Minimising a relaxed Willmore functional for graphs subject to boundary conditions

Klaus Deckelnick¹ ; Hans-Christoph Grunau¹ ; Matthias Röger² 

¹Otto-von-Guericke-Universität Magdeburg, Germany
²Technische Universität Dortmund, Germany

Interfaces and free boundaries, Tome 19 (2017) no. 1, pp. 109-140

Résumé

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 H2-regular graphs we show that bounds for the Willmore energy imply bounds on the surface area and on the height of the graph. 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 BV(Ω). 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 in L∞∩BV for the relaxed energy. Finally, we extend our results to Navier boundary conditions and more general curvature energies of Canham–Helfrich type.

Détail

DOI : 10.4171/ifb/378

Classification : 49-XX, 53-XX
Mots-clés : Graph, Willmore functional, boundary conditions, area estimate, diameter estimate, lower semicontinuity, relaxation, existence of a minimiser