Variational procedures for a class of exterior interface problems

AbstractThe problem under consideration is that of determining a function which is a solution of the Helmholtz equation in a planar region exterior to a simple closed curve and of an inhomogeneous Helmholtz equation inside the curve. Jump conditions on the function and its normal derivative across the cruve are given. The problem is first transformed into one involving the inner region only with a boundary condition which is non-local. This means that the solution at a point on the boundary is a functional of its values elsewhere. This second problem is further transformed into a variational form with all boundary conditions natural. It is shown that the variational problem has a solution. Finite dimensional approximate problems are defined and they are shown to have solutions converging to the solution of the variational problem