Interfaces and Free Boundaries

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Volume 6, Issue 3, 2004, pp. 351–359
DOI: 10.4171/IFB/104

Published online: 2004-09-30

The Wulff shape minimizes an anisotropic Willmore functional

Ulrich Clarenz[1]

(1) Bonn, Germany

The aim of this paper is to find a fourth order energy having Wulff shapes as minimizers. This question is motivated by surface restoration problems. In surface restoration usually a damaged region of a surface has to be replaced by a surface patch which restores the region in a suitable way. In particular one aims for $C^1$-continuity at the patch boundary. A fourth order energy is considered to measure fairness and to allow appropriate boundary conditions ensuring continuity of the normal field. Here, anisotropy comes into play if edges and corners of a surface are destroyed. In the present paper we define a generalization of the classical Willmore functional and prove that Wulff-shapes are the only minimizers.

Keywords: Wulff shapes, surface restoration, Willmore functional

Clarenz Ulrich: The Wulff shape minimizes an anisotropic Willmore functional. Interfaces Free Bound. 6 (2004), 351-359. doi: 10.4171/IFB/104