Existence And Uniqueness Of The Solution of Laplace’s Equation From A Model Of Magnetic Recording∗

A variational formulation is given for the solution of Laplace’s equation within an infinitely deep rectagular gap due to a source charge existing in a half space outside the gap. The problem arises in the modeling of the magnetic recording process. Ad hoc Fourier based solutions and finite element formulations are used by engineers to approximate solutions of the problem. However, despite the rel-atively simple equation and geometry, the appropriate variational formulation of the problem has never been given. The variational form is essential to confirm the veracity of numerical schemes applied to the problems as well as to provide the basis for numerical analysis of the schemes. The formulation will require two Dirichlet to Neumann maps. One map characterizes the interface from the half space to the gap. The second map truncates the solution domain within the gap itself. The maps will allow the problem to be formulated on a finite rectangular domain while properly representing the influence of the source charge which is external to the solution domain. The paper will offer the proper formulation of these boundary conditions and provide a proof of the existence and uniqueness of the solution to the variational problem.