Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations

Manno, Gianni and De Matteis, Giovanni (2014) Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure and Applied Analysis, 13 (1). pp. 453-481. ISSN 1534-0392

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Official URL: http://dx.doi.org/10.3934/cpaa.2014.13.453

Abstract

We compute the Lie symmetry algebra of the equation of Helfrich surfaces and we show that it is the algebra of conformal vector fields of R2. We also show that in the particular case of the Willmore surfaces we have to add the homothety vector field of R3 to the aforementioned algebra. We prove that a Helfrich surface that is invariant w.r.t. a conformal symmetry is a helicoid and that all such surface solutions satisfy one and the same system of ordinary differential equations obtained by symmetry reduction. We also show that for the Willmore surface shape equation the symmetry reduction leads to two systems of ODEs. Then we construct explicit solutions in the case of revolution surfaces. The results obtained can be extended to the study of PDE problems in 2 spatial dimensions admitting conformal Lie symmetries.

Item Type: Article
Uncontrolled Keywords: Symmetries of PDEs, invariant solutions, symmetry reduction, conformal vector fields, Willmore surfaces, Helfrich-Canham bending energy, vesicle shape, membrane systems
Subjects: G100 Mathematics
Department: Faculties > Engineering and Environment > Mathematics, Physics and Electrical Engineering
Depositing User: Ay Okpokam
Date Deposited: 24 Oct 2013 11:22
Last Modified: 10 Aug 2015 11:06
URI: http://nrl.northumbria.ac.uk/id/eprint/14361

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