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Minimising a relaxed Willmore functional for graphs subject to boundary conditions

by Klaus Deckelnick, Hans-Christoph Grunau, Matthias Roeger.

Series: 2015-06, Preprints

49Q10 Optimization of shapes other than minimal surfaces
53C42 Immersions (minimal, prescribed curvature, tight, etc.)

For a bounded smooth domain in the plane and smooth boundary data we consider the minimisation of the Willmore functional for graphs subject to Dirichlet or Navier boundary conditions. For regular graphs we show that bounds for the Willmore energy imply area and diameter bounds. We then consider the L1-lower semicontinuous relaxation of the Willmore functional, which is shown to be indeed its largest possible extension, and characterise properties of functions with finite relaxed energy. In particular, we deduce compactness and lower-bound estimates for energy-bounded sequences. The lower bound is given by a functional that describes the contribution by the regular part of the graph and is defined for a suitable subset of the space of functions of bounded variation.
We further show that finite relaxed Willmore energy implies the attainment of the Dirichlet boundary data in an appropriate sense, and obtain the existence of a minimiser for the relaxed energy. Finally, we extend our results to Navier boundary conditions and
more general curvature energies of Canham-Helfrich type.

Willmore functional, relaxation, boundary value problems