This paper deals with the direct boundary integral approach for solving the Dirichlet and Neumann problem for Laplace’s equation on domains with inward and outward peaks. These integral equations are obtained by direct reduction of representation formulas of harmonic functions to the piecewise smooth boundary and involve in the case of Dirichlet’s problem the single and for Neumann’s problem the double layer potential on curves with peaks.
To deal with the strong singularities of solutions the authors introduce suitable Sobolev type spaces with weight and study solvability and uniqueness problems in these spaces. Using deep results on conformal mapping and previously obtained results for the indirect approach and new mapping properties of the integral operators they describe the solutions of the homogeneous equations and prove uniqueness results for given boundary data from Sobolev type spaces.