On solvability of exterior boundary value problem with boundary operators of fractional order

Abstract

In this paper, in the class of regular harmonic functions we study the properties of some integro-differential operators, generalizing the operators of fractional differentiation in the sense of Hadamard. These operators convert regular harmonic functions in the same functions and are mutually inverse on regular harmonic functions. In the exterior of the unit ball studied the boundary value problem with the boundary operator fractional order. This problem generalizes the known Neumann problem on the boundary operators of fractional order. We prove a theorem on the existence and uniqueness of the solution of the problem. Obtained an integral representation of the solution of this problem.