New diffuse approximations of the Willmore energy, the mean curvature flow, and the Willmore flow
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Date
2023
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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.
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Keywords
Geometric measure theory, Phase-field approximations, Willmore energy, Gamma-convergence, De Giorgi type varifold solution for mean curvature flow, Blow-up