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Symmetric Willmore surfaces of revolution satisfying natural boundary conditions

by Bergner, M, Dall'Acqua, A, Fröhlich, S..

Series: 2009-14, Preprints

49Q10 Optimization of shapes other than minimal surfaces
53C42 Immersions (minimal, prescribed curvature, tight, etc.)
35J65 Nonlinear boundary value problems for linear elliptic equations
34L30 Nonlinear ordinary differential operators

We consider the Willmore-type functional
W_{\gamma}(\Gamma):= \int_{Gamma} H^2 \; dA -gamma \int_{Gamma} K \; dA,
where H and K denote mean and Gaussian curvature of a surface Gamma, and gamma \in [0,1] is a real parameter.
Using direct methods of the calculus of variations, we prove existence of surfaces of revolution generated by symmetric graphs which are solutions of the Euler-Lagrange equation corresponding to W_{gamma} and which satisfy the following boundary conditions: the height at the boundary is prescribed, and the second boundary condition is the natural one when considering critical points where only the position at the boundary is fixed. In the particular case gamma=0 the boundary conditions are arbitrary positive height alpha and zero mean curvature.

Natural boundary conditions, Willmore surfaces of revolution