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

**Series:** 2010-13, Preprints

- MSC:
- 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

**Abstract:**

We consider the family of smooth embedded rotationally symmetric annular type surfaces in R^3 having two concentric circles contained in two parallel planes of R^3 as boundary. Minimising the Willmore functional within this class of surfaces we prove the existence of smooth rotationally symmetric Willmore surface having these circles as boundary. When the radii of the circles tend to zero we prove convergence of these solutions to the round sphere.

**Keywords:**

Natural boundary conditions, Willmore surface, surface of revolution