ON THE EXISTENCE OF SOLUTIONS OF TWO-DIMENSIONAL HYPERSINGULAR INTEGRAL EQUATION IN THE CLASS OF FUNCTIONS WITH A SINGULARITY ON THE BOUNDARY OF A DOMAIN

A two-dimensional hypersingular integral equation in a convex bounded domain whose boundary is a smooth curve is considered. The equation contains an integral operator with an integral understood in the sense of a finite part according to Hadamard. The question of the existence of solutions having a power singularity in the neighborhood of the boundary of the domain is considered: the solution is sought in a class of functions represented as the ratio of a smooth function and the root of the distance from a point to the boundary. It is proved that when an integral operator with a power polar singularity of the third order acts on a function from the class in which the solution is sought, a function arises that is continuous in the sense of Helder on the entire domain. Further, the existence of solutions of the hypersingular equation having the specified power singularity in the neighborhood of the boundary of the domain is proved. A boundary condition is presented under which such a solution is unique.

integral equation, integral operator, hypersingular integral