On a $\mathrm{\Gamma }$-limit of Willmore functionals with additional curvature penalization term

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 $\mathrm{\infty }$ and rescaling suitably, we derive the limit functional in the sense of $\mathrm{\Gamma }$-convergence.