We consider the Willmore boundary value problem for surfaces of revolution where, as Dirichlet boundary conditions, any symmetric set of position and angle may be prescribed. Using direct methods of the calculus of variations, we prove existence and regularity of minimising solutions. Moreover, we estimate the optimal Willmore energy and prove a number of qualitative properties of these solutions. Besides convexity-related properties we study in particular the limit when the radii of the boundary circles converge to 0, while the "length" of the surfaces of revolution is kept fixed. This singular limit is shown to be the sphere, irrespective of the prescribed boundary angles.These analytical investigations are complemented by presenting a numerical algorithm based on C 1 -elements and numerical studies. They intensively interact with geometric constructions in finding suitable minimising sequences for the Willmore functional.

We consider the Willmore-type functionalwhere H and K denote mean and Gaussian curvature of a surface , and γ ∈ [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 γ 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 γ = 0 these boundary conditions are arbitrary positive height α and zero mean curvature.

We consider two-dimensional immersions of disc-type in R n . We focus well known classical concepts and study the nonlinear elliptic systems of such mappings. Using an Osserman-type condition we give a priori-estimates of the principle curvatures for certain graphs in R 4 with prescribed mean curvature.

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We study orthonormal normal sections of two-dimensional immersions in R n+2 , n ≥ 2, which are critical for a functional of total torsion and which we call Coulomb sections. In particular, we establish upper bounds for the torsion coefficients in the case of non-flat normal bundles. With these notes we continue a foregoing paper on surfaces in R 4 .