09-14
Symmetric Willmore surfaces of revolution satisfying natural boundary conditions
by Bergner, M; Dall'Acqua, A; Fröhlich, S.
Preprint series: 09-14, Preprints
- MSC:
- 49Q10 Optimization of the shape other than minimal surfaces, See also {73K40, 90C90}
- 53C42 Immersions (minimal, prescribed curvature, tight, etc.), See also {49Q05, 49Q10, 53A10, 57R40, 57R42}
- 35J65 Nonlinear boundary value problems for linear elliptic PDE; boundary value problems for nonlinear elliptic PDE
- 34L30 Nonlinear ordinary differential operators
Abstract: 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.
Keywords: Natural boundary conditions, Willmore surfaces of revolution
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