  We consider the axial compression of a thin elastic cylinder placed about a
hard cylindrical core. Treating the core as an obstacle, we prove upper and
lower bounds on the minimum energy of the cylinder that depend on its relative
thickness and the magnitude of axial compression. We focus exclusively on the
setting where the radius of the core is greater than or equal to the natural
radius of the cylinder. We consider two cases: the "large mandrel" case, where
the radius of the core exceeds that of the cylinder, and the "neutral mandrel"
case, where the radii of the core and cylinder are the same. In the large
mandrel case, our upper and lower bounds match in their scaling with respect to
thickness, compression, and the magnitude of pre-strain induced by the core. We
construct three types of axisymmetric wrinkling patterns whose energy scales as
the minimum in different parameter regimes, corresponding to the presence of
many wrinkles, few wrinkles, or no wrinkles at all. In the neutral mandrel
case, our upper and lower bounds match in a certain regime in which the
compression is small as compared to the thickness; in this regime, the minimum
energy scales as that of the unbuckled configuration. We achieve these results
for both the von K\'arm\'an-Donnell model and a geometrically nonlinear model
of elasticity.
