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

Abstract:

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