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

Gianni Manno, Giovanni De Matteis

    Research output: Contribution to journalArticlepeer-review

    6 Citations (Scopus)

    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.
    Original languageEnglish
    Pages (from-to)453-481
    JournalCommunications on Pure and Applied Analysis
    Volume13
    Issue number1
    DOIs
    Publication statusPublished - Jan 2014

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