Authors: | Knüttel, Sascha |
Title: | New diffuse approximations of the Willmore energy, the mean curvature flow, and the Willmore flow |
Language (ISO): | en |
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. |
Subject Headings: | Geometric measure theory Phase-field approximations Willmore energy Gamma-convergence De Giorgi type varifold solution for mean curvature flow Blow-up |
Subject Headings (RSWK): | Geometrische Maßtheorie Mittlere Krümmung Krümmungsfluss |
URI: | http://hdl.handle.net/2003/41858 http://dx.doi.org/10.17877/DE290R-23701 |
Issue Date: | 2023 |
Appears in Collections: | AG Biomathematik |
Files in This Item:
File | Description | Size | Format | |
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Dissertation_Knüttel.pdf | 1.41 MB | Adobe PDF | View/Open |
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