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Planar Wulff shape is unique equilibrium. (English) Zbl 1071.49025

Let ϕ be a continuous norm in 2 , i.e., ϕ: 2 + is a nonnegative, convex, homogeneous function on 2 . Let c:S 1 2 be a closed, connected, rectifiable curve parametrized by the arc length s, and set

ψ(c)= c ϕ(c ' (s))ds·

We say that c=c 0 is in equilibrium if for all smooth variations c t (s)+f(s,t), preserving the area enclosed by c, the derivative of ψ(c t ) is initially nonnegative.

The Wulff shape W ϕ of ϕ is the boundary of the unit ball in the dual norm ϕ * of ϕ. It is known that, for a given ϕ, among all curves enclosing the same area, the boundary of the ϕ * ball B (the Wulff shape) minimizes B ϕ(n)ds; here n is the unit normal of c(s). The main result of this paper is that, for any continuous norm on 2 , an equilibrium, closed, connected curve c must be a (scaled) Wulff shape (possibly with integer multiplicity if nonembedded curves are allowed).

MSC:
49Q05Minimal surfaces (calculus of variations)
53A10Minimal surfaces, surfaces with prescribed mean curvature
49Q10Optimization of shapes other than minimal surfaces
74E15Crystalline structure