Integral representations and boundary value problems for a linear third-order elliptic system with an interior supersingular point

Abstract

For a supersingular elliptic system, we find an integral representation of the solution and the corresponding inversion formula depending on the values of roots of the characteristic equation, which is of interest from both the theoretical and the practical viewpoint. All studies are carried out for the case in which the singular point is an interior point of the domain. Note that this case is most complicated. In the resulting integral representations, we clearly single out the singular part of the solutions, which permits analyzing their asymptotic behavior with respect to r. We study the influence of the supersingular point on the solvability of boundary value problems and find a well-posed statement of a number of Dirichlet and Riemann-Hilbert boundary value problems.