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

Giovanni De Matteis; Gianni Manno

doi:10.3934/cpaa.2014.13.453

Article Contents

2014, Volume 13, Issue 1: 453-481. Doi: 10.3934/cpaa.2014.13.453

This issue Previous Article Geometric conditions for the existence of a rolling without twisting or slipping Next Article

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

Giovanni De Matteis^1, and
Gianni Manno^2,

1.
Department of Mathematics and Information Sciences, Northumbria University, Pandon Building, Camden Street, Newcastle upon Tyne, NE2 1XE
2.
INdAM-COFUND Marie Curie Fellow, Mathematisches Institut, Friedrich-Schiller-Universität, Jena, 07737

Received: April 2013

Revised: April 2013

Published: January 2014

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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 $R^2$. We also show that in the particular case of the Willmore surfaces we have to add the homothety vector field of $R^3$ 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.

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Mathematics Subject Classification: 35B06, 35B07, 35Q92, 53A05, 58A20.

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