimensional Laplace Equation. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2019, vol. 19, iss. 1, pp. 16-23. DOI: 10.18500/1816-9791-2019-19-1-16-23

# Nonlocal Boundary-Value Problems in the Cylindrical Domain for the Multidimensional Laplace Equation

Correct statements of boundary value problems on the plane for elliptic equations by the method of analytic function theory of a complex variable. Investigating similar questions, when the number of independent variablesis greater than two, problems of a fundamental nature arise. Avery attractive and convenient method of singular integral equations loses its validity due to the absence of any complete theory of multidimensional singular integral equations. The author has previously studied local boundary value problems in a cylindrical domain for multidimensional elliptic equations. As far as we know, non-local boundary-value problems for these equations have not been investigated. This paper uses the method proposed in the author’s earlier works, shows unique solvabilities, and gives explicit forms of classical solutions of nonlocal boundary-value problems in the cylindrical domain for the multidimensional Laplace equation, which are generalizations of the mixed problem, the Dirichlet and Poincare problems. A criterion for uniqueness is also obtained for regular solutions of these problems is also obtained.

- Aldashev S. A. The correctness of the Dirichlet problem in cylindrical domain for the multidimensional Laplace equation. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2012, vol. 12, iss. 3, pp. 3–7 (in Russian).
- Aldashev S. A. Correctness of Poincare’s problem in a cylindrical region for Laplace’s multi-measured equation. News of the National Academy of Sciences of the Republic of Kazakhstan. Physical-mathematical Series, 2014, no. 3 (295), pp. 62–67 (in Russian).
- Aldashev S. A. The correctness of the Dirichlet problem in cylindrical domain for a class of multidimensional elliptic equations. Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform., 2012, vol. 12, iss. 1, pp. 7–13 (in Russian).
- Aldashev S. A. Well-Posedness of Poincare Problem in the Cylindrical Domain for a Class of Multi-Dimensional Elliptic Equations. Vestnik of Samara University. Natural Science Series, 2014, no. 10 (121), pp. 17–25 (in Russian).
- Aldashev S. A. The correctness of the local boundary value problem in cylindrical domain for the multidimensional Laplace equation. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2015, vol. 15, iss. 4, pp. 365–371 (in Russian). DOI: https:.doi.org/18500/18169791-2015-15-4-365-371
- Aldashev S. A. The correctness of the local boundary value problem in cylindrical domain for a class of multidimensional elliptic equations. Vestnik of Samara University. Natural Science Series, 2016, vol. 12 (123), iss. 1–2, pp. 7–17 (in Russian).
- Mikhlin S. G. Mnogomernye singulyarnye integraly i integral’nye uravneniya [Multidimensional singular integrals and Integral equations]. Moscow, Fizmatgiz, 1962. 254 p. (in Russian).
- Kamke E. Spravochnik po obyknovennym differencial’nym uravneniyam [Handbook of Ordinary Differential Equations]. Moscow, Nauka, 1965. 703 p. (in Russian).
- Bateman G., Erdei A. Vysshie transcendentnye funkcii [Higher transcendental functions]. Vol. 2. Moscow, Nauka, 1974. 295 p. (in Russian).
- Tikhonov A. N., Samarskii A. A. Uravneniya matematicheskoj fiziki[Equations of mathematical physics]. Moscow, Nauka, 1966. 724 p. (i n Russian).

- 841 reads