Boundary integral equations of the logarithmic potential theory for domains with peaks
Dirichlet and Neumann boundary value problems are considered for the Laplacian on a plane domain whose boundary is smooth except for a finite number of peaks. Peaks are points where arcs of the boundary meet at zero angle. Following the classical method, solutions are represented in the form of single and double layer potentials. This paper develops the theory of the corresponding boundary integral equations for the densities of the potentials. Fredholm theory is not applicable in the presence of peaks and an alternative approach is used, based on auxiliary boundary value problems. Theorems on unique solvability and asymptotic representations near peaks are presented. For a class of functions with a weak power singularity, the kernels of the integral operators are found.
scientific article; zbMATH DE number 915282