On a method to subtract off a singularity at a corner for the Dirichlet or Neumann problem

Let D D be a plane domain partly bounded by two line segments which meet at the origin and form there an interior angle π α > 0 π α > 0 . Let U ( x , y ) U(x,y) be a solution in D D of Poisson’s equation such that either U U or ∂ U / ∂ n ∂ U/∂ n (the normal derivative) takes prescribed values on the boundary segments. Let U ( x , y ) U(x,y) be sufficiently smooth away from the corner and bounded at the corner. Then for each positive integer N N there exists a function V N ( x , y ) {V_N}(x,y) which satisfies a related Poisson equation and which satisfies related boundary conditions such that U − V N U - {V_N} is N N -times continuously differentiable at the corner. If 1 / α 1/α is an integer