Uniqueness and existence of the solution to the static London-Maxwell equations in two dimensions

A theorem is given for the existence and uniqueness of the time-independent solution of the exterior-interior problem associated with determining the distribution of superconducting current, according to the London model, in simply- or doubly-connected two-dimensional regions. The proof of the corresponding theorem in three dimensions is outlined. A discussion is also given of the relationship between two "different" solutions which already exist for rectangular regions.