The Wulff shape minimizes an anisotropic Willmore functional

Abstract

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.