Integral-equation method in the mixed oblique derivative problem for harmonic functions outside cuts on the plane

The mixed problem for the Laplace equation outside cuts on the plane is considered. As boundary conditions, the value of the desired function on one side of each of the cuts and the value of its oblique derivative on the other side are prescribed. This problem generalizes the mixed Dirichlet-Neumann problem. By using the potential method, the problem reduces to a uniquely solvable Fredholm integral equation of the second kind. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 115–135, 2006.