The Neumann Problem in a 2-D Exterior Domain with Cuts and Singularities at the Tips

The Neumann problem for the harmonic functions in an exterior connected plane region with cuts is studied. The problem is considered with different conditions at infinity, which lead to different theorems on uniqueness and solvability. The existence of a classical solution is proved by potential theory. The problem is reduced to a Fredholm equation of the second kind, which is uniquely solvable. Explicit formulas for singularities of a gradient of the solution at the tips of the cuts are obtained. The results of the paper can be used to model the flow of an ideal fluid over several obstacles, including wings.