Wulff theorem

Definition 1 (Wulff shape).

Let ϕ:Rn[0,+) be a non-negative, convex, coercive, positively 1-homogeneous function. We define the Wulff shape relative to ϕ as the set

Wϕ:={xn:x,y1 for all y such that ϕ(y)1}

(where , is the Euclidean inner product in Rn.)

Theorem 1 (Wulff).

Let ϕ:Rn[0,+) be a non-negative, convex, coercive, 1-homogeneous function. Given a regular open set DRn we consider the following anisotropic surface energy:


where νD(x) is the outer unit normalMathworldPlanetmath to D, and σ is the surface areaMathworldPlanetmath on D. Then, given any set D with the same volume as Wϕ, i.e. |D|=|Wϕ|, one has Fϕ(D)Fϕ(Wϕ). Moreover if |D|=|Wϕ| and Fϕ(D)=Fϕ(Wϕ) then D is a translationMathworldPlanetmathPlanetmath of Wϕ i.e. there exists vRn such that D=v+Wϕ.

Title Wulff theorem
Canonical name WulffTheorem
Date of creation 2013-03-22 15:19:50
Last modified on 2013-03-22 15:19:50
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 8
Author paolini (1187)
Entry type Theorem
Classification msc 52A21
Related topic FinslerGeometry
Defines Wulff shape