Generalized minimizing movements for the varifold Canham–Helfrich flow-Reference-Cited by-同舟云学术

Generalized minimizing movements for the varifold Canham–Helfrich flow

Published:2024-01-13 Issue:0 Volume:0 Page:
ISSN:1864-8258
Container-title:Advances in Calculus of Variations
language:en
Short-container-title:

Author:

Brazda Katharina¹,Kružík Martin²^ORCID,Stefanelli Ulisse³^ORCID

Affiliation:

1. Faculty of Mathematics , University of Vienna , Oskar-Morgenstern-Platz 1, A-1090; and Vienna School of Mathematics (VSM) , Vienna , Austria

2. Czech Academy of Sciences , Institute of Information Theory and Automation , Pod vodárenskou veží 4, CZ-182 00 , Prague 8; and Faculty of Civil Engineering, Czech Technical University, Thákurova 7, CZ-166 29, Prague 6 , Czech Republic

3. Faculty of Mathematics , University of Vienna , Oskar-Morgenstern-Platz 1; and Vienna Research Platform on Accelerating Photoreaction Discovery, University of Vienna, Währingerstrasse 17, A-1090 Vienna , Austria ; and Istituto di Matematica Applicata e Tecnologie Informatiche E. Magenes, via Ferrata 1, 27100 Pavia, Italy

Abstract

Abstract The gradient flow of the Canham–Helfrich functional is tackled via the generalized minimizing movements approach. We prove the existence of solutions in Wasserstein spaces of varifolds, as well as upper and lower diameter bounds. In the more regular setting of multiply covered C 1 , 1

{C^{1,1}}

surfaces, we provide a Li–Yau-type estimate for the Canham–Helfrich energy and prove the conservation of multiplicity along the evolution.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,Analysis

Link

https://www.degruyter.com/document/doi/10.1515/acv-2022-0056/pdf

Reference58 articles.

1. W. K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417–491.

2. L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 19 (1995), 191–246.

3. L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed., Lect. Math. ETH Zürich, Birkhäuser, Basel, 2008.

4. J. W. Barrett, H. Garcke and R. Nürnberg, Parametric approximation of Willmore flow and related geometric evolution equations, SIAM J. Sci. Comput. 31 (2008), no. 1, 225–253.

5. J. W. Barrett, H. Garcke and R. Nürnberg, Computational parametric Willmore flow with spontaneous curvature and area difference elasticity effects, SIAM J. Numer. Anal. 54 (2016), no. 3, 1732–1762.