On the mixed problem for Laplace equation outside cuts, placed along a circumference in a plane

The boundary value problem for harmonic functions outside cuts lying on the arcs of a circumference is considered. The Dirichlet condition is given on one side of each cut and Neumann condition is specified on the other side. The problem is reduced to the Riemann–Hilbert problem for complex analytic function, which is solved in a closed form. An explicit solution of the original problem is obtained.