Abstract
We consider the Willmore functional on graphs, with an additional penalization of the area where the curvature is nonzero. 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.
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