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

Energy Bounds for a Compressed Elastic Film on a Substrate