The harmonic problem in the plane with cuts under boundary conditions of the third kind

Abstract

The Dirichlet and Neumann problems for harmonic functions in the plane with cuts of any shape were studied in [1] and [2], respectively; the Dirichlet problem was studied by the method of singular integral equations of the first kind, while the Neumann problem was studied by the method of hypersingular integral equations of the first kind. The problem with skew derivatives, whose special case is the Neumann problem, was studied for harmonic functions in the plane with cuts of any shape in [3, 4], where it was reduced to a uniquely solvable Fredholm integral equation of the second kind. Another generalization of the Neumann problem was considered in [5]. The mixed problem for harmonic functions with the Dirichlet condition on a part of the cuts and the condition with a skew derivative, which generalizes the Neumann condition, on the remaining cuts was studied in [6] by using a reduction to a uniquely solvable Fredholm integral-algebraic equation of the second kind in a Banach space. A special case of this problem [6] is the Dirichlet problem in the plane with cuts of any shape. The Dirichlet and Neumann problems in the plane with cuts for the Helmholtz equation were studied in [7, 8], where they were reduced to uniquely solvable Fredholm equations of the second kind. In this paper, we study the boundary value problem for harmonic functions in the plane with cuts under third-kind boundary conditions on both sides of the cuts. We prove the existence and uniqueness of a solution to this problem. We also obtain an integral representation for the solution in the form of potentials. The density in the potentials are determined from a uniquely solvable system of integral equations. The singularities of the gradient of the solution at the endpoints of the cuts are also studied.