"Gradient-free" diffuse approximations of the Willmore functional and Willmore flow

We introduce new diffuse approximations of the Willmore functional and the Willmore flow. They are based on a corresponding approximation of the perimeter that has been studied by Amstutz-van Goethem [{\em Interfaces Free Bound. 14 (2012)}]. We identify the candidate for the $\mathrm{\Gamma }$–convergence, prove the $\mathrm{\Gamma }$–limsup statement and justify the convergence to the Willmore flow by an asymptotic expansion. Furthermore, we present numerical simulations that are based on the new approximation.