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Planar Wulff shape is unique equilibrium. (English) Zbl 1071.49025
Let $\phi$ be a continuous norm in $\Bbb R^2$, i.e., $\phi:\Bbb R^2\to\Bbb R^+$ is a nonnegative, convex, homogeneous function on $\Bbb R^2$. Let $c: S^1\to\Bbb R^2$ be a closed, connected, rectifiable curve parametrized by the arc length $s$, and set $$ \psi(c)=\int_c \phi(c'(s))\,\, ds. $$ 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 $\psi(c_t)$ is initially nonnegative. The {\it Wulff shape} $W_\phi$ of $\phi$ is the boundary of the unit ball in the dual norm $\phi^*$ of $\phi$. It is known that, for a given $\phi$, among all curves enclosing the same area, the boundary of the $\phi^*$ ball $B$ (the Wulff shape) minimizes $\int_{\partial B} \phi(n)\, ds$; here $n$ is the unit normal of $c(s)$. The main result of this paper is that, for any continuous norm on $\Bbb R^2$, an equilibrium, closed, connected curve $c$ must be a (scaled) Wulff shape (possibly with integer multiplicity if nonembedded curves are allowed).

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