Geodesic
Nonlocal boundary-value problems in the cylindrical domain for the multidimensional Laplace equation
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 19 (2019) no. 1, pp. 16-23

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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 variables is greater than two, problems of a fundamental nature arise. A very 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. 
@article{ISU_2019_19_1_a1,
     author = {S. A. Aldashev},
     title = {Nonlocal boundary-value problems in the cylindrical domain for the multidimensional {Laplace} equation},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {16--23},
     publisher = {mathdoc},
     volume = {19},
     number = {1},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2019_19_1_a1/}
}
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S. A. Aldashev. Nonlocal boundary-value problems in the cylindrical domain for the multidimensional Laplace equation. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 19 (2019) no. 1, pp. 16-23. http://geodesic.mathdoc.fr/item/ISU_2019_19_1_a1/