Existence for Willmore surfaces of revolution satisfying non-symmetric Dirichlet boundary conditions

Abstract

In this paper, existence for Willmore surfaces of revolution is shown, which satisfy non-symmetric Dirichlet boundary conditions, if the infimum of the Willmore energy in the admissible class is strictly below {4\pi}. Under a more restrictive but still explicit geometric smallness condition we obtain a quite interesting additional geometric information: The profile curve of this solution can be parameterised as a graph over the x-axis. By working below the energy threshold of {4\pi} and reformulating the problem in the Poincaré half plane, compactness of a minimising sequence is guaranteed, of which the limit is indeed smooth. The last step consists of two main ingredients: We analyse the Euler–Lagrange equation by an order reduction argument by Langer and Singer and modify, when necessary, our solution with the help of suitable parts of catenoids and circles.

Lower semicontinuity for the Helfrich problem

“…Also some existence results can be more readily achieved in this class, see, e.g. [7,12] or [14] and solutions to (1.4) can be analysed in greater depths, e.g. the behaviour of singularities in [16].…”

Explicit formulas, symmetry and symmetry breaking for Willmore surfaces of revolution

“…This includes the derivation of explicit formulas, which turns out to be a nontrivial task. The key to achieve this goal is a thorough discussion of the function (7) Z(s; P ) = (γ 1 (s) − P 1 ) 2 + (γ 2 (s) − P 2 ) 2 2P 2 γ 2 (s) for all P = (P 1 , P 2 ) ∈ C with P 2 = 0, which for P ∈ H is a conformal invariant due to d H (γ(s), P ) = Arcosh(1 + Z(s; P )), see (3). We mention that in the context of wavelike elasticae we will study Z(·; P ) for complex numbers P ∈ C with P 2 = 0 despite the fact that it has a clear geometrical meaning only for P ∈ H. Now we recall the relevant information about circular and (asymptotically) geodesic elasticae.…”

“…Proposition 1. Let γ : R → H be parametrized by hyperbolic arclength and Z be given by (7). Then, for all P ∈ C 2 with P 2 = 0, the function Z(·; P ) satisfies…”

“…for 0 < k < 1, s * ∈ R and let φ, Z be defined as in (5), (7). We recall µ = 4(1 − k 2 )/(2 − k 2 ) 2 .…”

Boundary value problems for a special Helfrich functional for surfaces of revolution: existence and asymptotic behaviour

“…The Douglas (or Navier) boundary value problem was studied in [35,38]. In the class of surfaces of revolution existence results for the Willmore functional were obtained in [12,14,20]; see also references therein. In [16] existence of minimisers of a relaxed Willmore functional in the class of graphs over two-dimensional domains was proved.…”