  We consider the Willmore functional on graphs, with an additional
penalization of the area where the curvature is non-zero. Interpreting the
penalization parameter as a Lagrange multiplier, this corresponds to the
Willmore functional with a constraint on the area where the graph is flat.
Sending the penalization parameter to $\infty$ and rescaling suitably, we
derive the limit functional in the sense of $\Gamma$-convergence.
