  We prove that the Inverse Mean Curvature Flow of a non-star-shaped,
mean-convex embedded sphere in $\mathbb{R}^{n+1}$ with symmetry about an axis
and sufficiently long, thick necks exists for all time and homothetically
converges to a round sphere as $t \rightarrow \infty$. Our approach is based on
a localized version of the parabolic maximum principle.
  We also present two applications of this result. The first is an extension of
the Minkowski inequality to the corresponding non-star-shaped, mean-convex
domains in $\mathbb{R}^{n+1}$. The second is a connection between IMCF and
minimal surface theory. Based on previous work by Meeks and Yau and using
foliations by IMCF, we establish embeddedness of the solution to Plateau's
problem and a finiteness property of stable immersed minimal disks for certain
Jordan curves in $\mathbb{R}^{3}$.
