The mixed harmonic problem in a bounded cracked domain with Dirichlet condition on cracks

The mixed Dirichlet-Neumann problem for the Laplace equation in a bounded connected plane domain with cuts (cracks) is studied. The Neumann condition is given on closed curves making up the boundary of a domain, while the Dirichlet condition is specified on the cuts. The existence of a classical solution is proved by potential theory and boundary integral equation method. The integral representation for a solution is obtained in the form of potentials. The density in potentials satisfies the uniquely solvable Fredholm integral equation of the second kind and index zero. Singularities of the gradient of the solution at the tips of cuts are investigated.