Symmetric Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data

Abstract

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.

Error analysis for the approximation of axisymmetric Willmore flow by $C^1$-finite elements

“…We have used T = 1 as the length of the time interval. As initial condition u 0 we used an analytically motivated C 1 function that satisfies the boundary conditions (see [6]). To investigate the convergence behavior we performed a series of five computations where in the k-th calculation we used a boundary adapted grid consisting of NEL(k) := 3 · 2 k−1 elements with mesh-size h k = (12/5)2 −k .…”

“…Furthermore, one can use (1.2) in order to calculate local or global minima of W by considering the limits t → ∞. A rigorous proof of the existence of such minima subject to Dirichlet boundary conditions has been obtained in the axially symmetric case in [5], [6], while the parametric case is investigated in [19]. This paper is concerned with the numerical approximation of axially symmetric solutions of (1.2) subject to Dirichlet boundary conditions.…”

Symmetric Willmore surfaces of revolution satisfying natural boundary conditions

“…Existence and classical regularity of those axially symmetric Willmore surfaces with arbitrary symmetric Dirichlet boundary conditions were recently proved in [3,4]. With the paper at hand we continue these studies.…”

“…The proof of Theorem 1.1 is based on the existence results from [4] for symmetric Willmore surfaces of revolution satisfying Dirichlet boundary conditions u(±1) = α and ∓u (±1) = β for α > 0 and β ∈ R arbitrary. We construct a solution of (1.4) by minimising the Willmore energy for fixed α and variable β.…”

The Willmore boundary problem

“…The one-dimensional and rotational symmetric Willmore boundary problem with (1.3) and with Navier boundary conditions have recently been considered by Deckelnick and Grunau in [7][8][9] and together with Dall'Acqua, Fröhlich and Schieweck in [5,6] where explicit solutions were obtained in the one-dimensional case and an existence result was obtained for the rotational symmetric case.…”