  We study pattern formation in a compressed elastic film which delaminates
from a substrate. Our key tool is the determination of rigorous upper and lower
bounds on the minimum value of a suitable energy functional. The energy
consists of two parts, describing the two main physical effects. The first part
represents the elastic energy of the film, which is approximated using the von
K\'arm\'an plate theory. The second part represents the fracture or
delamination energy, which is approximated using the Griffith model of
fracture. A simpler model containing the first term alone was previously
studied with similar methods by several authors, assuming that the delaminated
region is fixed. We include the fracture term, transforming the elastic
minimization into a free-boundary problem, and opening the way for patterns
which result from the interplay of elasticity and delamination.
  After rescaling, the energy depends on only two parameters: the rescaled film
thickness, $\sigma$, and a measure of the bonding strength between the film and
substrate, $\gamma$. We prove upper bounds on the minimum energy of the form
$\sigma^a \gamma^b$ and find that there are four different parameter regimes
corresponding to different values of $a$ and $b$ and to different folding
patterns of the film. In some cases the upper bounds are attained by
self-similar folding patterns as observed in experiments. Moreover, for two of
the four parameter regimes we prove matching, optimal lower bounds.
