  This paper shows the unique solvability of elliptic problems associated with
two-phase incompressible flows, which are governed by the two-phase
Navier-Stokes equations, in unbounded domains such as the whole space separated
by a compact interface and the whole space separated by a non-compact
interface. As a by-product of the unique solvability of elliptic problems, we
obtain the Helmholtz-Weyl decomposition for two-phase incompressible flows.
