Analysis of singularities of gradient of a solution in the boundary value problem for the Laplace equation outside cut in a plane with mixed boundary condition

The mixed problem for the Laplace equation outside cut in a plane is considered. The Dirichlet condition is posed on one side of the cut and the skew derivative condition is posed on the other side. The gradient of the solution at the ends of the cut is studied in details. The asymptotic formulas are obtained. The effect of disappearing of singularity is discussed.