A consistency condition for the vector potential in multiply-connected   domains
  A classical problem in electromagnetics concerns the representation of the electric and magnetic fields in the low-frequency or static regime, where topology plays a fundamental role. For multiply connected conductors, at zero frequency the standard boundary conditions on the tangential components of the magnetic field do not uniquely determine the vector potential. We describe a (gauge-invariant) consistency condition that overcomes this non-uniqueness and resolves a longstanding difficulty in inverting the magnetic field integral equation. 
Charles L. Epstein, Zydrunas Gimbutas, Leslie Greengard, Andreas   Kl\"ockner, Michael O'Neil
physics.class-ph math.NA