# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Planar Wulff shape is unique equilibrium. (English) Zbl 1071.49025

Let $\varphi$ be a continuous norm in ${ℝ}^{2}$, i.e., $\varphi :{ℝ}^{2}\to {ℝ}^{+}$ is a nonnegative, convex, homogeneous function on ${ℝ}^{2}$. Let $c:{S}^{1}\to {ℝ}^{2}$ be a closed, connected, rectifiable curve parametrized by the arc length $s$, and set

$\psi \left(c\right)={\int }_{c}\varphi \left({c}^{\text{'}}\left(s\right)\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}ds·$

We say that $c={c}_{0}$ is in equilibrium if for all smooth variations ${c}_{t}\left(s\right)+f\left(s,t\right)$, preserving the area enclosed by $c$, the derivative of $\psi \left({c}_{t}\right)$ is initially nonnegative.

The Wulff shape ${W}_{\varphi }$ of $\varphi$ is the boundary of the unit ball in the dual norm ${\varphi }^{*}$ of $\varphi$. It is known that, for a given $\varphi$, among all curves enclosing the same area, the boundary of the ${\varphi }^{*}$ ball $B$ (the Wulff shape) minimizes ${\int }_{\partial B}\varphi \left(n\right)\phantom{\rule{0.166667em}{0ex}}ds$; here $n$ is the unit normal of $c\left(s\right)$. The main result of this paper is that, for any continuous norm on ${ℝ}^{2}$, an equilibrium, closed, connected curve $c$ must be a (scaled) Wulff shape (possibly with integer multiplicity if nonembedded curves are allowed).

##### MSC:
 49Q05 Minimal surfaces (calculus of variations) 53A10 Minimal surfaces, surfaces with prescribed mean curvature 49Q10 Optimization of shapes other than minimal surfaces 74E15 Crystalline structure
##### Keywords:
smooth variations; scaled Wulff shape; equilibrium