Corner singularities of solutions to Δ ±1/2 u=f in two dimensions

Abstract

Let Ω be a compact submanifold of a two dimensional Riemannian manifold X having a boundary ∂Ω which is piecewise smooth without sharp peaks. We study the pseudo-differential equations Op(|ξ|^±1)u=f in Ω in appropriate function spaces. Existence and uniqueness of solutions are proven. Moreover, it is shown that they possess an asymptotic expansion near each angular point provided the right hand side does. By a Mellin operator calculus we get information about the exponent of the asymptotics. A numerical algorithm is given which yields the leading singularity. Applications to the screen problem in electrodynamics, to the charge density problem in electrostatics and to the crack problem in elasticity are given.