Boundary-value problems for general elliptic systems in the plane

We consider boundary-value problems for general elliptic systems in a bounded multiply connected domain with a smooth boundary in the plane. This includes boundary-value problems for homogeneous elliptic systems and for systems elliptic in the sense of Petrovskii or Douglis-Nirenberg. The systems under consideration need not be properly elliptic. We obtain necessary and sufficient conditions for the system to be Noetherian and give index formulae with applications to geometry and hydrodynamics.