Elliott, Charles M. and Fritz, Hans and Hobbs, Graham (2017) Small deformations of Helfrich energy minimising surfaces with applications to biomembranes. MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES, 27 (8). pp. 1547-1586. ISSN 0218-2025, 1793-6314
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In this paper, we introduce a mathematical model for small deformations induced by external forces of closed surfaces that are minimisers of Helfrich-type energies. Our model is suitable for the study of deformations of cell membranes induced by the cytoskeleton. We describe the deformation of the surface as a graph over the undeformed surface. A new Lagrangian and the associated Euler-Lagrange equations for the height function of the graph are derived. This is the natural generalisation of the well-known linearisation in the Monge gauge for initially flat surfaces. We discuss energy perturbations of point constraints and point forces acting on the surface. We establish existence and uniqueness results for weak solutions on spheres and on tori. Algorithms for the computation of numerical solutions in the general setting are provided. We present numerical examples which highlight the behaviour of the surface deformations in different settings at the end of the paper.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | MEMBRANES; PROTRUSIONS; CURVATURE; DYNAMICS; VESICLES; WILLMORE; THEOREM; DRIVEN; FLOW; Surface deformations; Helfrich energy; point forces; PDEs on surfaces; existence and uniqueness of weak solutions; discretisation; surface finite element method |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 14 Dec 2018 13:16 |
| Last Modified: | 20 Feb 2019 11:08 |
| URI: | https://pred.uni-regensburg.de/id/eprint/1623 |
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