On the singularities of the gradient of the solution to the Dirichlet–Neumann problem outside a plane cut

The problem considered in this paper deals with the Laplace equation outside a cut of sufficiently arbitrary form. The Dirichlet condition is given on one side of the cut and the Neumann condition on the other. Using the integral representation for the solution, we obtain explicit asymptotic formulas describing the singularity of the gradient of the solution at the edges of the cut. We discuss the effect of the disappearance of the singularity.