“Gradient-free” diffuse approximations of the Willmore functional and Willmore flow

Article type: Research Article

Authors: Dabrock, Nils | Knüttel, Sascha^{; *} | Röger, Matthias

Affiliations: Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44227 Dortmund, Germany

Correspondence: [*] Corresponding author. E-mail: Sascha.Knuettel@math.tu-dortmund.de; Tel.: +49 231 755 5163; Fax: +49 231 755 5942.

Abstract: 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 [Interfaces Free Bound. 14 (2012)]. We identify the candidate for the Γ-convergence, prove the Γ-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.

Keywords: Willmore flow, phase-field model, diffuse interface, sharp interface limit

DOI: 10.3233/ASY-221810

Journal: Asymptotic Analysis, vol. 133, no. 1-2, pp. 185-226, 2023