New diffuse approximations of the Willmore energy, the mean curvature flow, and the Willmore flow

Abstract

In this thesis we derive a higher order diffuse approximation of the Willmore energy from contributions by Karali and Katsoulakis [KK07], who studied a diffuse approximation of mean curvature flow. We prove Γ–convergence in smooth limit points for the sum of diffuse perimeter and the higher order diffuse Willmore energy in dimensions 2 and 3. Moreover, we prove the convergence on arbitrary time intervals towards weak solutions of mean curvature flow. We also consider a gradient-free diffuse approximation of the Willmore energy in the sense of Γ–convergence which we derive from a gradient-free diffuse approximation of the perimeter by Amstutz and van Goethem [AVG12]. We prove the lim sup–property for the Γ–convergence towards a multiple of the Willmore energy. In addition, we consider L2-type gradient flows of both diffuse Willmore energies, and give an asymptotic convergence result. Formally these constitute diffuse approximations of mean curvature flow and Willmore flow. In a restricted class of diffuse phase-field evolutions, we prove that these gradient flows convergence towards rescaled mean curvature flow and rescaled Willmore flow, respectively. References [AVG12] S. Amstutz and N. Van Goethem. Topology optimization methods with gradient-free perimeter approximation. Interfaces Free Bound., 14(3):401–430, 2012. [KK07] G. Karali and M. A. Katsoulakis. The role of multiple microscopic mechanisms in cluster interface evolution. J. Differential Equations, 235(2):418–438, 2007.