A Navier boundary value problem for Willmore surfaces of revolution

by Deckelnick, K.; Grunau, H.-Ch.


Preprint series: 09-02, Preprints

53C42 Immersions (minimal, prescribed curvature, tight, etc.), See also {49Q05, 49Q10, 53A10, 57R40, 57R42}
34B15 Nonlinear boundary value problems
35J65 Nonlinear boundary value problems for linear elliptic PDE; boundary value problems for nonlinear elliptic PDE
35B32 Bifurcation, See also {58F14}


Abstract: We study a boundary value problem for Willmore surfaces of revolution, where the position and the mean curvature H=0 are prescribed as boundary data. The latter is a natural datum when considering critical points of the Willmore functional in classes of functions where only the position at the boundary is fixed. For specific boundary positions, catenoids and a suitable part of the Clifford torus are explicit solutions. Numerical experiments, however, suggest a much richer bifurcation diagram. In the present paper we verify analytically some properties of the expected bifurcation diagram. Furthermore, we present a finite element method which allows the calculation of critical points of the Willmore functional irrespective of their stability properties.

Keywords: Willmore surfaces, natural boundary value problem, surfaces of revolution, bifurcation, Clifford torus, Newton\'s method

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Letzte Änderung: 01.03.2018 - Ansprechpartner:

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