Approximation of a bidimensional problem with skin effect

The problem we are dealing with takes its origin in the electromagnetic shaping of melted metal. High frequency alternating currents are used in the process. They are running essentially on the conductor surface, so it implies skin effect since the electric and magnetic fields are strongly varying in the neighborhood of the conductor surface. We approach this phenomenon of thin layer by introducing a closed subspace Mα. Making use of asymptotic development techniques, we transform the problem under consideration into a series of singular integral equations on the boundary.