WELL-POSEDNESS OF POINCARE PROBLEM IN THE CYLINDRICAL DOMAIN FOR A CLASS OF MULTI-DIMENSIONAL ELLIPTIC EQUATIONS

The boundary value problems for second order elliptic equations in domains with edges are well studied. For elliptic equations, boundary-value problems on the plane were shown to be well posed by using methods from the theory of analytic functions of complex variable. When the number of independent variables is greater than two, diﬃculties of fundamental nature arise. Highly attractive and convenient method of singular integral equations can hardly be applied, because the theory of multidimensional singular integral equations is still incomplete. In this paper with the help of the method suggested by the author, the unique solvability is shown and explicit form of classical solution of Poincare problem in a cylindrical domain for a one class of multidimensional elliptic equations is received.

Download Full-text

Well-posed Boundary Value Problems for New Classes of Singular Integral Equations in Cylindrical Domains

Springer Proceedings in Mathematics & Statistics - Analysis and Partial Differential Equations: Perspectives from Developing Countries ◽

10.1007/978-3-030-05657-5_14 ◽

2019 ◽

pp. 219-241

Author(s):

Nusrat Rajabov

Keyword(s):

Boundary Value Problems ◽

Integral Equations ◽

Singular Integral ◽

Boundary Value ◽

Singular Integral Equations ◽

Cylindrical Domains ◽

Well Posed

Download Full-text

A CRITERION FOR THE UNIQUE SOLVABILITY OF THE POINCARE SPECTRAL PROBLEM IN A CYLINDRICAL DOMAIN FOR ONE CLASS OF MULTIDIMENSIONAL ELLIPTIC EQUATIONS

Journal of Numerical and Applied Mathematics ◽

10.17721/2706-9699.2019.3.01 ◽

2019 ◽

pp. 5-14

Author(s):

S. A. Aldashev

Keyword(s):

Integral Equations ◽

Singular Integral ◽

Unique Solvability ◽

Elliptic Equations ◽

Singular Integral Equations ◽

Spherical Functions ◽

Cylindrical Domain ◽

Two Dimensional ◽

Single Class ◽

Poincaré Problem

Two-dimensional spectral problems for elliptic equations are well studied, and their multidimensional analogs, as far as the author knows, are little studied. This is due to the fact that in the case of three or more independent variables there are difficulties of a fundamental nature, since the method of singular integral equations, which is very attractive and convenient, used for two-dimensional problems, cannot be used here because of the lack of any complete theory of multidimensional singular integral equations. The theory of multidimensional spherical functions, on the contrary, has been adequately and fully studied. In the cylindrical domain of Euclidean space, for a single class of multidimensional elliptic equations, the spectral Poincare problem. The solution is sought in the form of an expansion in multidimensional spherical functions. The existence and uniqueness theorems of the solution are proved. Conditions for unique solvability of the problem are obtained, which essentially depend on the height of the cylinder.

Download Full-text

CORRECTNESS OF THE LOCAL BOUNDARY VALUE PROBLEM IN A CYLINDRICAL DOMAIN FOR ONE CLASS OF MULTIDIMENSIONAL ELLIPTIC EQUATIONS

Vestnik of Samara University Natural Science Series ◽

10.18287/2541-7525-2016-22-1-2-7-17 ◽

2017 ◽

Vol 22 (1-2) ◽

pp. 7-17

Author(s):

S. A. Aldashev

Keyword(s):

Boundary Value Problem ◽

Boundary Value Problems ◽

Integral Equations ◽

Singular Integral ◽

Boundary Value ◽

Singular Integral Equations ◽

Classical Solutions ◽

Cylindrical Domain ◽

Local Boundary ◽

Criterion Of Uniqueness

Correctness of boundary value problems in a plane for elliptical equations has been studied properly using the method of the theory of analytic functions. At investigation of analogous problems, when the number of independent variables is more than two, there arise principle difficulties. Quite good and convenient method of singular integral equations has to be abandoned because there is no complete theory of multidimensional singular integral equations. Boundary value problems for second-order elliptical equations in domains with edges have been studied properly earlier. Explicit classical solutions to Dirichlet and Poincare problems in cylindrical domains for one class of multidimensional elliptical equations can be found in the author’s works. In this article,the author proved that the local boundary value problem, which is the generalization of Dirichet and Poincare problem, has only solution. Besides, the criterion of uniqueness of regular solution is obtained.

Download Full-text