  We consider a disk-shaped thin elastic sheet bonded to a compliant sphere.
(Our sheet can slip along the sphere; the bonding controls only its normal
displacement.) If the bonding is stiff (but not too stiff), the geometry of the
sphere makes the sheet wrinkle to avoid azimuthal compression. The total energy
of this system is the elastic energy of the sheet plus a (Winkler-type)
substrate energy. Treating the thickness of the sheet $h$ as a small parameter,
we determine the leading-order behavior of the energy as $h$ tends to zero, and
give (almost matching) upper and lower bounds for the next-order correction.
Our analysis of the leading-order behavior determines the macroscopic
deformation of the sheet; in particular it determines the extent of the
wrinkled region, and predicts the (nontrivial) radial strain of the sheet. The
leading-order behavior also provides insight about the length scale of the
wrinkling, showing that it must be approximately independent of the distance
$r$ from the center of the sheet (so that the number of wrinkles must increase
with $r$). Our results on the next-order correction provide insight about how
the wrinkling pattern should vary with $r$. Roughly speaking, they suggest that
the length scale of wrinkling should {\it not} be exactly constant -- rather,
it should vary a bit, so that the number of wrinkles at radius $r$ can be
approximately piecewise constant in its dependence on $r$, taking values that
are integer multiples of $h^{-\alpha}$ with $\alpha \approx 1/2$.
