New diffuse approximations of the Willmore energy, the mean curvature flow, and the Willmore flow
| dc.contributor.advisor | Röger, Matthias | |
| dc.contributor.author | Knüttel, Sascha | |
| dc.contributor.referee | Schweizer, Ben | |
| dc.date.accepted | 2023-06-12 | |
| dc.date.accessioned | 2023-06-29T14:59:45Z | |
| dc.date.available | 2023-06-29T14:59:45Z | |
| dc.date.issued | 2023 | |
| dc.description.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. | de |
| dc.identifier.uri | http://hdl.handle.net/2003/41858 | |
| dc.identifier.uri | http://dx.doi.org/10.17877/DE290R-23701 | |
| dc.language.iso | en | de |
| dc.subject | Geometric measure theory | de |
| dc.subject | Phase-field approximations | de |
| dc.subject | Willmore energy | de |
| dc.subject | Gamma-convergence | de |
| dc.subject | De Giorgi type varifold solution for mean curvature flow | de |
| dc.subject | Blow-up | de |
| dc.subject.rswk | Geometrische Maßtheorie | de |
| dc.subject.rswk | Mittlere Krümmung | de |
| dc.subject.rswk | Krümmungsfluss | de |
| dc.title | New diffuse approximations of the Willmore energy, the mean curvature flow, and the Willmore flow | de |
| dc.type | Text | de |
| dc.type.publicationtype | PhDThesis | de |
| dcterms.accessRights | open access | |
| eldorado.secondarypublication | false | de |