On Solvability of a Boundary Value Problem for Singular Integral Equations

2003 ◽  
Vol 26 (2) ◽  
pp. 235-244 ◽  
Author(s):  
Nguyen Van Mau ◽  
Nguyen Tan Hoa
Keyword(s):  
Boundary Value Problem ◽  
Integral Equations ◽  
Singular Integral ◽  
Boundary Value ◽  
Singular Integral Equations

scholarly journals Localized boundary-domain singular integral equations of the Robin type problem for self-adjoint second-order strongly elliptic PDE systems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Otar Chkadua ◽  
Sergey Mikhailov ◽  
David Natroshvili
Keyword(s):  
Boundary Value Problem ◽  
Integral Equations ◽  
Singular Integral ◽  
Boundary Value ◽  
Singular Integral Equations ◽  
Representation Formula ◽  
Potential Method ◽  
Factorization Method ◽  
Variable Coefficients ◽  
Second Order

AbstractThe paper deals with the three-dimensional Robin type boundary-value problem (BVP) for a second-order strongly elliptic system of partial differential equations in the divergence form with variable coefficients. The problem is studied by the localized parametrix based potential method. By using Green’s representation formula and properties of the localized layer and volume potentials, the BVP under consideration is reduced to the a system of localized boundary-domain singular integral equations (LBDSIE). The equivalence between the original boundary value problem and the corresponding LBDSIE system is established. The matrix operator generated by the LBDSIE system belongs to the Boutet de Monvel algebra. With the help of the Vishik–Eskin theory based on the Wiener–Hopf factorization method, the Fredholm properties of the corresponding localized boundary-domain singular integral operator are investigated and its invertibility in appropriate function spaces is proved.

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

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.

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

2017 ◽  
Vol 20 (10) ◽  
pp. 17-25
Author(s):  
S.A. Aldashev
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Singular Integral ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Singular Integral Equations ◽  
Cylindrical Domain ◽  
Second Order Elliptic Equations ◽  
Poincaré Problem ◽  
Well Posed

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.

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